We will present a recent diffuse interface model for a two-phase flow of viscous incompressible fluids taking the effect of a surfactant into account, which diffuses through the bulk phases and along the interface. The model was developed by Garcke, Lam and Stinner. We will discuss the existence of weak solutions for this model. This is a joint work with Harald Garcke and Josef Weber.

We discuss the calculus of non-smooth pseudodifferential operators with coefficients, which have a limiting regularity with respect to the spacial variable \(x\). Although the standard results on arbitrary compostions of pseudodifferential operators break down and the operators have limiting mapping properties, they can still be used to construct parametrizes for (parameter-)elliptic operators and various applications of it. We will discuss some applications and present a recent result on characterization of non-smooth pseudodifferential operators and spectral invariance.

Can the minimal period of closed Reeb orbits on a contact three-sphere be bounded from above in terms of the contact volume? I will discuss positive results and counterexamples related to this question, together with applications in symplectic and Finsler geometry. This talk is based on a joint work with B. Bramham, U. Hryniewicz and P. Salomão.

Ordinary homology and cohomology of simplicial complexes can be easily calculated by a computer, and has become an instrumental tool in computational topology. Many other interesting homotopy invariants are either provably uncomputable (like the fundamental group) or their complexity status is not known.

We present an algorithm which takes a finite simplicial complex \(X\) and in finitely many steps computes its complex topological K-theory group \(\mathrm{K}(X)\). Our approach is based on recent progress in computational homotopy theory, in particular on the computability of Postnikov systems and homotopy classes of maps.

Joint work with Marek Krcal and Uli Wagner (IST Vienna).

For a finite subset \(X\) of the unit circle and a fixed angle \(\alpha\) we consider the map \(f_\alpha:X\to X\) which takes every point \(x\) to the clockwise furthest element of \(X\) which is still in angular distance at most \(\alpha\) from \(x\). We are interested in the discrete dynamical system on \(X\) generated by \(f_\alpha\), and especially in its expected behaviour when \(X\) is a large random set. The first indication of how the model depends on \(\alpha\) was obtained through computer experiments. This setup is motivated by topological considerations. The number of periodic points and the lengths of orbits of \(f_\alpha\) determine the homotopy type of the so-called Vietoris--Rips complex of \(X\) at distance \(\alpha\), a geometric construction used commonly in computational topology. Joint work with Henry Adams and Francis Motta (Duke University).

In a recent paper the authors presented a general perturbation result for generators of strongly continuous semigroups. It is our aim to replace in case the unperturbed semigroup is analytic, the various conditions appearing in our former result by simpler assumptions on the domain and range of the operators involved. The power of our main result consists in the systematic treatment of various classes of PDE's.

The imaging technology Magnetic Particle Imaging (MPI) develops rapidly regarding specific hardware designs. As a result, different imaging sequences have been established each favoring different reconstruction methods. In this talk an overview of realized MPI sequences and published image reconstruction algorithms is given. Furthermore, the mathematical similarities and differences between the reconstruction methods will be investigated with respect to the applied imaging sequence. Since the increase of field of view size is an important topic a brief discussion on the resulting consequences for the reconstruction will be given to emphasize current research topics.

We present an application of the discontinuous Galerkin finite element method to the simulation of flow and transport processes in regional and coastal ocean. The talk discusses a number of discretization issues, numerical techniques for various physical parametrizations, and illustrates the performance of the method using several real-life problems.

We present some problems of the theory of the moment problem as related to infinite dimensional analysis, the theory of stochastic process and quantum (field) theory. Relations with spectral theory and some integrable systems are also discussed.

In this talk we present two applications of the theory of harmonic Maass forms. We explain how half-integer weight harmonic Maass forms serve as " generating series" for traces of CM values of integer weight harmonic Maass forms. This generalizes work by Zagier on the traces of the modular j-function.

Moreover, we show that there are special harmonic Maass form related to elliptic curves. Certain periods of these forms encode the vanishing of the central L-derivates of the Hasse-Weil zeta function of E.

We present the approximation of an infinite horizon optimal control problem for evolutive advection-diffusion equations. The method is based on a model reduction technique, using a Proper Orthogonal Decomposition (POD) approximation, coupled with a Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function of the corresponding control problem for the reduced system. We show that it is possible to improve the surrogate model by means of a Model Predictive Control (MPC) solver. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.

The Voronoi diagram of a finite set of points (sites) decomposes the d-dimensional space into cells such that, for all points in a cell, the Euclidean distance to the site within that cell is not larger than the distance to all other sites. In this talk we consider the inverse problem for rather general distance functions: recover the sites and parameters of the distance functions for a given tessellation.

While special cases have been studied in the past, there is much recent interest in the general problem originating from several applications. Along with an application from the field of tomographic imaging of polycrystalline structures, we present new general results that lead to an efficient inversion algorithm for Voronoi, Laguerre, and generalized power diagrams.

This is joint work with Andreas Brieden (Universität der Bundeswehr München) and Peter Gritzmann (TU München).

All attempts to internalise semisimplicial types in HoTT have failed, due to a coherence problem: how do we make precise the notion of an omega functor? This is not the only instance of such a problem. We suggest a two level system, which introduces pre types with a notion of strict equality and a universe of types with an extensional, univalent equality. This is inspired by Voevodsky¹s HTS but unlike HTS our approach is an extension of intensional type theory and can be easily formalised within existing systems like Agda. Our approach works for type valued pre shaves over a Reedy category. This relies on the assumption that the basic types like natural numbers agree with their pre type.

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. In this talk I will explain that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere. The talk is based on joint work with Wolfgang Ziller.

Derived Witt theory introduced by Paul Balmer immerses Witt groups of schemes into the realm of generalized cohomology theories providing a family of shifted Witt groups. These groups have some crucial basic properties similar to the ones of topological K-theory of real vector bundles with inverted 2: besides the fact that they are based on vector bundles equipped with a nondegenerate bilinear form, these groups are 4-periodic with the periodicity realized via multiplication by Bott element and the coefficients are concentrated in the 0 mod 4 degrees. Derived Witt theory is representable in the motivic stable homotopy category, and as I will show in the talk its algebras of stable operations and cooperations with rational coefficients have the same structure as in the case of topological K-theory of real vector bundles. In particular, rational stable operations are given by the values on the powers of Bott element. As an application one obtains a rational degeneration of a Brown-Gersten type spectral sequence for derived Witt theory. These results are inspired by the joint work with Ivan Panin and Marc Levine on motivic Serre's finiteness theorem.

In physics, Quantum Field Theory (QFT) has been very successful in describing fundamental particle physics. Mathematically, these theories still posses enormous challenges. So fare there are no mathematical definition of these theories. These QFT's have however analogous theories, which are considerably simpler and which can be defined mathematically. Among these are the so called Topological Quantum Field Theories (TQFT), who are purely topological in nature as indicated by their name. For TQFT's, the only non-trivial evolution happens when the space itself, on which the theories are considered, undergoes topological changes. This is atypical for general states of a proper physical QFT, however for the ground states of QFT, this is in general the physical expectation.

From a mathematical point of view, these theories are very exciting, since they can be mathematically defined, thus studied mathematically and one can in certain cases prove many of the physics predicted properties. Further, one can use these theories to provide new topological invariants and they give rise to representations of the symmetry groups of the underlying topological spaces, thus they become interesting from a purely mathematical point of view as well.

This has been particular successful in low dimensional topology and there has been a very interesting interplay between topology on the one hand and then properties of TQFT's on the other. This has lead to further insight into the influence of the global topology of space time on the possible ground states for general QFT's. In particular, as a somewhat unexpected by product, new efficient TQFT models for quantum computers has this way been identified.

In this talk we will review some of these TQFT's, discuss how they are linked to low dimensional topology and on the way touch on their applications towards quantum computing.

In this talk I will present a spectral decomposition of solutions to relativistic wave equations on a given Schwarzschild-black-hole background. To this end, the wave equation is Laplace-transformed which leads to a spatial differential equation with a complex parameter. This equation is treated in terms of a sophisticated Taylor series analysis. Thereby, all ingredients of the desired spectral decomposition arise explicitly, including quasi normal modes, quasi normal mode amplitudes and the jump along the branch cut. Finally, all contributions are put together to obtain via the inverse Laplace transformation the spectral decomposition in question.

The On-Line Encyclopedia of Integer Sequences (OEIS) is a collection of "mathematical fingerprints", containing sequences from many areas of mathematics. I will show how the OEIS can be a useful tool and give examples of interesting content and open problems posed. Stronger connections to other mathematical resources would be beneficial to the mathematical community. Specifically, the many formulas contained in the OEIS should be made available for the formula search mechanism of the database zbMATH.

I will discuss the question of when a topological complex vector bundle on a smooth complex variety admits an algebraic structure. A necessary condition for algebraizability is that the topological Chern classes of the bundle must be algebraic (i.e., they must lie in the image of the cycle class map from Chow groups to integral cohomology). A folk conjecture (that arguably can be attribute to P. Griffiths) states that if a topological complex vector bundle on a smooth complex affine variety has algebraic Chern classes, then it admits an algebraic structure. This conjecture is known to be true for varieties of dimension \(\leq 3\). I will explain joint work with Jean Fasel and Mike Hopkins that shows that the conjecture is false for \(n \geq 4\). In particular, we will construct a new obstruction to algebraizability for rank \(2\) vector bundes on smooth affine \(4\)-folds and give an example of a smooth complex affine \(4\)-fold that carries topological vector bundles with algebraic Chern classes yet for which this obstruction is non-zero.

We consider the question of encoding the trajectory of a stochastic process (lossy coding). Here, certain jump processes are considered, in particular Lévy processes, and we describe results concerning the rate of the coding error (quantization error). Further, we outline extensions to the coding of objects from stochastic geometry with some interesting open questions.

Hyperbolic PDO on Lorentzian manifolds appear naturally as field operators in QFT in CST, and their analysis yields directly to the field content. We will start from the separation of variables in a hyperbolic field equation by means of Fourier analysis, and the resulting mode decomposition of quantum states. Then we will go on to Paley-Wiener theorems and see how they translate the Hadamard property of a quasifree quantum state to the dual (momentum) space. FInally we will discuss more subtle topics on hyperbolic PDO and how they relate to Lorentzian geometry. Everywhere both available results and ongoing work will be presented.

In this work-in-progress talk, I will analyse the cubical model of homotopy type theory of Coquand et al. in functorial terms, making a few adjustments along the way. The basic category of cubical sets used is presheaves on the free cartesian category on a bipointed object, i.e. the Lawvere theory of bipointed objects. The presheaf category is the classifying topos for strictly bipointed objects. The Kan extension property familiar from algebraic topology is shown to be exactly what is required to model the Identity-elimination rule of Martin-Löf, and the closure of Kan objects under function spaces is ensured constructively by Coquand's uniformity condition, re-analysed as the existence of a certain natural transformation making natural choices of Kan fillers. A universe of Kan objects is given in the style of the recent "natural models" construction, based on ideas of Lumsdaine-Warren and Voevodsky.

In this paper we consider forward stochastic Volterra integral equations (FSVIE) in \(L^q(S,\mathcal{S},\mu)\), where \(\mu\) is finite measure. \(L^p\)-stochastic integrability in unconditional Martingal difference (UMD) Banach spaces is used for the stochastic integration. Stochastic integral is defined with respects to one dimensional Wiener process. By setting appropriate assumptions on coefficients of FSVIE in \(L^q\) on a given filtered probability space, its solubility is considered. We find the unique stochastic solution by using fix-point theorem in Banach spaces. Some properties of solution are also being discussed.

Atiyah, Patodi and Singer constructed the relative K-theory class \([\alpha]\) associated with a flat unitary vector bundle over a closed manifold. This class is related to the spectral invariant rho of a Dirac operator by the so called index theorem for flat bundles, which computes the pairing between \([\alpha]\) and the K-homology class \([D]\) of the Dirac operator. The pairing is in turn equal to a type II spectral flow, as proved by Douglas, Hurder and Kaminker.

In this talk we will focus on the operator algebraic point of view on these relative invariants by showing how the construction of \([\alpha]\) can be seen as a consequence of Atiyah's \(L^2\) index theorem. We will also give new relative K-theory construction obtained in joint work with Paolo Antonini and Georges Skandalis that generalize the class \([\alpha]\) to a noncommutative setting.

We study an atom with a finite number of energy levels, coupled to an infrared-divergent boson field (the Spin-Boson Model). Without any regularization of the coupling function, we prove existence of the ground state energy and construct the ground state. Our method is based on the multi-scale analysis introduced by A. Pizzo in 2003, but we use the continuous version of it developed by V. Bach and M. Könenberg (2006). The main difficulty is the infrared divergence in the coupling function that behaves as \(|k|^{-{1 \over 2}}\) (\(|k|\) being the norm of the momentum of the Boson), since such models do not necessarily admit ground states. Assuming some symmetries, it is proved by D. Hasler and I. Herbst in 2011 that the ground state exists in the model at stake, using the spectral renormalization group analysis introduced by V. Bach, J. Fröhlich and I.M. Sigal in 1998. On the contrary, in the present work we construct the ground state projection as a limit of projections \(P_t, \, t \in [0, \infty)\), corresponding to infrared-regularized models, by proving that the norm of the derivative \(\dot P_t\) is integrable.

Modular forms for the full modular group are the first and easiest examples of automorphic forms. They are given by holomorphic functions in the upper half-plane possesing a Fourier expansion (q-series) whose coefficients are of arithmetical interest. Using complex analysis one can prove a lot of relations between these functions which then yield relations between their Fourier coefficients. In this talk we want to discuss a purely combinatorial approach to prove such relations, without using any complex analysis, which was inspired by the theory of multiple zeta values. Multiple zeta values are generalizations of the classical Riemann zeta values appearing in different areas of mathematics and theoretical physics. There are a lot of Q-linear relations between these real numbers which are called double shuffle relations. The goal of this talk is to introduce a space of certain q-series which has the modular forms as a subspace and whose algebraic structure is similar to the one of multiple zeta values. We will see that these q-series also fulfill a variation of the double shuffle relations which then enables us to recover well-known relations of modular forms in a purely combinatorial way.

We prove an index theorem for the Dirac operator on compact Lorentzian manifolds with spacelike boundary. Unlike in the Riemannian situation, the Dirac operator is not elliptic. But it turns out that under Atiyah-Patodi-Singer boundary conditions, the kernel is finite dimensional and consists of smooth sections. The corresponding index can be expressed by a curvature integral, a boundary transgression integral and the eta-invariant of the boundary just as in the Riemannian case. There is a natural physical interpretation in terms of particle-antiparticle creation. This is joint work with Alexander Strohmaier.

An algorithm for computing control Lyapunov functions for nonlinear affine, asymptotically controllable systems is presented. It is based on a simplicial triangulation and the ansatz with continuous, piecewise affine (CPA) functions. Due to the missing regularity of the ansatz functions nonsmooth versions of the weak infinitesimal decrease condition of the control Lyapunov function using subdifferentials have to be used.

The characterizing conditions for a CPA control Lyapunov function are stated in all vertices of the triangulation and lead to a mixed integer linear optimization problem in which the values of the CPA function can be computed. By incorporating error bounds on this CPA interpolation the calculated CPA function is a control Lyapunov function and not an approximate one.

A first numerical example, problems with the decay condition formulated with Clarke's subdifferential and possible improvements are discussed.

A feasible flow in a network \(N = (V, A, u)\) is any non-negative function \(x : A \rightarrow R\) which satisfies that \(x_{ij} \leq u_{ij}\) for all arcs \(ij\in A\). In many cases \(N\) is also equipped with a so-called balance vector \(b : V\rightarrow R\) and then a feasible flow must also satisfy that at every vertex \(v\in V\) the sum of the flow on outgoing arcs from \(v\) minus the sum of the flow on incomming arcs at \(v\) must equal \(b(v)\). The theoretical and algorithmic aspects of network flows are well understood and flows form a very useful tool for modelling problems, as a machinery to prove results or develop polynomial algorithms for (di)graphs. Examples are: Menger's theorem, Halls theorem, finding subdigraphs with prescribed in- and out-degrees etc. There are polynomial algorithms for testing whether a given network \(N = (V, A, u, b)\) has a feasible flow and to find such a flow when it exists. In this talk we consider an extension of the flow model which allows us to model a large number of different problems which cannot be modelled in the standard flow model. Two flows \(x, y\) in a network \(N \) are arc-disjoint if \(x_{ij} * y_{ij} = 0\) for every arc \(ij\). A very natural problem, which is also interesting from an applications point of view, is as follows: given a network \(N = (V, A, u, b)\); does \(N\) have a pair of arc-disjoint feasible flows \(x, y\). This innocent sounding problem contains several problems that cannot be modelled by standard flows as special cases: arc-disjoint linkages, number partitioning and arc-disjoint spanning subdigraphs with prescribed degrees. Hence the arc-disjoint flow problem is NP-complete in general and it is interesting to focus on special cases. A branching flow from a root s in a network \(N = (V, A, u)\) is a flow \(x\) whose balance vector is \(-1\) at all vertices except \(s\) where it is \(n-1\). Here \(n = |V|\). If all capacities are \(n-1\), then a feasible branching flow exists if and only if \(D = (V, A)\) has an out-branching from \(s\) and there are \(k\) arc-disjoint branching flows in \(N\) if and only if \(D\) has \(k\) arc-disjoint out-branchings. By Edmonds' branching theorem, this can be checked in polynomial time using a polynomial algorithm for the maximum flow problem. When the capacities are (much) smaller than \(n-1\), the structure of the arc-set carying non-zero flow in a faesible branching flow may become quite complicated, but we can still determine the existence of a feasible branching flow in polynomial time using any polynomial maxflow algorithm. However, if we want to determine the existence of a pair of arc-disjoint branching flows the problem becomes NP-complete when capacities are bounded by any constant. We will discuss the complexity of the problem when the capacities are all \(n-k\) for some \(k\). Our analysis reveals interesting structure of feasible branching flows in network with these capacity bounds.

When the body gets infected by a pathogen the immune system develops pathogen-specific immunity. Induced immunity decays in time and years after recovery the host might become susceptible (S) again. Exposure to the pathogen in the environment, that is, contact with infectives (I), boosts the immune system thus prolonging the time in which a recovered individual is immune (R). Such an interplay of within host processes and population dynamics poses significant challenges in rigorous mathematical modeling of immuno-epidemiology. In the first part of the talk we propose a framework to model SIRS dynamics, monitoring the immune status of individuals and including both waning immunity (W) and immune system boosting. Our model is formulated as a system of two ordinary differential equations (ODEs) coupled with a partial differential equation for the immune population. We prove basic properties of this model, such as existence and uniqueness of a classical solution and the stability of the unique disease-free stationary solution. In the second part of the lecture we show how to obtain, under particular assumptions on the general model, known examples such as large systems of ODEs for SIRWS dynamics, as well as SIRS with constant and state-dependent delay, which we shall consider in detail.

This is a joint work with G. Röst.

Es werden zwei Projekte aus dem Bereich E-Learning vorgestellt, an deren Entwicklung die Hamburger Hochschulen Uni HH, HCU, TUHH und HAW beteiligt waren. Der OMB+ (Online Mathematik Brückenkurs +) ist ein Online-Kurs, mit dessen Hilfe Studieninteressierte noch vor Studienbeginn die Schulmathematik selbständig wiederholen können. Mit dem Hamburger Orientierungstest wurde ein kurzer diagnostischer Online-Test bereitgestellt, der Studieninteressierten eine Rückmeldung darüber gibt, in welchen Bereichen der Schulmathematik sie Nachholbedarf haben. Als Grundlage für den zu vermittelnden Stoff diente der Mindestanforderungskatalog Mathematik der baden-württembergischen COSH-Gruppe. Teilnahme an Orientierungstest und OMB+ sind kostenlos und können von zuhause erfolgen, es soll jedoch auch das Angebot geben, diese in Gänze oder in Teilen in den Schulunterricht zu integrieren.

Affine processes are common generalizations of continuous state and continuous time branching processes with immigration and Ornstein-Uhlenbeck type processes. Roughly speaking, the affine property means that the logarithm of the characteristic function of the process at any time is affine with respect to the initial state. Let us consider an affine two-factor model given by the jump-type SDE \[ \mathrm{d} Y_t = (a-bY_t)\,\mathrm{d} t + \sqrt[p]{Y_{t-}}\,\mathrm{d} L_t,\qquad t\geq 0,\\ \mathrm{d} X_t = (\alpha-\gamma X_t)\,\mathrm{d} t + \sqrt{Y_t}\,\mathrm{d} B_t,\qquad t\geq 0, \] where \(a>0\), \(b, \alpha, \gamma \in \mathbb{R}\), \(p\in(1,2]\), \((L_t)_{t\geq 0}\) is a spectrally positive \(p\)-stable process with Lévy measure \(C_p z^{-1-p}\mathbf 1_{\{z>0\}}\), where \(C_p:=(p\Gamma(-p))^{-1}\) (\(\Gamma\) denotes the Gamma function) if \(p\in(1,2)\), and a standard Wiener process if \(p=2\), and \((B_t)_{t\geq 0}\) is an independent standard Wiener process. Note that the first coordinate is nothing else but a so-called \(p\)-root process with \(p\in(1,2]\), which, in case of \(p=2\), is also called a CIR process. Provided that \(a>0\), \(b>0\) and \(\gamma>0\), we prove that the affine model above has a unique strictly stationary solution in both cases \(p\in(1,2)\) and \(p=2\). Further, in case of \(p=2\), supposing that \(a>0\), \(b>0\) and \(\gamma>0\), the ergodicity is also shown together with the fact that the unique strictly stationary solution is absolutely continuous having finite mixed moments. In case of \(p\in(1,2)\), the question of ergodicity remains open, however, we will briefly describe a possible approach for proving such a result.

Im Rahmen einer Kooperation des DZLM mit dem Kultusministerium in NRW wurde die staatliche Lehrerfortbildungsreihe "GTR kompakt" durchgeführt, um Lehrkräfte auf einen sinnvollen Einsatz digitaler Werkzeuge im Mathematikunterricht der Oberstufe vorzubereiten.

Die Konzeption der Fortbildung wurde von einer Gruppe von Lehrenden aus Schule und Universität gemeinschaftlich und theoriebasiert entwickelt. Im Mittelpunkt stand, die Möglichkeiten und Gefahren des Rechnereinsatzes insbesondere im Bereich der Analysis aufzuzeigen und sinnvolle Unterrichtswege anzubahnen. Gezielter Vorstellungsaufbau, Repräsentationswechsel, verstehensorientierte Aufgaben und schülerzentriertes Unterrichten markieren dabei zentrale Aspekte im Bereich der Möglichkeiten. Gezieltes Entwickeln händischer Fertigkeiten und mathematischer Nomenklatur trotz vorkommender "Rechnersprache" sowie Veränderungen im Lehrprozess stehen für besondere Problemfelder beim Rechnereinsatz, die in der Fortbildung gezielt bearbeitet wurden. "GTR kompakt" wurde im Rahmen zweier Begleitstudien in ihrer Wirksamkeit auf Lehrer- und auf Schülerebene untersucht.

Im Vortrag werden die Grundzüge der Konzeption sowie der Begleitstudie mit ersten Erkenntnissen vorgestellt.

The \(\Phi\)-variation of a stochastic process may be viewed as a measurement of its smoothness, and it plays an important role in integration theory, rough paths theory and Fourier analysis. The special cases of bounded variation and bounded p-variation were introduced by Jordan and Wiener respectively, and the general definition of \(\Phi\)-variation goes back to Young. A classical result by Lévy states that the sample paths of a Brownian motion are of bounded p-variation if and only if \(p>2\). This result has been improved by Taylor who showed that the ''correct'' \(\Phi\)-variation of the Brownian motion is \(\Phi(x)=x^2/log(log(1/x))\). Furthermore, Dudley and Norvaisa has characterized the correct \(\Phi\)-variation of the fractional Brownian motion. On the other hand, there does not exists a correct \(\Phi\)-variation function for a stable non-Gaussian Lévy process. In this talk we will, in particular, derive the correct \(\Phi\)-variation of a class of self-similar Gaussian chaos processes with stationary increments called Hermite processes. This class includes the fractional Brownian motion and Rosenblatt process as special cases. Our technique relies on metric entropy methods for stochastic processes with exponential moments.

This talk is based on joint work with Michel Weber, IRMA, Université de Strasbourg.

For real entire functions with exclusively negative zeros important connections to totally non-negative (TNN) matrices exist. Characterization of those entire functions whose Taylor expansion \(\sum a_{k}z^k, a_0>0\), generates totally non-negative matrices \((a_{j-i})_{i,j=0}^{\infty}\), is a consequence of the AESW-theorem (theorem of Aissen, Edrei, Schoenberg, and Whitney). The structured matrices \(H(g,h)\), exhibited by Hurwitz in his variant of the Hermite-Jacobi approach to quadratic forms, are known to have all minors non-negative if and only if the polynomial \(f(z)=h(z^2)+zg(z^2)\) with positive coefficients has all zeros in the closed left half-plane, i.e., if and only if \((g,h)\) is a generalized positive pair with positive coefficients. Recent work by Holtz and Tyaglov (2012) connected total non-negativity of an infinite supermatrix \(\hat{H}\) of \(H(g',g)\) with the zero-location of \(g\) via continued fractions and factorizations of infinite TNN matrices. Dyachenko (2014) gave a complete characterization of those series generating a TNN matrix \(\hat{H}\). In connection with transforms of the Riemann \(\Xi\)-function the TNN minors yield a set of coefficient inequalities (exponentially growing with the number of considered coefficients) bearing on the Riemann hypothesis.

We contribute the following: For real entire functions \(f\) of the form \(f(z)=e^{\beta \cdot z} g(z), \beta \geq 0\), where \(g\) is of genus zero, we generalize the Holtz-Tyaglov result on polynomials, and exhibit its dependency from the AESW-theorem. Our analytic proof reveals for the first time the intimate connection of \(\hat{H}\) and \(H(f',f)\). Hence, we generalize our results to the case of a positive pair \((f_1,f_2)\) with positive coefficients; this yields an independent simple, self-contained proof of Dyachenko's result for the considered functions. We proceed and show that the componentwise (Schur-Hadamard) product of matrices \(\hat{H}_i\) and \(\hat{H}_j\) is totally non-negative, thus extending a result of Garloff and Wagner (1996) to the supermatrices \(\hat{H}\), and entire functions. We find an essential set of minors in \(\hat{H}\) which determines total non-negativity considering Grommer's seemingly different characterization of exclusively negative zeros in terms of Markov moments (which can be related to the AESW-theorem via Mittag-Leffler expansions). The Grommer characterization is computationally transformed here to reveal an essential set of minors in \(\hat{H}\), in the polynomial and, more important, in the transcendental case. This yields a transcendental, computational characterization of correctness of the Riemann hypothesis involving only essential minors. We show that the Laguerre-Turán inequalities (involving three consecutive coefficients) discussed in this connection since Pólya's question of 1927, and the successive four term improvement by Craven-Csordas (2002), are a weaker necessary criterion for the Riemann hypothesis than the four term inequality from the first non-trivial of our essential minors.

A pseudo-Riemannian solvmanifold is a homogeneous space \(M\) with pseudo-Riemannian metric \(\mathrm{g}\) on which a connected solvable Lie group \(G\) of isometries acts transitively and almost effectively. We show that \(M=G/\Gamma\), where \(\Gamma\) is a discrete cocompact subgroup of \(G\), and that \(\mathrm{g}\) is induced by a biinvariant pseudo-Riemannian metric on \(G\).

I will present some recent work with Martin Hils and Rahim Moosa on the model theory of holomorphic discrete dynamics on compact complex manifolds. I will describe a classification of minimal dynamics (including the Zilber trichotomy via the Canonical Base Property), and make some remarks on quotient structures (geometric eliminination of imaginaries; failure of full EI due to failure of 3-uniqueness in CCM).

Metric rigidity of holomorphic maps (as that of smooth maps between Riemannian mannifolds) generally requires some kind of non degeneracy assumptions. Thus holomorphic maps in complex projective spaces are congruent if they have the same first fundamental form. In Hermitian symmetric target spaces of higher rank however, the maps should be full in the sense that their osculating space exhausts the ambient tangent space.

In Grassmannians this can be resolved by fixing the second fundamental form as well, but this over determines the map. For holomorphic maps into Grassmannians, we determine a complete set of invariants and some of the arising relations. Most of this also works for harmonic maps.

The numerical approximation of multi-scale problems like hydrodynamic turbulence requires approximation schemes that not only offer fast convergence rates for smooth solutions and well-resolved cases, but also controllable approximations errors and inherent stability for under-resolved flows. One family of approximation methods that combines accuracy and robustness for convection-dominated problems with excellent parallelization efficiency are high order discontinuous Galerkin schemes, based on an element-wise variational formulation with local testfunctions. The implementation of the associated projection operators demands particular attention, as inexact integration of non-linearities (of flux functions and transformation metrics) leads to an efficient implementation, but can cause aliasing instabilities.

In this talk, we will present a highly efficient discontinuous Galerkin framework for solving the compressible Navier-Stokes equations in complex domains. We will focus on the non-linear instabilities through inexact projection and discuss different remedial strategies in terms of accuracy, stability and implementation efficiency. We will conclude by presenting a novel, spatially and temporally adaptive de-aliasing strategy suitable for both continuous and discontinuous Finite Element formulations.

Historische Untersuchungen zur Entwicklung der Mathematik in England in der frühen Neuzeit haben sich bisher fast ausschliesslich mit zentralen und bekannten Figuren wie Wallis, Brouncker oder Newton befasst. In diesem Vortrag wird der Blick auf die englischen Provinzen gerichtet, speziell auf die Grafschaft Somerset. Der dort lebende Algebraiker Thomas Strode hat eine bemerkenswerte Arbeit über Kegelschnitte verfasst, in der auch Ideen aus Kinckhuysens Algebra Aufnahme finden. Über Strodes Apollonius analyticus berichtet Oldenburg auch mehrfach in seiner umfangreichen Korrespondenz mit Leibniz.

The dynamical object which we study is a compact invariant set with a suitable hyperbolic structure. Stability of hyperbolic attractors was studied by Pliss and Sell. They assumed that the neutral and the stable linear spaces of the corresponding linearized systems satisfy Lipschitz condition. They showed that if a perturbation is small, then the perturbed system has a hyperbolic attractor \(M\), which is homeomorphic to the hyperbolic attractor \(K\) of the initial system, close to \(K\), and the dynamics on \(M\) is close to the dynamics on \(K\). At the same time, it is known that the Lipschitz property is too strong in the sense that the set of systems without this property is generic. Hence, there was a need to introduce new methods of studying stability of hyperbolic attractors without Lipschitz condition. In our talk we will show that even without Lipschitz condition there exists a continuous mapping \(h\) such that \(h(K) = M\).

In this talk, we consider exponential functionals of two one-dimensional independent Lévy processes, as they appear as stationary distributions of generalized Ornstein-Uhlenbeck processes. Hereby the integrating Lévy process will be assumed to be a subordinator. In particular, we present an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping, which maps the law of the integrating Lévy process to the law of the corresponding exponential functional, where the other Lévy process remains fixed. We study the behaviour of the range of this mapping for varying characteristics of the Lévy process in the integrand. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range.

Singular is a computer algebra system with a focus on polynomial computations; it depends on an existing codebase of several hundreds of thousands lines of code written in efficient, but low-level C/C++. In this talk, we describe how we transformed this codebase to make it thread-safe and discuss the mechanisms we introduced to facilitate multi-threaded programming in Singular.

Discontinuous Galerkin models have recently been used to produce accurate and ro- bust solutions of the shallow water equations for various geo-scientific applications. They can be easily formulated to be mass-conservative, are extendable to higher-order accuracy and have a local stencil, the latter being advantageous for parallelization. On the other hand, certain aspects are still under heavy development, such as the accurate treatment of wetting and drying events. The presented inundation scheme utilizes slope-limiting techniques that do not influence the stability of the scheme and are free from additional parameters. It can be shown to be mass-conservative, positivity-preserving and well-balanced for the still water state at rest.

To reduce the computational effort for complex flow situations a dynamically adaptive mesh is used, and problem-dependent refinement indicators are introduced to resolve local features of interest. Furthermore, the patch-wise mesh manipulation strategy that we employ, keeps the mesh conforming throughout the simulation, which further simplifies the computations. We will discuss the efficiency of our adaption strategy and its effects on the overall accuracy of the simulation. Numerical test cases demonstrate the applicability of our model to quasi-realistic scenarios.

One of the intrinsic structures induced by a smooth embedding of a curve in a conformal manifold (of dimension at least 3; for an ambient space of dimension 2 a Möbius structure is additionally required) is a projective structure (another iduced structure is a conformal structure, but this is trivial). For periodic curves with period 1, the moduli space of projective structures is a locally 1-dimensional, non-Hausdorff space (for a smooth loop in a Riemannian manifold the moduli space of lengths is also 1-dimensional), and it is given by the conjugacy class of \(\tilde R(1)\), where \(R\) is the fundamental solution of the linear system associated to the Hill's equation characterizing the projective structure, and \(\tilde R\) is its lift to the universal covering of \(\mathrm{SL}(2,\mathbb{R})\). We show that a large part of the moduli space can be realized by embeddings of plane curves in euclidean spaces, in particular we give examples of non-homgeneous projective structures realized this way.

The question of how to glue local observables for an Abelian gauge theory to global ones is addressed at the kinematic level using the framework of homotopy theory. Starting from functors providing chain complexes that describe configurations and observables for an Abelian gauge theory on contractible manifolds, we present a procedure to extend those functors to non-contractible manifolds by means of homotopy (co)limits. This approach turns out to be flexible enough to encode also the relevant topological information (non-trivial principal bundles, flat connections, ...). Furthermore, on contractible manifolds, the extended functors are shown to agree with the original ones up to natural quasi-isomorphisms.
Joint work with A. Schenkel and R.J. Szabo. Pre-print: arXiv:1503.08839 [math-ph].

Each quasi-order, and thus, each order, can be easily presented as a category. In turn, the category of orders with monotone maps is a reflective subcategory in the category of quasi orders. By imposing appropriate structural conditions, it is possible to present well quasi-orders in a purely categorical way, and it happens that well quasi-orders form a full subcategory of quasi-orders, as well orders form a full subcategory of orders. More interestingly, the subcategory of well orders is reflective in the category of well quasi-orders.

By expressing the properties of well quasi orders as logical propositions, it becomes natural to consider them inside the internal language of the presheaf topos over the category of well quasi orders. In this context, the above mentioned relations between categories provide ways to transport theorems from a subtopos to the including topos and vice versa.

The purpose of the present contribution is to systematically illustrate the overall picture, and to characterise which properties, and to what extent, are subject to the transport process above.

In this talk I will discuss weak universes, with any small type only being weakly equivalent to something inside the universe. The motivation is that it turns out to be much easier to construct models of homotopy type theory with weak univalent universes and that such weak universes are just as well-behaved as ordinary universes. For example, they also lead to models of CZF: to give a nice categorical proof of this fact we introduce a new categorical construction which we call a "homotopy exact completion". This is ongoing joint work with Ieke Moerdijk.

Computational technology impacts network and research activities of scientists. The presented collection of the C.A.S.E. Computer Museum tracks the interdisciplinary relationship between computational, methodical and informational advances. We aim to document historical challenges and mainta in reproducibility. The collection includes a wide range of computers, mechanical and digital calculators, peripherals, software and text books related the development of mathematics and statistics in particular. Its use in teaching is intended to visualize increases in knowledge and capabilities over time that result from the interaction between computers, methods and data.

We study the concept of locally controlled invariant submanifolds for nonlinear descriptor systems. In contrast to classical approaches, we define controlled invariance as the property of solution trajectories to evolve in a given submanifold whenever they start in it. It is then shown that this concept is equivalent to the existence of a feedback which renders the closed-loop vector field invariant in the descriptor sense. This result is motivated by a preliminary consideration of the linear case.

Local controlled invariance leads to the concept of output zeroing submanifolds. We show that the outcome of the differential-algebraic version of the zero dynamics algorithm yields a maximal output zeroing submanifold. The latter is then used to characterize the zero dynamics of the system. In order to guarantee that the zero dynamics are locally autonomous (i.e., locally resemble the behavior of an autonomous dynamical system), sufficient conditions involving the locally maximal output zeroing submanifold are presented.

Most of the large networks studied in applications (sociology, internet, biology ...) exhibit a notion of communities, that is nodes having the same connection patterns between them and toward the rest of the network. Several approaches are possible to take communities into account when describing a random graph model, one of them being to consider mixtures of Erdős-Rényi graphs. The present talk will consist in the presentation of such models, and in particular of a generalization allowing a node to be part of several communities. The statistical tools devoted to parameter inference and classification of the nodes will be investigated, as well as perspectives for the application on very large graphs.

In this talk we present a graph-theoretical method to approximate local attractors for continous-time dynamical systems, and consequently use a Massera-like construction in order to construct a continuous piecewise-affine (CPA) function on a simplicial complex, which approximates a Lyapunov function for the system. We present some sufficient conditions for such a CPA functions to be an actual Lyapunov function for a given system, and finally we give some examples.

Nambu's (1973) extension of Hamiltonian mechanics is applied to Geophysical Fluid Dynamics by including several conservation laws in the dynamical equations. Ideal hydrodynamics is formulated in a Nambu representation in two and three dimensions using enstrophy and helicity as second conservation laws in addition to the total energy (Névir and Blender, 1993). Noncanonical Hamiltonian mechanics is embedded in Nambu mechanics if a Casimir function can be incorporated as a conservation law. The Nambu representations of the quasigeostrophic equations, the shallow water model, the Rayleigh-Bénard equations, and the baroclinic atmosphere are reviewed. Salmon (1995) suggested the design of conservative numerical codes based on a Nambu formulation. Gay-Balmaz and Holm (2013) have used the Nambu approach to parameterise the selective decay in 2D hydrodynamics. To derive the Nambu brackets for two-dimensional systems a geometric approach is suggested (Blender and Badin, 2015). As results, 2D hydrodynamics and Rayleigh-Bénard convection emerge. Approximations are obtained by the definition of constitutive conservation laws.

(Based on joint work with Benedikt Löwe) With a set theoretic multiverse I essentially mean the collection of "all" models of a fixed set theory with a relation on this collection representing the ability to get from one model to another related model via a given set theoretic model construction (e.g. via generic extensions, symmetric extensions or inner models). The modal logic of a given multiverse shall be the set of all basic modal formulas which are valid in every set theoretic model with arbitrary interpretation of the propositional variables as set theoretic sentences and of the modal operators in terms of the relation on the multiverse.

Of course it is not immediately clear how we can formalize the above notions inside of set theory itself. The aim of my talk will be to present an approach for a set theoretic framework that allows us to treat arbitrary set theoretic multiverses along the above lines. I will also show how known results from the modal logic of forcing -- which were established in a purely syntactic framework -- lift to this rather semantic new framework.

Modellieren ist eine der sechs allgemeinen Kompetenzen, die Schülerinnen und Schüler anhand von Oberstufeninhalten (weiter)entwickeln sollen und die im Abitur verbindlich überprüft werden sollen. Gängige Abituraufgaben enthalten i.A. nur bescheidene Modellierungsanforderungen. Im Vortrag soll erörtert werden, inwiefern diese Beschränkungen den Rahmenbedingungen des Abiturs geschuldet sind und welche Modellierungsanforderungen man sinnvoll im Abitur verlangen kann oder sollte.

The classical right (or left, respectively) Clifford parallelism in real projective \(3\)-space can be described by using Hamilton's quaternions. Analogously, an octonion division algebra \(O\) over a field \(F\) gives rise to a right (or left, respectively) Clifford parallelism of lines in \(\mathrm{PG}(7,F)\). However, in contrast to the quaternion case, here the parallelism depends on the choice of a base point.

More exactly, there is a \(1\)--\(1\) correspondence between the point set of \(\mathrm{PG}(7,F)\) and the set \(\Pi^+\) of all right Clifford parallelisms in \(\mathrm{PG}(7,F)\). We show that this bijection can be seen as an isomorphism of point-line geometries \(\mathrm{PG}(7,F)\to(\Pi^+,\mathcal{C}^+,\ni)\), where \(\mathcal{C}^+\) is the set of all parallel classes of all right Clifford parallelisms. A similar result holds for the left Clifford parallelisms.

The spaces of Clifford parallelisms can also be interpreted on the hyperbolic quadric in \(\mathrm{PG}(7,E)\) given by the split octonions over a quadratic extension \(E/F\) contained in \(O\)

An orthoscheme is a special simplex first explained by L. Schläfli in 1859. An orthoscheme in \(d\)-dimensional space of constant curvature generates at first a Napier cycle and then a Napier cycle type. To a Napier cycle type belongs a set of special permutations, called geometric permutations. As a represent of these permutations a special permutation of these ones can be chosen, called excellent permutation. This permutation generates of course also the Napier cycle type. The intersection of all the orthoschemes of a Napier cycle is called the hyperbolic kernel belonging to this Napier cycle.

In 1936 H.S.M. Coxeter and G. T. Bennett formulated a geometric connection between elements of elliptic rectangular triangles (orthoschemes) and a special configuration of semi-circles over a line (Coxeter-Bennett configuration), called Coxeter Theorem. Coxeter extended these results to the \(d\)-dimensional elliptic case and then to the hyperbolic case. His hints to the case of a Minkowskian space find a generalization of this theorem here. Thus a generalization of Napier's rule can be given. Generally for Minkowskian spaces of arbitrary dimension the connection between types of orthoschemes and permutations can be described. For proving some important assertions the knowledge of the structure of geometric permutations is used. The theory for hyperbolic kernels is a first application of geometric permutations. A second application of geometric permutations is the theory of self-dual \(2-\)colored necklaces with \(2n\) beads searched by R. W. Robinson and E. M. Palmer in 1984 and later by W. J. A. Sloane in 1995. Here can by shown that the number of hyperbolic kernel types of dimension \(d\) agrees with the number of necklace types having \(2(d+3)\) beads and beings self-dual and \(2\)-colored.

A topological surface \(F_g\) admits a unique smooth structure, but there is a whole space of complex structures. This space is the moduli space \(\mathfrak M_g\). Riemann started the study of these spaces, but they are far from beeing completely understood. One approach to detect the homology of \(\mathfrak M_g\) is to find a suitable triangulation \(C\) translating its geometry into combinatorial data. In my talk, I will sketch one of these models which allows computer aided computations.

We give an overview of the four Skorokhod topologies. Each topology naturally suggests a particular embedding of discrete time processes into continuous time. Within this framework we compare the convergence of embedded Markov chain approximations.

To every spin lens space \(L\) we associate an affine lattice that fully characterizes the isometry class of \(L\). The multiplicities of the eigenvalues of the Dirac operator on \(L\) are connected to the size of the intersection of this lattice with the norm-one spheres by a simple formula. We use this formula to obtain an isospectrality criterion for lens spaces which leads to the identification of several isospectral families. This is joint work with Emilio Lauret.

The study of the combinatorial diameter of polyhedra is a classical open problem in the theory of linear optimization. In a new approach to this field, we introduce a hierarchy of so-called circuit diameters, which generalize the combinatorial diameter and provide lower bounds on it. In contrast to traditional edge walks, circuit walks take steps along potential edge directions, so in particular they can walk through the interior of a polyhedron. We examine the structure of this hierarchy in detail, prove similarities and differences between the many diameter notions, and exhibit for which of these classes the Hirsch conjecture bound holds and for which it is open. Finally, we turn to some classes of polyhedra to highlight the insight gained from these studies.

In Deutschland unterrichten Lehrerinnen und Lehrer das Fach Mathematik, die dazu formal nicht qualifiziert sind. In vielen Fällen haben sie andere Fächer an der Hochschule studiert und sind mit der Mathematik lediglich in der eigenen Schulzeit in Berührung gekommen. Es stellt sich die Herausforderung, dieses Phänomen nicht nur defizitorientiert zu begreifen: Was macht fachfremd unterrichtende Lehrkräfte jenseits mangelndem fachlichen und fachdidaktischen Wissen aus und wie verstehen und gestalten sie mathematisches Lehren und Lernen? Schließlich ist es von Interesse, Erkenntnisse darüber zu gewinnen, wie diese Lehrerinnen und Lehrer durch Interventionsmaßnahmen unterstützt werden können. Es wird eine qualitative, empirische Studie vorgestellt, für die 21 fachfremd unterrichtende Mathematiklehrerinnen und -lehrer der Sekundarstufe I zu ihrem Bild von Mathematik und von Mathematikunterricht interviewt wurden. Außerdem wurden 5 Unterrichtsstunden von 4 Lehrpersonen der Stichprobe videographiert.

The local rate of mass transfer of a gaseous component from a gas bubble rising in an ambient liquid, say, is strongly influenced by the presence of surface active substances, so-called surfactants. This is due to both a change of the local hydrodynamics due to Marangoni stresses because of inhomogeneous surface tension and a barrier effect resulting from the coverage of the interface by surfactant molecules. We give a thermodynamically consistent mathematical model for these phenomena, based on continuum thermodynamics employing sharp-interface balances and an appropriate form of the entropy principle. This is complemented by first numerical results on mass transfer under such conditions.

This is joint work with Chiara Pesci and Holger Marschall (Darmstadt)

We will discuss 2 closely related families of structures whose existence follows from weak forms of AC and whose non-existence follows from the principle that every invariant colouring of the set of infinite sets of natural numbers is Ramsey.

The packing/covering conjecture, a unifying generalisation of the base packing, base covering, union and intersection theorems (and so also of Menger's theorem) to infinite matroids, is the most important open problem in infinite matroid theory. For example, it implies the Erdős-Menger Conjecture, recently proved by Aharoni and Berger. I'll explain why it is so important and talk about what progress has been made in proving special cases.

It has been known since the 1990's that harmonic maps from Riemannian or Lorentzian surfaces into Symmetric spaces admit loop group generalizations of the classical Weierstrass representation (Riemannian) or d'Alembert solution of the wave equation (Lorentzian). These allow one to construct solutions to the various geometric problems that are associated, via the Gauss map, to harmonic maps. The utility of these representations is obstructed by the loss of geometric information in the loop group decomposition that relates the harmonic map to the "Weierstrass" data. Recently, special types of Weierstrass data have been introduced that contain full geometric information along a curve. In this presentation, we will discuss recent applications of this technique to the construction of all equivariant Willmore surfaces and the study of singularities of constant curvature surfaces.

We discuss several geometric flows that arise as renormalization group flows in perturbative quantum field theory. One example has become known in mathematics as second order renormalization group flow, which is a non-linear deformation of the Ricci flow. We will study general properties of these flows from the point of view of global analysis and differential geometry. In particular, we will point out new analytic and geometric phenomena that occur in the investigation of these flows.

The image reconstruction problem of magnetic particle imaging consists of the determination of the magnetic particle density function from the measured voltage signal induced by an applied magnetic field. The relation between the particle distribution and the measured signal is described by the system function which contains information about particle dynamics, experimental setup, and the measurement parameters. In practice, a discrete version of the system function is determined experimentally.

Since the image reconstruction is sensitive to noise, regularization methods are necessary. Currently, regularization strategies such as classical Tikhonov regularization, truncated singular value decompositions as well as iterative methods such as Kaczmarz method or conjugate gradient algorithm are applied. We will focus on variational regularization methods with sparsity constraints for MPI which incorporate more adequate a priori information on the solution such as total variation norm for preserving edges in the image or sparsity promoting \(L_{1}\) -norms of shearlet coefficients. We will present numerical results both for simulations as well as real data.

From practical applications in the area of wind and water power management we have prediction models with uncertain data which results in optimization problems with probabilistic constraints, so called chance constraints. Suppose such a problem can be transformed to a deterministic one by using the analytic expression of the distribution function of the probabilistic part, and that this distribution function has a reasonable numeric approximation, as this is the case for multivariate normal-distributed values. Why not using an SQP method to solve this problem? In general the accuracy of the numeric approximation of a high dimensional multivariate cdf does not fit the needs of SQP solver for smooth nonlinear problems. The talk shows which problems arises and how to deal with them, especially with the non-smooth character of the approximation in use. Numeric results also shows the effect of parallelization and automatic differentiation

Classical modular forms are holomorphic functions that are meromorphic at the cusps and satisfy nice modular symmetries. Harmonic Maass forms are real-analytic generalizations thereof in that instead of being holomorphic they are annihilated by the weight \(k\) Laplace operator. These functions generalize Maass waveforms, which are of weight \(0\) and decay in the cusps. Recently there has been an active interest in harmonic Maass forms, as their holomorphic parts (so-called mock modular forms) naturally occur in various areas of math- ematics and physics. The probably most famous example of mock modular forms are given by Ramanujan's mock theta function which he introduced in his last letter to Hardy shortly before he died.

In this talk I instead consider Maass forms which are also allowed to have poles in the upper half plane and give various applications of such functions including Hardy-Ramanujan type formulas for meromorphic modular forms. All this is joint work with Ben Kane.

Um die Vergleichbarkeit der Anforderungen an die zentrale Matura in Österreich begründen zu können, wurde ein dreidimensionales Modell entworfen und empirisch geprüft mit den Kompetenzbereichen Operieren, Modellieren und Argumentieren. Im Vortrag werden die theoretischen Hintergründe (Modellierungsansätze) vorgestellt und die aktuellen Abituraufgaben in Österreich und Deutschland (exemplarisch) miteinander verglichen.

In parallel to the Gross-Kohnen-Zagier theorem, Zagier proved that the traces of the values of the j-function at CM points are the coefficients of a weakly holomorphic modular form of weight 3/2. Later this result was generalized in different directions and also put in the context of the theta correspondence. We recall these results and report on some newer aspects.

The moment problem has a direct application in polynomial optimization, where one wants to optimize the value a polynomial can attain over a given set. Several interesting problems in polynomial optimization turn out to be hard, but a suitable method to approximate these problems is the so-called Lasserre relaxation, i.e. one replaces positive polynomials by sums of squares. This results in a semidefinite program (SDP) which can be dualized via conic duality, resulting in an SDP where one optimizes over linear functionals. In this step the moment problem plays a crucial role: If the optimizing linear functional turns out to be a moment function, i.e. its moments are the moments of a positive measure, then the relaxation is actually exact and one obtains the optimal value of the original polynomial optimization problem.

The classical moment problem for linear functionals on polynomials in commuting variables has a long and fruitful history, whereas the investigation of the moment problem in non-commuting variables is relatively new. The latter is closely related to operator theory and one can end up in possibly infinite-dimensional spaces. In this talk we will introduce the non-commutative equivalent(s) of the moment problem and show how the theory can be applied to polynomial optimization problems arising in quantum theory.

Abiturprüfungen gerade im Fach Mathematik stehen schnell und häufig im Fokus der öffentlichen Aufmerksamkeit. Dabei ist im vergangenen Jahr auch die Art und Weise, wie die Hamburger Abituraufgaben konzipiert sind, über Presseveröffentlichungen in die öffentliche Diskussion geraten. Die Spannweite der Einschätzungen dieser Aufgaben ist dabei beträchtlich: Manche sehen eine zunehmende Trivialisierung der Anforderungen, andere betrachten die Hamburger Aufgaben als bundesweit vorbildlich und richtungsweisend. Unbestritten ist, dass die heutigen Anforderungen einen anderen Charakter haben als die früherer Jahre. Während vor gut zehn Jahren stärker theorie- und insbesondere fertigkeitsbezogene Anforderungen zu bewältigen waren, liegt der Schwerpunkt in den letzten Jahren deutlich im Bereich der Realitätsbezüge, wobei Übertragungsleistungen zwischen verbal und mathematisch beschriebener Realität eine wichtige Rolle spielen. In dem Vortrag werden Kritikpunkte aufgegriffen, Stärken und Schwächen in der aktuellen Aufgabenkonzeption gegenübergestellt und Perspektiven aufgezeigt.

Professionelle Kompetenz von Mathematiklehrkräften setzt sich aus verschiedenen Komponenten zusammen. Dabei spielen zum einen Wissen—hier speziell mathematisches, mathematikdidaktisches und pädagogisches Wissen—als auch Wahrnehmung und Können im Kontext dieser Wissensaspekte eine zentrale Rolle.

Im Vortrag wird dargelegt, wie die verschiedenen Kompetenzkomponenten methodisch erfasst werden können. Dazu werden Konzepte und Realisierungen aktueller empirischer Studien darlegt und auf Itemebene illustriert. Ausgewählte Ergebnisse werden präsentiert.

I will present and discuss a few examples and some applications of well quasi-orders and better quasi-orders, mostly arising in the theory of linear orders. I will try to clarify the connections between them, and point to some related questions stemming from descriptive set theory.

In computational biology, a fundamental question is the extraction of evolutionary information from DNA sequences. We consider protein sequences here and we shall describe how a precise mapping between the one-dimensional representation of a protein (its sequence) and its three-dimensional representation (its structure) revealed important biological information on protein-protein binding sites and on mechanical and allosteric properties of proteins. Coupled with a reductionist physical model of molecular interaction, this mapping has been fundamental for discriminating protein partners, and considerably advancing on the problem of a computational reconstruction of a protein-protein interaction (PPI) network.

We recall that PPI are at the heart of the molecular processes governing life and constitute an increasingly important target for drug design. Given their importance, it is vital to determine which protein interactions have functional relevance and to characterize the protein competition inherent to crowded environments. Suitable mathematical approaches appear necessary to properly address these questions.

In this talk we consider an optimal control problem of the evolutionary Navier-Stokes system in two spatial dimensions. The control is distributed and submitted to bound constraints. The cost is the tracking functional, but it does not include the Tikhonov's regularization term. The numerical approximation of this problem is considered. The lack of coercivity of the cost functional in the control variable makes the analysis more complicated in several aspects. First, the sufficient second order optimality conditions, which are the main tool for the error estimates, are not the standard ones. Second, the strong convergence of the optimal controls for the numerical discretization approximation is not clear at all. This convergence can be proved for bang-bang optimal controls. In this case, we prove some error estimates for the difference between the discrete and the continuous optimal states.

Heteroclinic cycles and networks occur as prototypes for stop-and-go dynamics in a wide range of applications from geophysics to neurosciences. They consist of finitely many equilibria \(\xi_j\) and connecting trajectories \([\xi_j \to \xi_{j+1}] \subset W^{u}(\xi_j) \cap W^{s}(\xi_{j+1})\), and may be structurally stable in systems with symmetry. In this talk we consider simple heteroclinic networks in \(\mathbb{R}^4\)–constructed from simple, non-homoclinic, robust cycles. There are few ways by which such cycles can be joined to form a network, and we provide a complete list of these. Using the stability index from Podvigina and Ashwin (Nonlinearity 24, 887-929, 2011), we describe non-asymptotic stability properties of individual cycles and derive information about stability of the entire network as well as nearby dynamics. This strongly depends on the equivariance of the system–networks with seemingly identical geometry, but different symmetry groups, display very different stability configurations. This talk will be divided into parts one and two.

We show that the Schwarzschild spacetime is the only static vacuum asymptotically flat general relativistic spacetime that possesses a suitably geometrically defined photon sphere. We will present two proofs, both extending classical static black hole uniqueness results. Part of this work is joint with Gregory Galloway.

In a typical participating life insurance contract, the insurance company is entitled to a share of the return surplus as compensation for the return guarantee granted to policyholders. This call-option-like stake gives the insurance company an incentive to increase the riskiness of its investments at the expense of the policyholders. This conflict of interests can partially be solved by regulation deterring the insurance company from taking excessive risk. In a utility-based framework where default is modeled continuously by a structural approach, we show that a flexible design of regulatory supervision can be beneficial for both the policyholder and the insurance company.

I will introduce two new classes of forcing notions, which are intermediate between \(\sigma\)-centered and c.c.c., and strongly proper and proper respectively. Forcings in these classes have nice and interesting properties, including not adding random reals, not adding uncountable anti-cliques in open graphs, the \(\omega_1\) approximation property, iterability, and other properties. Many classical forcing notions fall into these newly defined classes.

This is a joint work with Jindrich Zapletal.

Learning or memory formation are associated with the strengthening of the synaptic connections between neurons according to a pattern reflected by the input. According to this theory a retained memory sequence is associated to a dynamic pattern of the associated neural circuit. In this work M. Krupa and myself have considered a class of network neuron models, known as Hopfield networks, with a learning rule which consists of transforming an information string to a coupling pattern in the form of a robust heteroclinic cycle for an approximate system. I will explain this idea and present results which show a tight connection between existence of the heteroclinic cycles and the structure of the coupling.

Fully-discrete approximations of the Allen-Cahn equation are considered. In particular, we analyze a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space), and we prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. Stability estimates in the natural energy norms are proved using an appropriate duality argument, combined with a boot-strap technique. Great care is exercised in order to quantify the dependence upon \(1/\epsilon\) of various constants appearing in these estimates. In particular, the polynomial dependence upon \(1/\epsilon\) is demonstrated. The applicability of our estimates in a optimal control setting is also demonstrated.

In this lecture I will discuss the theory of soap films, or minimal surfaces, and their role in mathematics. These objects appear as fundamental tools in problems coming from different fields, like mathematical physics, topology, complex geometry, conformal geometry and others. In particular I will describe how they come up in our resolution, together with André Neves, of the Willmore conjecture (1965). This was about the quest to find the best torus of all.

We consider a reaction diffusion equation that models biological processes with a substance fixed at each spatial point that can be in one of two states. Moreover, points may switch state according to a hysteresis law. Points in different states segregate the domain into several subdomains and switching implies that these subdomains are separated by free boundaries.

For bounded domains in \(\mathbb{R}^n\), numerical results reproduce the observed experimental patterns, but for \(n \geq 2\), existence and uniqueness of solutions as well as their continuous dependence on initial data have not been rigorously addressed. We will present recent progress on these questions under the assumption that the solution has non-vanishing derivative at the free boundary.

My talk is motivated by the problem of renormalization of QFT on real analytic spacetimes. In a first part, I will explain how we are let to study and regularize families of complex powers of analytic functions of the form \[\prod_{i=1}^p (f_j+i0)^{\lambda_j} \] where \(f_j\) are real analytic functions and \(\lambda_j\) are complex powers. I will show that \(\prod_{i=1}^p (f_j+i0)^{\lambda_j}\) is a distribution valued in meromorphic functions with linear poles, the proof relies on Hironaka's resolution of singularities and recent results (2015) of Guo--Paycha--Zhang. This allows me to regularize \(\prod_{i=1}^p (f_j+i0)^{\lambda_j}\) at integer values for \(\lambda_j\). Then I will show how a detailed functional analytic study of the family \(\prod_{i=1}^p (f_j+i0)^{\lambda_j}\) allows to renormalize QFT on analytic spacetimes following the Epstein--Glaser method generalizing the work of Düutsch-Fredenhagen-Keller-Rejzner.

Existence of Hadamard states for a free field theory on a globally hyperbolic spacetime has been proven via a metric deformation argument, proposed by Fulling, Narkowich and Wald in the eighties. The main deficiency of this scheme is the complete loss of any control on the invariance of the state under the action of the background isometries. In order to account for them, one needs to resort to specific construction schemes which are often valid for a given free field with a fixed value of the mass and, if present, of the coupling to scalar curvature. Via an extended version of the M\oslash;ller operator, we show that, these isometry invariant Hadamard states can be deformed to Hadamard states for any value of the mass and of the coupling to scalar curvature. Furthermore the invariance under any spacelike isometry is preserved, while, for the timelike ones, a kind of adiabatic procedure is necessary. (Joint work with Nicoló Drago, Genoa; arXiv:1506.09122.)

We construct a symmetric monoidal stable model category of proper \(G\)-spectra where \(G\) is any Lie group. The homotopy category of this model category is generated as a triangulated category by the \(G\)-orbits with compact isotropy and admits restriction functors to genuine \(H\)-spectra for any compact subgroup \(H\) of \(G\). When \(G\) is discrete, a proper \(G\)-spectrum gives rise to a \(G\)-Mackey functor by taking homotopy groups. If \(G\) has enough bundle representations, then on finite proper \(G\)-CW complexes we identify the cohomology theory represented by the sphere \(G\)-spectrum as Lück's equivariant stable cohomotopy. Further we will provide an algebraic model for rational proper \(G\)-spectra for a discrete group \(G\). If time permits we will also discuss relations to equivariant K-theory. All this is joint work with Degrijse, Hausmann, Lück and Schwede.

Let \(G\) be an infinite discrete group. A classifying space for proper actions of \(G\) is a proper \(G\)-CW-complex \(X\) such that the fixed point sets \(X^H\) are contractible for all finite subgroups \(H\) of \(G\). In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper \(G\)-spectra and study finiteness properties of such a stable classifying space for proper actions. We investigate when \(G\) admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the smallness of the sphere spectrum in the homotopy category of proper \(G\)-spectra and to classical finiteness properties of the Weyl groups of finite subgroups of \(G\). If \(G\) is virtually torsion-free, we show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of \(G\) thus providing a geometric interpretation of the virtual cohomological dimension of a group. We also present and example of a group that admits a stable classifying space for proper actions of strictly smaller dimension than the dimension of any classifying space for proper actions.

We study Lissajous curves in the 3-cube, that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single 1-d Fast Chebyshev Transform (by using the well-known Chebfun package), and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Typical applications are in the framework of Lissajous sampling for MPI (Magnetic Particle Imaging).

Joint work with L. Bos (University of Verona - Italy) and M. Vianello (University of Padova - Italy)

We study asset allocation decisions of an investor that has the opportunity to invest in an illiquid asset that is only traded at time 0. We use a generalized martingale approach to find the optimal terminal wealth and to determine the optimal amount invested in the illiquid asset. We also characterize optimal trading strategies via Clark?s formula and provide a simple representation in terms of a liquidity-related derivative. As an application, we? study optimal asset allocation with fixed-term deposits and fixed-term defaultable investments. We demonstrate that the presence of such investment opportunities can have a significant impact on asset allocation: CRRA agents with realistic values of relative risk aversion optimally allocate more than 40% of their wealth to illiquid assets if these yield a moderate excess return of 100 basis points over the money market account.

Es ist wohlbekannt, dass man ein diskretes Logarithmusproblem als Basis für einen Pseudozufallsgenerator benutzen kann. Die ursprüngliche Konstruktion von Blum und Micali ist allerdings vom praktischen Standpunkt aus recht ineffizient.

Der "Dual Elliptic Curve Deterministic Random Bit Generator" ist ein von NIST standardisierter Generator ähnlicher Art, dessen Sicherheit auf dem ersten Blick auf dem diskreten Logarithmenproblem in elliptischen Kurven beruht. Dieser Generator ist allerdings unsicher. Außerdem enthält der Standard, wenn man ihn wörtlich nimmt, eine offensichtliche "Hintertür". Letzteres wurde im Jahr 2013 im Zuge der Enthüllungen von Edward Snowden bekannt.

Aufgrund dieser Gegebenheiten liegt die Konklusion nahe, dass man das diskrete Logarithmenproblem in elliptischen Kurven nicht als Basis für einen Generator verwenden sollte.

In dem Vortrag wird aufgezeigt, dass dieser Schluss allerdings voreilig ist. In der Tat ist es möglich, einen recht effizienten Generator zu erhalten, dessen Sicherheit auf einer Standardannahme über die Schwierigkeit des bekannten Diffie-Hellman-Problems beruht.

Der Vortrag beruht auf einer Zusammenarbeit mit Domingo Gomez in Santander.

The Kähler/Kähler correspondence is a special case of the hyper-Kähler/quaternionic Kähler correspondence, the latter of which relates the supergravity \(c\)-map to the rigid \(c\)-map. We describe the K/K correspondence in a more general context as the special case of a twist construction which was established by A. Swann, analyse how the Ricci curvatures are related, and present new twists which might relate the supergravity \(r\)-map to the rigid \(r\)-map.

It is common practice to consider the generic version of large cardinals defined with an elementary embedding, but what happens when such cardinals are really large? The talk will concern a form of generic I0 and the consequences of this extravagant hypothesis on the "largeness" of the power set of \(\aleph_\omega\). This research is a result of discussions with Hugh Woodin.

We generalize a method of constructing the smallest (w.r.t. set inclusion) monotonic (w.r.t. a "designated statistic") confidence region for a general parameter of interest in an arbitrary model on a totally preordered sample space. This method, dating back to 1957 and due to Robert J. Buehler, is originally known from reliability theory, but is applicable to statistical interval estimation rather generally, and deserves, in our opinion, more attention.

Apart from establishing some of the optimality properties of Buehler confidence regions, we briefly address the problem of selecting meaningful designated statistics by presenting some examples.

In this talk we consider the impact of a very simple and small spatial heterogeneity on the existence of localized patterns in a system of PDEs in one spatial dimension. The existence problem for a `localized defect pattern' reduces to the problem of finding a heteroclinic orbit in an ODE in `time' \(x\), for which the vector field for \(x > 0\) differs slightly from that for \(x < 0\), under the assumption that there is such an orbit in the unperturbed problem. We show that both the dimension of the problem as well as the nature of the linearized system near the endpoints of the heteroclinic orbit has a remarkably rich impact on the existence these defect solutions.

A stunning feature of Einstein's equations of general relativity is the onset of singularities in finite time from perfectly regular initial data. This happens, for instance, in the dynamical formation of black holes. In most cases, however, a rigorous treatment of this phenomenon is hopeless at the present stage of research. Consequently, one resorts to simpler model problems. In the last decade there was tremendous progress in the study of singularity formation for nonlinear wave equations and I will report on some of the most important results.

We characterize exchangeability of infinitely divisible distributions in terms of the characteristic triplet. This is applied to stable distributions and self-decomposable distributions, and a connection to Lévy copulas is made. We further study general mappings between classes of measures that preserve exchangeability and give various examples which arise from discrete time settings, such as stationary distributions of AR(1) processes, or from continuous time settings, such as Ornstein-Uhlenbeck processes or Upsilon-transforms.

Sub-critical graph classes are special block-stable graph classes that are defined by an analytic condition on their generating functions. However, they appear quite frequently as minor closed graph classes. For example, series-parallel graphs (\(Ex(K_4)\)) or outerplanar graphs (\(Ex(K_4,K_{2,3})\)) are sub-critical and in general it is conjectured that a minor closed graph class is sub-critial if at least one of the excluded minors is planar. In particular it is known that planar graphs \(Ex(K_5, K_{3,3})\) are not sub-critical.

During the last few years random sub-critical graph classes have been intensively studied and many characteristics (degree distribution, maximum degree, maximum block, scaling limit etc.) have been characterized. The purpose of this talk is to give a survey of these results and to present also new ones (for example on the size of maximal independent sets or on a central limit theorem for subgraph counts).

In 1968, V. E. Zakharov derived the Nonlinear Schrödinger equation for the two-dimensional water wave problem in the absence of surface tension, i.e., for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this talk we give a rigorous proof that the wave packets in the two-dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the Nonlinear Schrödinger equation.

Although Václav Hlavatý himself considered his achievements in differential geometry, especially in connection with relativity theory and unified field theory, more important, he was one of the few mathematicians who were closely involved in politics. This did not come out of the situation created in emigration, where he was deeply involved in the activities of Czechoslovak emigré community in the USA (1948-1969), but is apparent also in the earlier period. In this contribution, I will pay attention especially to the changes in the style of education of doctoral students in the turbulent years after WWII, when Hlavatý was in strong opposition to mass breeding of doctoral students in mathematics.

Well quasi orders, a "frequently discovered concept" to quote Kruskal, are of immense importance in combinatorial set theory and logic in general, including computer sciences. One can associate several different ordinal-valued ranks to such orders and in general, they are well understood. With one exception, that of the rank of the tree of incomparable sequences, called the width. We have studied this rank in a joint work with Schmitz and Schnoebellen and the talk will present some of our results.

Many high-dimensional data analysis problems require some dimension reduction process before sophisticated analysis tools can be used. Standard approaches design one single projector of some predefined rank. One may ask what to do if several projectors can be chosen. We study the choice of collections of projectors of varying ranks that be used simultaneously.

It is a classical observation that a small geodesic ball at a point of positive scalar curvature contains more volume than a round ball in Euclidean space that has the same surface area. In this talk, I will describe several global effects of non-negative scalar curvature on the large-scale isoperimetric structure of asymptotically flat manifolds, including seminal contributions by H. Bray, G. Huisken, J. Qing, R. Schoen, G. Tian, and S.-T. Yau. I will then discuss consequences of these phenomena for the space-times evolving from such manifolds according to the Einstein equations. My presentation will include recent joint work with S. Brendle, A. Carlotto, O. Chodosh, and J. Metzger.

In this talk I will first recall the classical Brauer algebra and present a somewhat suprising fact relating the representation theory of Brauer algebras to evaluations of Kazhdan-Lusztig polynomials for special orthogonal Lie algebras.

To explain this connection, I will introduce a category of representation for these Lie algebras with the Brauer algebra appearing as the endomorphism ring of a specific module. Using a graded analog of this category I will explain how to equip this endomorphism ring with a grading and transfer this grading onto the Brauer algebra. This grading is inherently linked to Kazhdan-Lusztig theory and explains why the corresponding graded representation theory of this graded Brauer algebra is controlled by Kazhdan-Lusztig polynomials. Finally I want to link the branching behaviour of the family of Brauer algebras to the categorification of a representation for a quantum symmetric pair. All of this is based on joint work with C. Stroppel.

Mathematical models in many applications across the physical and life sciences involve partial differential equations on complex evolving domains. Often the domain is unknown leading to free boundary problems. I will discuss the variational formulation of parabolic equations posed on evolving surfaces. These surface PDEs are usually coupled to equations for the surface involving geometric quantities and surface energy functionals such the area or Willmore functionals. I will discuss their numerical approximation using evolving surface finite elements. The context will be applications in cell biology involving biomembranes and cell motion.

Kurze Charakterisierung Digitaler Werkzeuge (DGS, TK, F-Plotter, CAS) und Lernumgebungen im MU. Dynamische Visualisierungen durch dynamische Software (z.B. GeoGebra), hier vorzugsweise DGS und Funktionenplotter.

Hauptteil: Beispiele aus

• Geometrie (Euler-Gerade, Parabel aus Brennpunkt-Leitlinie)
• Algebra (Binomische Formel, Pythagoras indisch oder Stuhl der Braut, Heron-Verfahren)
• Funktionen (Elementare Optimierung, Funktionenlupe Differenzialrechnung).

Bei den Geometrie-Beispielen steht der Aspekt der Dynamisierung im Vordergrund, eine visuelle Repräsentation ist durch eine Zeichnung oder Konstruktion ja schon gegeben.

Bei den beiden ersten Algebra-Beispielen geht es darum, einen abstrakten, formelmäßigen Sachverhalt graphisch zu repräsentieren (visualisieren) und zu erleben, welche Vorteile eine Dynamisierung dabei noch bringt. Beim Heron kann man erleben, wie man mit einer geeigneten Lernumgebung aus einer völlig einfachen Idee einen Algorithmus entwickeln kann.

Bei den Funktionen steht im Vordergrund, wie man aus einem anschaulichen Herangehen Funktionsgraphen erzeugen bzw.untersuchen kann. Dies ist ein schülernaher Ansatz, der durchaus konträr zum üblichen Vorgehen im MU ist (mit Vorrang von Algebra & Kalkül).

Abschließend seien noch (vorwiegend für die Diskussion) Gefahren und offene Fragen genannt, die der weiteren Erforschung wert sind.

Gefahren:

• Ersetzt die Visualisierung die Mathematik?
• Visualisiert die Visualisierung das, was sie visualisieren soll? Oder schafft sie neue Vorstellungen, die sich dazwischen schieben?
• Wird die Visualisierung vielleicht selbst zum eigenen Gegenstand?

Fragen:

• Was ändert sich durch den Einsatz dynamischer Software und dynamischer Lernumgebungen beim Lernen von Mathematik?
• Lernen die Schüler mehr, besser?
• Gibt es durch dynamische Software & Visualisierungen andere Vorstellungen und Bilder von Mathematik bei den Schülern?
• Welche Fehler sind zu vermeiden?

Das Funktionenmikroskop von A. Kirsch war ein Klassiker für die Erarbeitung eines Grundverständnisses von Differenzierbarkeit im Sinne der "Idee der 'lokalen Glättung' des Graphen bei fortwährender Vergrößerung". Zunächst ein aufwändiger Foliensatz in Lehrerhand, konnte die Grundidee des 'Hineinzoomens' später mit gängigen Funktionenplottern digital umgesetzt werden.

In dem Vortrag wird nun eine interaktive digitale "Funktionenlupe" vorgestellt, Funktionenmikroskop 2.0 gewissermaßen, die mit zwei Graphikfenstern und Ortslinien einen entdeckenden, anschaulichen und kalkülfreien Zugang lokal zur Steigung des Funktionsgraphen und global zur Ableitung der Funktion sowie zur Krümmung bietet.

Die Funktionenlupe ist eine interaktive Lernumgebung für Schüler und ermöglicht beim Einstieg in die Analysis einen zunächst kalkülfreien Aufbau von Grundvorstellungen.

Nonlinear delay differential equation problems appear in all scientific disciplines. Classical problems such as chatter instabilities in mechanical engineering, abnormal physiological controls, and lasers experiencing delayed feedbacks are now systematically investigated [1]. Although we may simulate these problems numerically, asymptotic approaches based on natural limits of some of the parameters (large delay, low or high feedback levels, multiple time scales) are needed to substantiate analytically specific dynamical phenomena caused by the delay. Lasers and optical oscillators are particularly interesting devices because the effects of a delayed feedback can be studied both theoretically and experimentally. In this presentation, we concentrate on time-periodic square-wave (SW) oscillations. SW oscillations of scalar delay differential equations exhibiting a large delay have been rigorously studied in the 1980's. They result from Hopf bifurcations and the plateau lengths are nearly equal to one delay. Here we show both mathematically and experimentally the possible existence of periodic SWs with different plateau lengths and periods close to one delay. We also found that multiple stable SWs of different periods may coexist for the same values of the parameters [2-3]. The model equations are second or higher order delay differential equations and Hopf bifurcations are the basic mechanisms for the SWs.

1. T. Erneux "Applied delay Differential Equations" Springer (2009)
2. G. Friart, G. Verschafelt, J. Danckaert, and T. Erneux, All-optical controlled switching between time-periodic square waves in diode lasers with delayed feedback, Optics Letters 39, 6098-6101 (2014)
3. L. Weicker, T. Erneux, D. P. Rosin and D. J. Gauthier, Multi-rhythmicity in an optoelectronic oscillator with large delay, Phys. Rev. E91, 012910 (2015)

There are two types of compact symmetric spaces (other than Lie groups): the classical ones which come in 7 infinite series, and the 12 exceptional ones. Classical symmetric spaces are just Grassmannians \({\bf G}_k({\bf K}^n)\) over the real, complex and quaternion numbers (\({\bf K} = {\bf R},{\bf C},{\bf H}\)) and further spaces of self-reflective subspaces of Grassmannians (fixed spaces of certain involutions); e.g. \({\bf O}_{2n}/U_n = \{{\bf C}{\bf P}^{n-1}\subset {\bf G}_2({\bf R}^{2n})\}\) = the set of all subspaces congruent to \({\bf C}{\bf P}^{n-1} \subset {\bf G}_2({\bf R}^{2n})\). The 12 exceptional spaces should be somehow related to the octonions \({\bf K}={\bf O}\), but the relation is still not fully understood. Among these spaces the role of the Grassmannians is played by the so called Rosenfeld planes of dimension 16, 32, 64, 128; all others are spaces of reflective subspaces. Boris Rosenfeld (around 1956) tried to describe these spaces as projective planes over \({\bf O}\otimes{\bf K}\) for \({\bf K} = {\bf R},{\bf C},{\bf H},{\bf O}\), but his attempt failed. However, there are several reasons for a tight relation between the Rosenfeld planes and projective planes:

1. the isotropy representation is the (slightly enlarged) spin representation on \(({\bf O}\otimes{\bf K})^2\),
2. there is Vinberg's infinitesimal description in terms of \(3\times3\) matrices over \({\bf O}\otimes{\bf K}\),
3. there are ``projective lines''.

The evolution of two fluid phases in a porous medium is considered. The fluids are separated from each other and also the wetting phase from air by interfaces which evolve in time. It is shown that the problem can be reduced to an abstract evolution equation. A generalized Rayleigh--Taylor condition characterizes the parabolicity regime of the problem and allows to establish a general well-posedness result and to study stability properties of flat steady states. If surface tension effects on the interface between the fluids are included and if the more dense fluid lies above, bifurcating finger-shaped equilibria exist, which are however all unstable.

The Helmholtz equation or reduced wave equation appears, in particular, as a model for the propagation of acoustic waves. We will present in this talk existence and multiplicity results concerning nonlinear versions of the Helmholtz equation in the whole space. The asymptotic behavior of the solutions we obtain will also be discussed. Our main tool is a dual variational approach in Orlicz spaces which, combined with estimates for the resolvent Helmholtz operator, allows to prove the existence of real-valued solutions for nonlinearities which are not necessarily homogeneous.

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability follows by an energy argument. We prove in addition that the development of non-CMC initial data close to the background contains a CMC hypersurface, which in turn implies that stability holds for arbitrary perturbations. Furthermore, we construct a one-parameter family of initial data such that above a critical parameter value the corresponding development is future and past incomplete.

Let \(N\geq 1\) and \(s\in (0,1)\). In the present work we characterize bounded open sets \(\Omega\) with \(\mathrm{C}^2\) boundary (not necessarily connected) for which the overdetermined problem \(( -\Delta)^s u = f(u)\) in \(\Omega\), \(u=0\) in \(\mathbb{R}^N\setminus \Omega\) and \((\partial_{\eta})_s u=\mathrm{const}\) on \(\partial \Omega\) has a nonnegative and nontrivial solution. Here \(\eta\) is the outer unit normal vector field along \(\partial\Omega\) and for \(x_0\in\partial\Omega\) \[ \left(\partial_{\eta}\right)_{s}u(x_{0})=-\lim_{t\to 0}\frac{u(x_{0}-t\eta(x_0))}{t^s}. \] Under mild assumptions on \(f\), we prove that \(\Omega\) must be a ball. In the special case \(f\equiv 1\), we obtain an extension of Serrin's result in 1971. The fact that \(\Omega\) is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.

Atmospheric mid-latitude circulation is dominated by a zonal, westerly flow. Such a flow is generally symmetric, but it can be occasionally broken up by blocking anticyclones. The subsequent asymmetric flow can persist for several days. In this paper, we apply new mathematical tools in order to reexamine the dynamical mechanisms responsible for the transitions between zonal and blocked flows. By analyzing several blocking indices, we discard the general claim that mid-latitude circulation features two distinct stable equilibria or chaotic regimes, in favor of a simpler mechanism that is well understood in dynamical systems theory: we identify the blocked flow as an unstable fixed point (or saddle point) of a single basin chaotic attractor, dominated by the westerlies regime. We also analyze the North Atlantic Oscillation and the Arctic Oscillation atmospheric indices, whose behavior is often associated with the transition between the two circulation regimes, and investigate analogies and differences with the bidimensional blocking indices. We find that the Arctic Oscillation index, which is a proxy for a global average of the Tibaldi-Molteni blocking index, keeps track of the presence of unstable fixed points. On the other hand, the North Atlantic Oscillation index is representative only of local properties of the North Atlantic blocking dynamics.

Let \(k\) be a field and \(X\) a smooth affine variety over \(k\). If \(Z \subseteq X\) is a closed subvariety equipped with a trivialization of its conormal bundle, then we will give cohomological criteria for \(Z\) to be a complete intersection in \(X\). Along the way, we will compare the Euler class groups as defined by Nori, Bhatwadekar and Sridharan with the Chow-Witt groups introduced by Barge and Morel and give a conditional answer to a conjecture of Murthy. Our method relies on the geometry of smooth split quadrics.

The presence of a \(G_2\)-structure on a manifold \(M\) is equivalent to the existence of a certain 3-form \(\varphi\) on \(M\). Different classes of special \(G_2\)- structures can be described by the behavior of the 3-form \(\varphi\). For example, a \(G_2\)-structure is called calibrated if \(\varphi\) is closed, and cocalibrated if \(\varphi\) is coclosed, that is, if \(\ast\varphi\) is closed with \(\ast\) denoting the Hodge star operator. In the latter case, if \(\ast\varphi\) is proportional to \(d\varphi\), then the \(G_2\)-structure \(\varphi\) is said to be nearly parallel.

As it was shown in [2] the behavior of the Ricci tensor associated to the metric \(g_{\varphi}\) is closely related with the behavior of the \(G_2\)-structure \(\varphi\). By the results in [3] no compact 7-dimensional manifold can support a calibrated \(G_2\)-structure \(\varphi\) whose underlying metric \(g_{\varphi}\) is Einstein unless \(g_{\varphi}\) has holomomy contained in \(G_2\). However, 7-dimensional manifolds with a nearly parallel \(G_2\)-structure are always Einstein.

Using warped products, we show how to construct manifolds endowed with special \(G_2\)-structures from manifolds endowed with different classes of \(\mathrm{SU}(3)\)-structures in such a way that the Einstein condition on the corresponding metric is preserved along this construction.

1. Besse A., Einstein manifolds, Springer, Berlin, Heidelberg, New York, 1987.
2. Bryant R. L., Some remarks on \(G_2\)-structrures, Proceedings of Gokova Geom.- Topology Conference 2005, 75-109, Gokova Geometry/Topology Conference (GGT), Gokova, 2006.
3. Cleyton R. Ivanov S., On the geometry of closed \(G_2\)-structures, Commun. Math. Phys. 270 (2007), 53-67.
4. O'Neill B., Semi-Riemannian Geometry with Applications to Relativity, Pure and Appl. Math. 103, Academic Press, New York, 1983.

It is well known that, for any given bounded domain \(\Omega\) with a ``nice'' boundary, \(BV(\Omega)\) embeds in \(L^1(\partial \Omega)\), in the sense that the total variation of a function \(u\) bounds the \(L^1\) norm of \((u-c)\) through a constants \(K\) which depends on \(\Omega\). About \(c\) various choices can be made. We consider the cases where c is the median or the mean value of the trace of \(u\) over the boundary of \(\Omega\). We prove that balls achieve the least embedding constant \(K\) in both inequalities. Uniqueness of such minimizers is also discussed in details. Some of the tools used in the proof are: modified Cauchy area formula, characterization of sets of constant brightness, characterization of sets of constant projection.

This is a joint work with A. Cianchi, C. Nitsch and C. Trombetti.

The split property expresses the way in which local regions of spacetime define subsystems of a quantum field theory. It is known to hold for general theories in Minkowski space under the hypothesis of nuclearity. In this talk, the split property will be discussed for general locally covariant quantum field theories in arbitrary globally hyperbolic curved spacetimes, using a spacetime deformation argument to transport the split property from one spacetime to another. It is also shown how states obeying both the split and (partial) Reeh--Schlieder properties can be constructed, providing standard split inclusions of certain local von Neumann algebras. Sufficient conditions are given for the theory to admit such states in ultrastatic spacetimes, from which the general case follows. A number of consequences are described, including the existence of local generators for global gauge transformations, and the classification of certain local von Neumann algebras. I will also show that, in the locally covariant context, theories with a finite splitting distance (distal split property) must in fact obey the split property. The interpretation of this last result is that theories that obey the distal split property, but not the split property, either fail to obey the timeslice axiom, or do not admit locally quasiequivalent state spaces.

I will report on several occurences of intersection matrices associated to geometric and representation theoretic problems in positive characteristics. These matrices govern the failure of Lusztig's formula for irreducible characters of reductive algebraic groups in small characteristics, and were used by Geordie Williamson in his recent counterexamples to Lusztig's conjecture.

We describe a group based protocol on a secure password exchange.

In this talk we continue our investigation of circuit diameters. We show the potential of augmentation along circuits as an approach for solving optimization problems. Thereto we study families of polyhedra whose circuit diameter is much lower than their combinatorial diameter, which indicates the possible benefits of circuit algorithms compared to the Simplex method. We further demonstrate that several well-known efficient algorithms actually are circuit augmentation algorithms.

Symplectic forms taming complex structures on compact manifolds are strictly related to a special type of Hermitian metrics, known in the literature also as "pluriclosed" metrics. I will present some general results on "pluriclosed" metrics and their link with symplectic geometry for solvmanifolds. Moreover, I will show for certain 4-dimensional non-Kaehler 4-manifolds some recent results about the Calabi-Yau equation in the context of symplectic geometry.

Given a simplicial sphere or an oriented matroid we can ask if those objects are polytopal and if this is the case are they inscribable? This question can be rephrased as "is a certain semialgebraic set empty?". In many cases we can answer this question numerically with the help of nonlinear optimization and then obtain exact solutions from the numerical solutions. As application of this method we present a description of all 3-spheres with small valence, and an attempt to find the exact number of simplicial 4-polytopes with 10 vertices.

The Jeffrey-Kirwan residue formula computes the intersection pairings on a symplectic quotient \(M // G\) as the residues of certain meromorphic differential forms associated to the fixed point set \(M^T\), where \(T\) is a maximal torus of the compact Lie group \(G\). Key ingredients of the proof are equivariant integration and localization. We extend these techniques to the setting of K-contact manifolds and obtain an analogous residue formula. This is based on joint work with Lana Casselmann.

In this talk we present the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure ?? to be quantized. Then we show that the discrete functionals, de???ning the discrete quantizers as their minimizers, actually ?? -converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-de???nite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which already found applications in the asymptotic analysis of corresponding gradient ???ows. To model situations where the given probability is affected by noise, we additionally consider a modi???ed energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of ?? -convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses.

We consider variations on the commutative diagram consisting of the Fourier transform, the Sampling Theorem and the Paley-Wiener Theorem. We start from a generalization of the Paley-Wiener theorem and consider entire functions with specific growth properties along half-lines. Our main result shows that the growth exponents are directly related to the shape of the corresponding indicator diagram, e.g. its side lengths. Since many results from sampling theory are derived with the help from a more general function theoretic point of view (the most prominent example for this is the Paley-Wiener Theorem itself), we motivate that a closer examination and understanding of the Bernstein spaces and the corresponding commutative diagrams can - via a a limiting process to the straightline interval [-A,A] - yield new insights into the Lp(R)-sampling theory. This is joint work with Gunter Semmler, TU Bergakademie Freiberg, Germany.

Many results in classical Ramsey theory and the theory of well-quasi-orderings can be proven by topological means, or be expressed as topological phenomena.

Thierry Coquand has given many examples of such topological expressions of these combinatorial phenomena which are classically equivalent to statements in point-free topology but which are, thus formulated, constructively provable.

We shall discuss these results from the viewpoint of Gelfand duality of commutative \(C^*\)-algebras, the latter being provable in constructive mathematics, when adequately phrased, and having, therefore, interesting computational content.

In this talk, we will consider bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least \(r\) infected neighbours becomes infected and remains so forever. Assume that initially \(a(t)\) vertices are randomly infected, where \(t\) is the total number of vertices of the graph. Suppose also that \(r < m\), where 2m is the average degree. We determine a critical function \(a_c(t)\) such that when \(a(t) >> a_c(t)\) complete infection occurs with high probability as \(t\) grows, but when \(a(t) << a_c (t)\), then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to \(a(t)\).

This is joint work with Mohammed Abdullah (Huawei Labs, Paris).

I will recall recent results about PBW filtration on the universal enveloping algebra of a simple complex Lie algebra and on their simple modules. Using these results we obtain a flat degeneration of the (partial) flag variety, which has been identified as Schubert variety as well as as quiver Grassmannian.

Generalizing this, I will introduce a three families of degenerations, one of the classical flag variety considered as a Schubert variety, one of the flag variety considered as a highest weight orbit, and one of the flag variety considered as a quiver Grassmannian and study their relations. Finally, interesting combinatorics such as string polytopes, torus fixed points and moment graphs related to these degenerations will be discussed.

We study Dirichlet forms on bounded open subsets of euclidean spaces describing diffusion processes. By introducing speed measures supported on proper subsets we obtain so-called singular diffusions. For the process this corresponds to a time change allowing for jumps. This setup yields an analytic description of a jump-diffusion process, for example on Koch's snowflake.

The twist construction of Andrew Swann is a method to produce new interesting examples of geometric structures out of well-known ones and generalizes, e.g., the \(hK/qK\) correspondence and the construction of nilmanifolds from a torus by repeatedly adding differential relations. In this talk, we present a generalization of the twist construction: the shear construction. We present some interesting new examples of geometric structures obtained by the shear which cannot be obtained by the twist and indicate how the repeated application of the shear produces all 1-connected solvable Lie groups from \(\mathbb{R}^n\).

We describe a market model for trading a single risky asset, in which a large investor seeks to liquidate his position in an infinite time horizon, while maximizing expected proceeds. Trading large orders has an adverse effect on the asset's price, which is determined by the investor's current volume impact and is multiplicative in relation to the current price. The volume impact is a deterministically mean-reverting process whenever no trade occurs. We justify why the proceeds should have a certain form, heuristically by describing a multiplicative limit order book, and also by drawing a link to Marcus type SDEs. The martingale optimality principle suggests that the two dimensional state space of volume impact and number of held assets is separated by a free boundary into a wait- and a sell-region. We derive this free boundary using classical calculus of variations and prove optimality. If time permits, we discuss a variant of our model with stochastic volume impact, in which case verification of optimality reduces to showing certain analytic properties of Hermite functions, some of which remain to be fully proven.

We discuss results and open questions on the stability and the asymptotic behaviour of solutions to Einstein's field equations with positive cosmological constant and discuss their role in the context of the recently proposed conformal cyclic cosmological model.

Tomographic reconstruction from incomplete data plays an important role in many practical applications (due to technical restrictions), and is utilized as a technique for dose reduction, e.g., in x-ray imaging. The underlying mathematical problem is known to be severely ill-posed, i.e., even small measurement errors can cause huge reconstruction errors. As a result, some features of the original object cannot be reconstructed reliably (invisible singularities) if a significant portion of the data is missing and additional artifacts (added singularities) can be generated that can degrade the reconstruction quality even further. Both phenomena can be mathematically characterized in terms of their orientations by using microlocal analysis. To integrate this type of information into practical reconstruction algorithms, sophisticated tools are needed that can make this directional information practically accessible. In this talk, we will discuss that highly directional systems can be used for the implementation of microlocal characterizations into practical algorithms. In addition to that, we also show that by using directional systems an orientation sensitive and edge preserving regularization can be achieved.

Magnetic Particle Imaging (MPI) is an emerging imaging modality that measures the magnetization response of paramagnetic nanoparticles to determine its spatial distribution. In this talk, we consider a 2D situation in which the data is generated by using a constantly rotated and shifted field free line (FFL). In this imaging setup, the measured signals incorporate an averaging along these FFL¹s, leading to a mathematical formulation of the reconstruction problem that is a combination of a 2D Radon transform and a 1D MPI model. Therefore, one of the main steps in the reconstruction algorithms is the inversion of the Radon transform. However, in most practical situations, the generated field free lines are not ideal and appear to be rather curved. In this talk, we present a reconstruction approach which takes into account the curved nature of the FFL¹s and investigate the effectiveness of this method in numerical experiments.

The topology of the Milnor fibre of an isolated singularity holds important information about the singularity. For hypersurface singularities and complete intersections, it is well understood. But for the slightly more complicated case of Cohen-Macaulay codimension 2 singularities, systematic studies have only started recently. In this talk, I shall present an approach to computing the Betti numbers of the Milnor fibre (in the range of good dimensions). Applying this to the list of simple 3-fold singularities within this class, certain patterns are revealed, which I am also going to explain in the talk (joint work with Matthias Zach).

The generalized continuous wavelet transforms that this talk is about are constructed by choosing a suitable matrix group, the so-called dilation group \(H\). Wavelet systems associated to this group then arise by picking a suitable wavelet, dilating it by elements of \(H\), and translating arbitrariily. The wavelet transform of a signal (a function or tempered distribution) then arises by taking scalar product with the elements of the wavelet system.

In higher dimensions, there is a large variety of suitable matrix groups to choose from. One of the pertinent questions in connection with the associated transforms is whether they are able to efficiently encode important properties of the analyzed functions, specifically smoothness behaviour. So far, these questions have only been studied for a few isolated groups.

For example, it is well-known that the homogeneous Besov spaces in any dimension are related to the continuous wavelet transform associated to the similitude group. Another, celebrated, example is the shearlet group and its associated systems of wavelets, called shearlets. Shearlet coorbit space norms, obtained by imposing weighted mixed \(L^p\) norms on the shearlet coefficients, can be understood as a quantification of (directional) global smoothness, whereas local directional smoothness (or roughness) features such as the wavefront set are captured by local decay behaviour of the shearlet coefficients.

In this talk I will give an overview of recently developed methods that allow to study the above-mentioned properties of wavelet systems in a unified and comprehensive framework, for a large variety of dilation groups. The new techniques provide far-reaching extensions of the above-mentioned results for shearlets.

The addictive power of many drugs including ethanol relies on their ability to change synaptic transmission in the brain's reward system. As part of this system we consider glutamatergic synapses on medium spiny neurons (MSN) in the Nucleus Accumbens which are modulated by dopaminergic input from the midbrain.

Depending on their input the strength of these synapses changes to increase or decrease synaptic transmission and thereby mediate a coupling between incentives and behavior. In case of addictive drugs this plasticity is altered which leads to craving for the drug and expecting pleasure from consumption of the drug.

Ethanol interferes with the functions of several receptors known to be respsonsible for the plastic changes at the synapses such as glutamate receptors of NMDA type and dopamine receptors of D1 or D2 type. Physiologically, it has for instance been observed that stimulations which under normal conditions induce long term depression (decrease of synaptic strength) can instead induce long term potentiation (increase of synaptic strength) at the same synapses when ethanol is present at sufficiently high doses.

To understand the alterations of the synaptic response due to the presence of ethanol we propose a model for the synaptic transmission in the Nucleus accumbens where the expected effects of ethanol on each kind of receptor is taken into account. The model comprises a fast time scale describing electric currents through receptor channels and a slow time scale for the resulting modifications of proteins like phosphorylation. On an even lower time scale, structural changes in the composition of the synapse will be responsible for prolonged potentiation or depression of the synapse.

To validate our model we use data of single cell measurements obtained from brain slices of mice with and without added ethanol. In a first step we try to capture the acute effects of ethanol on the synaptic transmission. Moreover, we shall describe the changes in synaptic plasticity due to the presence of ethanol.

From a dynamical systems point of view we obtain a slow-fast system where certain inputs to the fast system may or may not be able to induce changes of the slow system leading to shifts of equilibrium values. These shifts can then be interpreted as expressions of synaptic plasticity.

I will introduce several new examples of isospectral but non-diffeomorphic nilmanifolds. These nilmanifolds are constructed from Clifford modules. Their classification leads us to such examples, not only just pairs, but any given number of such manifolds.

Alexandrov spaces (with curvature bounded below) are a natural synthetic generalization of Riemannian manifolds. In this talk I will discuss recent developments on the geometry and topology of Alexandrov spaces with isometric actions of compact Lie groups.

The theory of Weihrauch degrees is about representing classical theorems of analysis in Baire space and comparing their strength (measured as the Weihrauch degree). In this talk, we are exploring a version of this theory for generalized Baire space. The first step in this generalization is that of finding a suitable generalization of the real numbers on which we can prove generalized version of theorems from classical analysis. Due to the fact that the real numbers are the only complete ordered field and that completeness is crucial in most theorems of analysis, classical topological approaches fail in the generalized context. For this reason, in generalizing results from classical analysis, different tools have to be used. The first part of the talk will be devoted to the presentation of these tools and to the construction of an extension of the real numbers on which they can be used to prove basic theorems form analysis (i.e. Intermediate Value Theorem). In the second part of the talk we will be focusing on generalizing notions from computable analysis and investigate how this new framework can be used to characterize the strength of the generalized version of a basic theorem of analysis we presented in the first half of the talk.

I will review and discuss some aspects of Voevodsky's univalence axiom. First, I will illustrate how recent work of Cisinski provides a streamlined proof of the validity of the univalence axiom in the simplicial model. Secondly, I will describe how, in analogy with the notion of a univalent fibration, it is possible to define a notion of univalent dependent type, giving some examples

We investigate the problem of optimal topologies of fluid domains. In a given container we search for a topology of a fluid domain, filling a given proportion of the container, such that a functional of the resulting velocity field inside this domain is minimized. Here the velocity owes to the Navier--Stokes system. The problem is handled by both using a porosity approach and a phase field concept. The Navier--Stokes system is solved on the whole domain, where the phase field serves as an indicator function for the two phases, namely the fluid domain (high porosity) and the dense domain (low porosity). The phase field itself is obtained by a gradient flow for a specific inner product. In this talk we sketch the underlying concept and investigate numerically the properties of the overall concept.

Das schlechte Abschneiden deutscher Schülerinnen bei TIMSS-Beweisaufgaben K10 und K18 beruht nach Reiss & Heinze (2004) nicht auf einem Mangel an Faktenwissen - viel mehr mangelt es an der Fähigkeit, Argumente korrekt zu begründen und zu einer Beweiskette zu verknüpfen. Beweisfindung und -darstellung sind Phasen im Boeroschen Beweismodell , die nach Heinze & Reiss (2004) unterrichtlich zu wenig praktiziert und von den Lernern daher nur unzureichend beherrscht werden. Wir betrachten hierzu zwei heuristische Hilfsmittel, die an Schulbuchinhalte Thema Beweisen anknüpfen. Im Schulbuch "Neue Wege" (Lergenmüller & Schmidt (2007) ,S. 72) wird zur Beweisorganisation erläutert, wie man Beweise im Zweispaltenformat aufschreibt, so dass Argumente verkettet und Begründungen ausgewiesen werden können - versehen mit dem Hinweis "Die Beweisfigur und die übersichtliche Darstellung der Beweisschritte stehen meist nicht am Anfang des Beweisens, sie entwickeln sich oft erst nach vielen Ansätzen mit Versuch und Irrtum." Eine über dieses heuristische Basisprogramm hinausgehende Hilfestellung zur Beweisfindung wird indes nicht gegeben. Im Schulbuch "Elemente der Mathematik" (Griesel et al. 2008, S.145) wird die Beweisfindung an Hand der Metapher "Flussüberquerung" als Überbrückung der Kluft von Voraussetzung und Behauptung durch sukzessive Interpolation von Zwischenaussagen dargestellt. Hieran knüpft der Lösungsgraph nach Pólya und König an - er ist ein heuristisches Hilfsmittel, das Beweise in einem ikonisch-symbolischen Format repräsentiert. Stärker als im Zweispaltenbeweis werden so die Gliederung der Argumentation der Zusammenhang, der einzelnen Argumente und der Beweisfluss insgesamt deutlich. Während die lineare Anordnung der Argumente im Zweispaltenformat Reihenfolgeentscheidungen erzwingt, die naturgemäß teilweise willkürlich sind, hebt der relationale Charakter des Graphen stärker die logische Struktur hervor. Der Lösungsgraph eignet sich daher dazu, in der Rückschauphase dem Beweisaufbau zu veranschaulichen und den Nutzen heuristischer Impulse für die Lösungsfindung hervorzuheben. Zudem ist er für den Prozess des Verallgemeinerns nutzbar. Wir konkretisieren unsere Überlegungen am Satz des Thales und seiner Verallgemeinerung zum Umfangswinkelsatz.

• Griesel, H.; Postel, H. & Suhr, F. (Hrsg., 2008): Elemente der Mathematik 8, Braunschweig: Schroedel.
• Heinze, A. & Reiss, K. (2004): The teaching of proof at lower secondary level - a video study. ZDM - International Journal on Mathematics Education 36(3), 98 - 104.
• Lergenmüller, A & Schmidt, G. (2007): Mathematik Neue Wege 8, Arbeitsbuch für Gymnasien Nds., Braunschweig: Westermann
• Reiss, K. & Heinze, A. (2004) Knowledge acquisition in students' argumentation and proof processes. In G. Törner, R. Bruder, N. Neill, A. Peter-Koop & B. Wollring (Eds.), Developments in Mathematics Education in German-Speaking Countries. Selected Papers from the Annual Conference on Didactics of Mathematics, Ludwigsburg 2001 (pp. 107-115). Hildesheim: Franzbecker.

We present an adaptive strategy for solving unsteady compressible flows by a discontinuous Galerkin method. The underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids. This provides information on the difference between successive refinement levels that may become negligibly small in regions where the solution is smooth. Applying thresholding, the data is compressed thereby triggering local grid adaptation. Furthermore, this information is used as an additional indicator for limiting.

In recent years, regularization methods for linear ill-posed problems with sparsity constraints have been discussed widely in literature. One specific approach is regularization with a Besov space penalty, which, under certain conditions, can be described in a simple way using a wavelet basis. Convergence of the solutions has been analysed assuming deterministic worst-case error bounds of the error between the noisy measurements and the true data. In the talk we will exchange this with an explicit stochastic noise model, i.e., allow arbitrarily large measurement errors, but with low probability. We use a specific metric to lift deterministic results into the stochastic setting. We will prove convergence of the solutions with respect to the variance of the error and, using a new parameter choice rule, derive convergence rates. The theoretical results are illustrated in one dimensional and two dimensional examples.

Starting from the Zakharov/Craig-Sulem formulation for the water waves problem with and without surface tension (gravity-capillary and gravity waves, respectively), we are interested in the macroscopic manifestation of the interaction of different weakly amplitude-modulated plane waves of the linearized problem when amplitude, macroscopic space and macroscopic time have the same scaling coefficient. Apart from the formal derivation of the corresponding modulation equations, we present results concerning their justification in the case of purely gravity waves, which are based on recent work of Alvarez-Samaniego and Lannes on the long-time well-posedness of the water waves problem of finite depth.

Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method, in its original form, does not include a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding.

In this talk, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a Lyapunov function. Then we interpolate this function by a continuous piecewise affine (CPA) function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction.

This is joint work with Sigurdur Hafstein (Reykjavik University, Iceland).

The determination of the basin of attraction of an equilibrium or periodic can be achieved by different methods. In this talk, we discuss a local method, which does not require any knowledge of the position of the equilibrium or periodic orbit, using a contraction metric. A contraction metric is a Riemannian metric with a local contraction property. It can be used to prove existence and uniqueness of a periodic orbit or equilibrium and determine a subset of its basin of attraction. We discuss both how contraction metrics can be used to determine the basin of attraction and converse theorems, ensuring the existence of such contraction metrics.

The isothermal Navier-Stokes-Korteweg (NSK) model is a well-known diffuse interface model describing compressible multi-phase flows. Its stability analysis is far from straightforward in case it derives from a non-convex energy density, which is standard in the modelling of multi-phase flows. We explain how the relative entropy method, a classical tool in the stability analysis of compressible fluid flows, can be modified as to provide a stability estimate for the NSK model with non-convex energy density.

We also outline how this result can be used to show convergence of solutions of a lower order model to solutions of the NSK model in some limit. Our interest in this limit arises from numerical considerations as solutions of the lower order model can be approximated much easier than solutions of the NSK model.

The word problem and word search problem for finitely presented groups are the basis of several proposed cryptosystems. In this talk we discuss the feasibility of sampling hard instances of these problems for given groups.

Based on the relation between kinetic theory and non-linear hyperbolic equations, we derive a general kinetic-induced infinite moment system for a spatially one-dimensional non-linear balance law. The derivation is based on an artificial Boltzmann-like transport equation with a BGK-relaxation. Using Chapman-Enskog-like asymptotic expansion techniques, it will be shown that at each order of \(\varepsilon\), a scale-induced closure is possible, which results in a finite number of moment equations.

The obtained result is then applied to the one dimensional Burgers' equation in order to study its numerical and spectral properties.

As is well known, Einstein's field equations of General Relativity impose no topological obstruction on the Cauchy hypersurface. Hence there is generally some topological freedom in modelling initial data corresponding to a specific physical situation, like, e.g., a collection of black holes momentarily at rest. This gives rise to interesting topological considerations connected with the physically motivated question concerning the structure of configuration spaces in General Relativity. A simple 2-hole example serves to illustrate the somewhat surprising richness of structure and allows to speculate about its possible physical implications.

In this talk I will consider various constructions of categorical models of dependent type theory, and look at how the fibrations involved interact with the structure required for identity types and enrichment.

We show in this talk that the symplectic group is a maximal symmetry group for covariance properties of Weyl pseudodifferential operators. We thereafter address the case of Shubin and Jordan operators, and show that the corresponding pseudodifferential calculi are covariant under the action of certain subgroups of the symplectic group. We relate these symmetry properties to those of the Wigner transform in the Weyl case, and to the Cohen class in the more general case.

Every upward closed subset A in a wqo has a finite basis. Computing such a finite basis is a basic step in the verification of so-called well-structured transition systems. Generalizing a lemma of Valk and Jantzen (1985), we show that, in wqos with the so-called effective complement property, that question reduces to the simpler question of the decidability whether A meets a given order ideal. The proof is elementary. The poset of ideals itself is a completion of the original wqo, and we show that it has concrete, computable presentations in many interesting cases.

We study isolated singularities of a space embedded in a smooth Riemannian manifold from a differential geometric point of view. While there is a considerable literature on bi-lipschitz invariants of singularities, we obtain a more precise (complete asymptotic) understanding of the metric properties of certain types of singularities. This involves the study of the family of geodesics emanating from the singular point. While for conical singularities this family of geodesics, and the exponential map defined by them, behaves much like in the smooth case, the situation is very different in the case of cuspidal singularities, where the exponential map may even fail to be locally injective. We also study a mixed conical-cuspidal case. Our methods involve the description of the geodesic flow as a Hamiltonian system and its resolution by blow-ups in phase space.

In den Bildungsstandards im Fach Mathematik für die allgemeine Hochschulreife gibt es fachspezifische Hinweise zur Gestaltung der schriftlichen Prüfungsaufgabe im Fach Mathematik. Auf dieser Grundlage arbeiten die Länder am Aufbau eines gemeinsamen Pools von Abiturprüfungsaufgaben für das Fach Mathematik ab dem Jahr 2017. Dieser Aufgabenpool soll Qualität und Vielfalt von Prüfungsaufgaben in den Ländern sowie gleichzeitig die notwendige Vergleichbarkeit sichern. Im Vortrag werden Qualitätskriterien für die Erstellung von Prüfungsaufgaben auf der Basis der Bildungsstandards vorgestellt und mit Hilfe von Beispielaufgaben erläutert. Dabei wird auch der Einsatz digitaler Mathematikwerkzeuge berücksichtigt.

Software is an emerging field of mathematical research and knowledge. The Open Access database wMATH containing information about nearly 10.000 software packages is one of the most comprehensive information services on mathematical software. Its unique feature is the linking of software with publications which describe or apply the software. The publication-based approach allows to create and update information about the content and other features of mathematical software in an efficient semi-automatic way. Therefor, the information of the database swMATH are analysed systematically. swMATH is a project of the German research campus MODAL and will be provided by FIZ Karlsruhe. The talk gives an overview of nthe state of the art and planned developments of the swMATH service.

We suggest a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymer chains. The polymer chains are modeled by dumbbells subjected to generic elastic spring-force potentials. Their density and orientation are described by a Fokker-Planck-type equation which is coupled to a Cahn-Hilliard and a momentum equation for phase-field and gross velocity/pressure. Henry-type energy functionals are used to describe different solubility properties of the polymers in the different phases or at the liquid-liquid interface.

Taking advantage of the underlying energetic/entropic structure of the system, we prove existence of a weak solution globally in time in the case of FENE-potentials. We discuss extensions of the model to take the interaction between polymer and fluid interface orientation into account ("amphiphilic surfactant") .

Finally, as a by-product of our general modeling approach, we suggest a two-phase visco-elastic model of Oldroyd-B-type.

This is based on a joint work with S. Metzger.

For a bounded smooth domain \(\Omega\subset \mathbb{R}^2 \) and a smooth boundary datum \(\varphi:\overline{\Omega}\to \mathbb{R}\) we consider the minimisation of the Willmore functional \[ W(u) := \frac{1}{4} \int_{\Omega} H^2 \; \sqrt{1+ | \nabla u |^2} \, dx \] for graphs \(u: \overline{\Omega}\to \mathbb{R}\) with mean curvature \(H:=\operatorname{div}\left(\frac{\nabla u }{ \sqrt{1+ | \nabla u |^2}}\right)\) subject to Dirichlet boundary conditions, i.e. in the class \[ \mathcal{M}:=\{u\in H^2 (\Omega): (u-\varphi) \in H^2 _0(\Omega)\}. \] Making use of a celebrated result by L. Simon [3] we first show that in this class, bounds for the Willmore energy imply area and diameter bounds. Examples show that stronger bounds in terms of the Willmore energy are not available. This means that \(L^\infty\cap BV(\Omega)\) is the natural solution class where, however, the original Willmore functional is not defined. So, we need to consider its \(L^1 \)-lower semicontinuous relaxation. Our main result states that this relaxation coincides on \(\mathcal{M}\) with the original Willmorefunctional so that the relaxed functional is indeed its largest possible \(L^1\)-lower semicontinuous extension to \(BV(\Omega)\). Moreover, finiteness of the relaxed energy encodes attainment of the Dirichlet boundary conditions in a suitable sense. Finally, weobtain theexistence of a minimiser in \(L^\infty\cap BV(\Omega)\) for the relaxed/extended energy. The major benefit of our non-parametric approach is the validity of a-priori diameter and area bounds, which are not available in the general setting of R. Schätzle's work [2]. On the other hand we need to leave open most of the regularity issues.

(This joint work with Klaus Deckelnick (Magdeburg) and Matthias Röger (TU Dortmund).)

1. Klaus Deckelnick, Hans-Christoph Grunau, Matthias Röger, Minimising a relaxed Willmore functional for graphs subject toboundary conditions, Preprint 2015, arxiv:1503.01275.
2. Reiner Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37, 275--302,2010.
3. Leon Simon,Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1, 281--326, 1993.

Simplicial triangulations of \(\mathbb{R}^n\) have a wide range of applications. Of particular interest to us, is using a triangulation to facilitate the introduction of so-called Continuous- and Piecewise-Affine (CPA) functions for their use in the numerical generation of Lyapunov functions for dynamical systems.

Using a standardized triangulation with the \(\mathbb{Z}^n\) lattice its vertices, we consider a class of transformations which distribute the vertices in a desired way, maintaining certain symmetry properties around the origin. A new family of simplices is generated as follows: For each simplex in the original triangulation, a new convex combination of the mapped vertices is generated. We are interested in examining when this new family of simplices represents a proper simplicial triangulation. Furthermore, we prove that not only is the new triangulation proper for our transformation, but its simplices also remain non-degenerate in a sense which is particularly meaningful for our application.

The usage of topological and geometrical concepts in signal and data analysis has seen multiple developments in the last few years. In this talk, we present an overview of some selected topics in these fields. We begin with a basic review of ideas in signal processing using frame theory as a main tool that generalizes time-frequency analysis and wavelets transforms. We explain basic concepts on manifold learning and dimensionality reduction as modern tools for data analysis. A related topic is persistent homology, which provides new analysis strategies using concepts from homology. Noncommutative algebras provides a powerful machinery integrating topological and algebraic constructions. We illustrate the application of these tools using examples from both audio signal processing and image analysis.

How easy is it to generate a naked singularity from analytic initial data in general relativity with well-behaved matter (such as a scalar field or perfect fluid)? The surprise answer is that one only needs to fine-tune any one generic parameter of the initial data to the collapse threshold. In this sense, naked singularities are codimension-1 generic. The underlying theory involves self-similar spacetimes that have only one growing perturbation, and dynamical systems theory. Open questions concern vacuum relativity and the role of angular momentum.

Das Thema Aufmerksamkeit von in Mathematik-Lehrveranstaltungen ist deswegen besonders wichtig, weil Lernende leicht den Anschluss verlieren können und dann darauffolgendes Lehrmaterial nicht verstehen. Allerdings ist noch zu wenig erforscht, in welchem Umfang und zu welchen Zeitpunkten Lernende mehr bzw. weniger aufmerksam sind. Entsprechende wissenschaftliche Untersuchungen gibt es in der Literatur zwar seit Jahrzehnten, doch erstens sind die Ergebnisse in der Regel nicht mathematikspezifisch und zweitens ziehen neue wissenschaftliche Erkenntnisse einige der älteren Ergebnisse in Zweifel.

In diesem Vortrag werden erstens theoretische Grundlagen und historisch wichtige Studien zu Aufmerksamkeit von Studierenden in Hochschulvorlesungen vorgestellt, verglichen und diskutiert. Zweitens werden Methoden diskutiert, wie Aufmerksamkeit speziell in Mathematik-Lehrveranstaltungen untersucht werden kann. Erste Ergebnisse eines speziell dafür geschaffenen Untersuchungsinstruments (eingesetzt in Hochschulvorlesungen) werden vorgestellt und das Instrument selbst kritisch beleuchtet.

We consider continuous and discrete reaction-diffusion equations with hysteresis which is given at every spatial point. Such equations arise when one describes hysteretic interaction between several diffusive and nondiffu- sive substances. In the talk, we will discuss mechanisms of appearing global and local spatio-temporal patterns due to hysteresis as well as their intercon- nection. This is a joint work with Sergey Tikhomirov (Max Planck Institute for Mathematics in the Science, Leipzig, Germany).

In the talk we present CAP, which is a realization of categorical programming written in GAP. CAP makes it possible to compute complicated mathematical structures, e.g., spectral sequences. This can be achieved using only a small set of basic algorithms given by the existential quantifiers of ABELian categories, e.g., composition, kernel, direct sum. In the talk we will explain the concepts of categorical programming and give a demonstration of the functionalities of CAP.

In their breakthrough paper, Knutson and Tao proved the saturation conjecture about the representation theory of \(\mathrm{GL}_n(\mathbb{C})\): for partitions \(\lambda, \mu, \nu\) and an integer \(N\) the irreducible representation \(V_\nu\) occurs as a subrepresentation of \(V_\lambda \otimes V_\mu\) if \(V_{N\nu}\) occurs inside \(V_{N\lambda} \otimes V_{N\mu}\).

They reformulated the conjecture in terms of the hives polytope \(H(\lambda,\mu,\nu)\), and show that \(H(\lambda,\mu,\nu)\) is non-empty only if it contains a point with all coordinates integral.

De Loera and McAllister observe that this would follow if the defining matrix of \(H(\lambda,\mu,\nu)\), considered as a vector configuration, had a unimodular cover. Based on computer experiments they conjecture that this homogenized hives matrix even has a unimodular triangulation.

In this talk, I will argue that the original proof by Knutson and Tao already implies the existence of a regular unimodular triangulation.

Mit einem neuen Förderprogramm stellt die DFG im Zeitraum 2013-2015 das System der Sondersammelgebiete auf "Fachinformationsdienste für die Wissenschaft" um. Seit Januar 2015 betreiben die SUB Göttingen und die TIB Hannover den "Fachinformationsdienst Mathematik", welcher die bisherigen Sondersammelgebiete "Reine Mathematik" an der SUB Göttingen und "Angewandte Mathematik" an der TIB Hannover abgelöst hat. In diesem Vortrag soll erläutert werden, mit welchem Konzept beide Bibliotheken gemeinsamen an die Ausgestaltung des neuen Fachinformationsdienstes für die Mathematik gehen.

The Principle of Perturbative Agreement, as introduced by Hollands & Wald, is a renormalisation condition in quantum field theory on curved spacetimes. This principle states that the perturbative and exact constructions of a field theoretic model given by the sum of a free and an exactly tractable interaction Lagrangean should agree. We develop an alternative proof of the validity of this principle in the case of scalar fields and quadratic interactions without derivatives. Afterwards, we prove a generalisation of the Principle of Perturbative Agreement and show that considering an arbitrary quadratic contribution of a general interaction either as part of the free theory or as part of the perturbation gives equivalent results. Motivated by the thermal mass idea, we use our findings in order to extend the construction of massive interacting thermal equilibrium states in Minkowski spacetime developed by Fredenhagen & Lindner to the massless case.

When an ordinary differential equation is diffusively coupled to a zero vector field (doubling the space dimension) then a dynamical system with quiescent phases is obtained, similarly for maps coupled to the identity. The question how the dynamics of the coupled system is related to that of the original system has been partially answered. Such results can be extended to delay equations.

There are several ways to connect quiescent phases to vector-valued delay equations.

First, one can see a delay equation as a retarded ordinary differential equation and couple it to zero as in the o.d.e. case. Secondly, one can look at a delay equation as a dynamical system in some function space (space of histories) and do the same thing there - with a different result. Thirdly, one can start from a system with quiescent phases with distributed exit times and get a system of delay equations. In this case the delay is the length of the quiescent phase. Finally, one can consider a Gurtin-MacCamy population model with quiescent phases and a maturation period. Then one gets the same result as in the first approach and the delay is the maturation period.

In all these ways one arrives at systems of vector-valued delay equations - of different types and with different stability conditions - which show similar behavior. If the rates (of going quiescent and returning to the active phase) are the same for all dependent variables, then quiescent phases act stabilizing (against Hopf bifurcations). For unequal rates there may be excitation phenomena unless the Jacobian matrix of the underlying o.d.e. system has some strong stability properties. For two dependent variables exact algebraic conditions for strong stability can be given.

Our database represents basic biographic informatio n and scientific contributions of more than 100 statisticians and mathematicians. It cover s the period from the 16th century until current time and the number of biographies is const antly growing, whereas the information on scientific contributions is updated regularly. Diff erent criteria and principles for database creation are explained, and applications in teachin g are demonstrated. The current BBI interface allows the connection between different s cientists as well as their contributions in a modern style suitable for teaching purposes.

After the introduction of random operators to nuclear physics by Eugene Wigner in 1955, random quantum systems have grown in popularity. Wigner's idea was to consider families of Hamiltonians that underlie a certain probability distribution to describe overly complicated systems. Of particular interest are, of course, the spectra of these Hamiltonians. In this talk we consider random, in general non-self-adjoint, tridiagonal operators on the Hilbert space of square-summable sequences, which can be used to describe quantum particles on a lattice. In particular, we are interested in the so-called Feinberg-Zee random hopping matrix, that, despite its simple appearance, seems to have a very complicated spectrum.

Input-to-state-stability is a measure of robustness of the stability of an attractor in dynamical systems. Recently, the CPA method to compute Lyapunov functions for nonlinear systems using linear programming was adapted to computing ISS Lyapunov functions. We will discuss this new algorithm.

Lyapunov functions give important information on the basin of attraction and robustness of attractors. Their generation for nonlinear systems is, however, a difficult task. In the talk a generally applicable method to compute continuous and piecewise affine Lyapunov functions for nonlinear systems via linear programming is described.

Es geht in diesem Vortrag um die "vorwärts treibende Kraft", mit der bewegliche Darstellungen der DMS (Dynamischen Mathematik-Systeme) das Verstehen von Mathematik fördern. Damit muss auch die Lehre von Mathematik die allzu statische Sicht überwinden und wirklich "vorankommen", in eine gute Zukunft gehen. In drei Themen sollen Beispiele diesen Mehrwert von "Dynamik" zeigen. Zu einem frei beweglichen Polynom entsteht als Ortslinie die Ableitung, die dann aber auf dynamische Änderungen des Polynoms zwangläufig in der ihr eigenen Weise antworten muss. Wer die Antwort voraussagen kann, hat etwas verstanden. Zwei weitere Beispiele aus der Analysis zeigen eine dynamische Hinführung zur e-Funktion und zum "Hauptsatz". Mit der polar-kartesisch-gekoppelten Darstellung wird mit zwei synchronen Graphikfenstern ein vertieftes Verständnis von Polarkoordinaten vorgestellt. Das Problem des Durchlaufs eines Kurvenpunktes kann durch vergleichendes Argumentieren gelöst werden. Auf geometrische Weise wird die Reflexion achsenparalleler Strahlen an einer Parabel realisiert. Eine verblüffende Dynamisierung lässt dann die Leitgerade "aus dem Nichts" erscheinen. Die dynamische Betrachtung visualisiert nicht nur schon vorhandene mathematische Aussagen, sondern bringt neue Erkenntnisse - bei den Lernenden - hervor. Darum muss die Mathematiklehre sich nun auch selbst bewegen.

We discuss the connections between the cycle space and the cut space of transitive graphs. In particular, we will see that the cut space of a transitive graph \(G\) is a finitely generated \(\mathrm{Aut}(G)\)-module as soon as the same holds for the cycle space.

In addition, we discuss accessibility in transitive locally finite graphs: when does there exist some positive integer n such that any two ends can be separated by removing at most n vertices? We use our previously mentioned result to see that this is the case if the cycle space is generated by cycles of bounded length. It turns out that this condition on the cycle space is satisfied by various natural classes of graphs.

Supermanifolds are generalisations of manifolds, whose algebras of functions also contain anticommuting elements. They have been applied, e.g., in index theory or classical field theory and hence, it is interesting from a geometric and physical point of view, to study variational problems and PDEs on these spaces.

We will first describe an approach to mapping spaces that allows for a satisfactory construction of a variational calculus on supermanifolds. To study spaces of solutions, one may either reduce the problem to PDEs on a smooth manifold ("component decomposition") or generalise techniques from analysis in order to obtain an intrinsic PDE-theory on supermanifolds. We will discuss (closed) supergeodesics to illustrate the first strategy and present simple, explicit examples which highlight the influence of the underlying geometry on the associated spaces of solutions. We will finally look at hyperbolic equations and indicate, that some analytic methods (e.g. energy methods) can be carried over to supermanifolds. In good cases, such tools yield well-behaved (infinite-dim.) solution spaces which can be used as phase spaces in fermionic classical field theory (joint work with I. Khavkine).

We will demonstrate a new computer algebra package written in the programming language Julia.

Nemo is designed to provide generic algorithms for a variety of different generic rings which are constructed over base rings provided by various computer algebra libraries, such as Singular, Flint, Antic and others.

We will discuss what is currently available in Nemo, compare Nemo with various other systems and outline our current plans for the future of Nemo.

After the discovery of quantum groups by Drinfeld and Jimbo in the 1980ies, the study of Hopf algebras and tensor categories became a quickly developing field in pure mathematics. The structure of pointed Hopf algebras appeared to be closely related to those in Lie theory, a fact which motivated N. Andruskiewitsch and H.-J. Schneider around 1998 to initiate a powerful method to classify pointed Hopf algebras [AS98]. The basic object in this program is the Nichols algebra of a braided vector space (or a Yetter-Drinfeld module).

In this talk, we introduce a new method to determine all Nichols algebras of diagonal type over arbitrary fields.

Es gibt verschiedene fachliche Perspektiven, aus denen ein mathematischer Unterrichtsgegenstand betrachtet werden kann. Dazu gehören die Schulmathematik vom höheren Standpunkt im Sinne Felix Kleins, die didaktische Phänomenologie mathematischer Strukturen im Sinne Freudenthals und die epistemologische Detailanalyse. Diese Sichtweisen sind je für sich wichtig und erst ihr Zusammenspiel erzeugt das rechte Hintergrundwissen für den Unterricht. Der Vortrag möchte diese Auffassung an Beispielen entfalten.

We discuss a new approach to the derivation of sharp interface models for fluid-fluid interactions. The resulting models also describe the interaction between the fluid-fluid interface and a solid surface. This leads to a new perspective on the Dussan-Davis experiment and to Huh's and Scriven's paradox.

Lévy semistationary processes of the form \[X_t=\int_{-\infty}^t g(t-s)b_s dL_s,\] where \(g\) is a deterministic kernel and \(b\) is predictable, have been proposed for modeling the velocity in a turbulent flow in 2005. Since then, various properties of these processes have been successfully studied, amongst others the limiting behavior of the power variation \[V(p)_n=\sum_{i=1}^n |X_{i/n}-X_{(i-1)/n}|^p,\] for \(n\to \infty\) for the case where \(L\) is a Brownian motion. We will now present a limit theory for the case where the integrator is a pure jump Lévy process, leading to some surprising results.

A number of higher-order Newton's method have been suggested in the last decade. All confirm the conjecture by Kung-Traub from 1974 according to which an optimal iterative method based on \(n + 1\) evaluations may achieve a maximum convergence order of \(2n\). We derive many well-known methods by means of an alternative convergence theory. This theory aims to identify the secant that connects initial guess and root. The existence of this secant is provided by the mean value theorem. The classical Newton's method serves as an auxiliary step defining a success parameter similar to the trust region in optimization. The slope of the corresponding secant is approximated as a power series in this success parameter. It is demonstrated that roots of quadratic and asymptotically even cubic functions may be computed in arbitrary accuracy.

The Bianchi IX cosmological model is homogeneous but anisotropic. The dynamics on the Ringström's attractor is a network of heteroclinic orbits that supports the Belinsky-Khalatnikov-Lifschitz conjecture: a universe tumbling from one Kasner state to the next in a specific chaotic manner. We introduce a parameter in this model, whose variation from the critical value corresponding to Bianchi IX changes the dynamics dramatically. Below the critical value, heteroclinic chains generically end up after finally many iterations, while chaos of BKL type survives only on a fractal Cantor set of measure zero. Above the critical value, chaos is in some sense generic, but not of BKL-type because the concept of era loses its meaning. We will give interpretations of the parameter introduced and explain how the methods of symbolic dynamics used here can be applied also for FLRW models with scalar fields.

The MAPKinase cascade is part of a signaling network in many organisms. In each layer of the cascade, a protein is phosphorylated or dephosphorylated via enzymatic reactions. In other words, phosphate groups are attached to or detached from the protein. A single layer of the cascade is called a (multiple) futile cycle. The fully phosphorylated form of the protein is the enzyme for the phosphorylation on the layer below. We show that the dual futile cycle shows bistability - a feature that explains a property called "good switch" by biologists. Furthermore we show that oscillations appear in a cascade of at least two layers. The results are based on bifurcation theory and geometric singular perturbation theory. This talk reports about joint work with Alan Rendall.

High-order WENO (i.e., weighted essentially non-oscillatory) methods are widely used for the approximation of hyperbolic partial differential equations. A common approach to use WENO methods on multidimensional Cartesian grids consists in applying a one-dimensional WENO method in each direction. This spatial discretization is typically combined with a Runge-Kutta method in time, i.e., during each stage of a Runge-Kutta method one-dimensional WENO schemes are used in a dimension-by-dimension fashion.

However, it is known that finite volume WENO methods based on a dimension-by-dimension approach retain the full order of accuracy (of the one-dimensional method) for linear multidimensional problems, but they are only second order accurate for the approximation of nonlinear multidimensional problems.

In my talk, I will present a simple modification of finite volume WENO methods, which leads to the full spatial order of accuracy by using only one-dimensional polynomial reconstructions in a dimension-by-dimension approach.

Furthermore, I will discuss the use of this method on adaptively refined grids.

This is recent joint work with Pawel Buchmüller and Jürgen Dreher.

In this talk we will review some new inequalities obtained for the elastic energy defined, for any regular closed curve \(\gamma\) in the plane by \(E( \gamma)=\frac{1}{2} \int_\gamma C^2 \,ds\) where \(C\) is the curvature and \(s\) the curvilinear abscissa. We denote by \(\Omega\) the bounded domain whose boundary is \(\gamma\). We prove, in particular, that the disk minimizes \(E(\gamma)\) among sets of given area. On the contrary, if we consider a constraint on the inradius, the disk is no longer the minimizer. We also consider a Blaschke-Santalò diagram for convex domains involving area, inradius and elastic energy.

The Positive Mass Conjecture for asymptotically Euclidean manifolds has been proved in some special cases (e.g., for manifolds of dimension at most 7 or for spin manifolds) but the general case is still subject to current research. In this talk we present a surgery result might help to give a proof in the general case. This is joint work with Emmanuel Humbert (Université de Tours, France).

One of the most important cryptographic assumptions is arguably the Decisional Diffie-Hellman Assumption, asking to tell whether a group element is of the form \(g^{ab}\), given \(g,g^a,g^b\) in some cyclic group. This assumption does not to hold in groups that allow a symmetric pairing, so we need to generalize and replace DDH by other similar assumptions if we want to enjoy the functionalities that groups with such pairing offer. In general, the type of assumption that we consider is of the form that some solving a elementary problem from linear algebra like teling whether a vector \(v\) is in a subspace described as the image of a matrix, is infeasible if we are only given the inputs "in the exponent". Security analysis in an appropriate model boils down to the analysis of generic polynomial relations between the exponents. For the case of membership in the image of a matrix, the ideal formed by these relations is typically generated by determinants, which allows for an extremely analysis. Additionally, taking this algebraic point of view allows to interpret the vectors \(v\) themselves as polynomials, giving additional structure that leads to some interesting applications, efficiency improvements allows to gain new insights into existing constructions.

Many dynamical patterns in Hamiltonian lattices can be described by (nonlinear) traveling waves. In this talk we present a refined asymptotic analysis for the high-energy limit of such waves.

(joint work with Karsten Matthies, University of Bath).

We address the solution of nonlinear optimal control problems by sequential quadratic programming (SQP) methods in function space. A composite-step trust-region framwork is employed for globalization. In each substep, a quadratic programming problem needs to be solved, possibly subject to a trust-region constraint. We discuss the efficient solution of those problems by tailored preconditioned Krylov subspace methods in function space. Numerical results will be included.

We want to investigate when an infinite graph has a Hamilton cycle. To overcome the problem what an infinite cycle should be, we use a definition which depends not just on the graph itself but on a topological space consisting of the graph together with its ends. To be more precise, we look at the Freudenthal compactification of the graph. This enables us to extend theorems about the existence of Hamilton cycles in finite graphs to locally finite graphs. In particular we extend a theorem of Oberly and Sumner and, partially, a theorem of Asratian and Khachatrian to locally finite graphs.

In the framework of the German-Indonesian Tsunami Early Warning System, the tsunami modeling group at AWI developped the simulation code TsunAWI which discretises the non-linear shallow water equations on an unstructured finite element mesh. A modular extension including a non-hydrostatic correction of the pressure term can be invoked to improve the simulation e.g. in regions with steep bathymetry. However, the non-hydrostatic pressure term requires the solution of a large sparse system of linear equations in each time step. In this talk, we investigate several numerical solver techniques e.g., sequential and parallel preconditioning methods applied to the Krylov subspace method FGMRES(m), domain decomposition techniques, and resorting algorithms. An emphasis will be put on the parallel Schur and restrictive additive Schwarz preconditioner that proved to provide very good convergence and computational efficiency for non-hydrostatic TsunAWI as well as for the sparse linear system arising in the ocean model FESOM. The correspondend solver components were also implemented in the framework of the pARMS solver library.

We present a simple criterium which allows to decide whether a semilinear optimal control problem admits a unique global solution. For the discretized problem this criterium can be checked exactly. We present several numerical examples with unique global solutions.

This is joint work with Ahmad Ali (Hamburg) and Klaus Deckelnick (Magdeburg).

In Free Material Optimization, the design variable is the full material tensor of an elastic body. Written in matrix notation one obtains a control-in-the-coefficients problem for the material tensor. In this talk we discuss recent results in the finite element analysis in Free Material Optimization. We employ the variational discretization approach, where the control, i.e., the material tensor, is only implicitly discretized. Using techniques from the identification of matrix-valued diffusion coefficients, we derive error estimates depending on the coupling of the discretization and Tikhonov regularization parameters. Furthermore, this approach allows to also take into account a noise level on the measured data. Numerical examples supplement our analytical findings.

We consider a strictly stationary and ergodic sequence of random elements \((X_t)\) taking values in some Hilbert space. Such a setting is broad enough to cover most practically relevant functional time series models. Our target is then to study the weak convergence of the discrete functional Fourier transforms of the observations under sharp conditions. As an application we discuss detection of a possibly periodic mean curve of the time series. The talk is based on joint work Clément Cerovecki (ULB).

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.

Back in 1994 Thomas Streicher and myself discovered the groupoid interpretation of Martin-Löf's type theory which is now seen as a precursor of Homotopy Type Theory and in fact anticipated some simple cases of important ideas of Homotopy Type Theory, notably a special case of the univalence axiom. I will explain how and why we found the groupoid interpretation, our motivations and results. I will also present some less well-known results about *extensional* Martin-Löf type theory and speculate how this might relate to homotopy type theory.

I will review some recent progress in the black hole stability problem including a proof of the linear stability of the Schwarzschild spacetime under gravitational perturbations (joint work with Dafermos and Rodnianski).

One of the challenges in analysis of geophysical systems is to learn about the causality relations in the considered systems on a certain level of resolution - and to distinguish between the true causality from simple statistical correlations.

Proper inference of such causality relations, besides giving an additional insight into such processes, can allow improving the respective mathematical and computational models. However, inferring such relations directly from geophysical equations/models is hampered by the multiscale character of the underlying processes and the presence of unresolved/sub-grid scales.

Implications of missing/unresolved scales for this problem will be discussed and an overview of methods for data-driven causality inference will be given. Recently-introduced data-driven multiscale causality inference framework for discrete/Boolean data will be explained and illustrated on analysis of historical climate teleconnection series and on inference of their mutual influences on monthly scale.

The talk will be based on the recently published paper

S. Gerber and I. Horenko. "On inference of causality for discrete state models in a multiscale context", Proceedings of the National Academy of Sciences of USA (PNAS), 111 (41), 14651-14656, 2014.

Sei \(G\) eine endliche auflüsbare Gruppe und \(\nu_0(G):=F(G)\) die Fitting-Untergruppe von \(G\), also der maximale nilpotenten Normalteiler von \(G\). Für \(i\geq 0\) sei \(\nu_{i+1}(G)\) der kleinste Normalteiler von \(F(G)\), so dass \(\nu_i(G)/\nu_{i+1}(G)\) ein direktes Produkt elementar-abelscher Gruppen ist, welche von \(F(G)\) zentralisiert werden. Dann ist \(G\geq \nu_0(G) \geq \nu_1(G) \dots\) die \(F\)-zentral-Reihe von \(G\). Die Lünge \(c\) dieser Reihe ist die \(F\)-Klasse von \(G\). Ein \(F\)-Nachfolger von \(G\) ist eine Gruppe \(H\), so dass \(H/\nu_c(H)\cong G\) und \(\nu_{c}(H)\neq 1=\nu_{c+1}(H)\) gelten.

Man kann nun die Isomorphietypen endlicher auflüsbarer Gruppen einer festen Ordnung \(N\in\mathbb{N}\) algorithmisch klassifizieren, in dem man sukzessive \(F\)-Nachfolger einiger geeigneter (bekannter) endlicher auflüsbarere Gruppen bestimmt, wobei die Ordnungen der Ausgangsgruppe alle kleiner \(N\) sind (siehe ``The construction of finite solvable groups revisited'' (B. Eick und M. Horn, J. Algebra 408, 2014).

Hierbei ist ein Schlüsselschritt die Bestimmung von \(F\)-Überlagerungsgruppen von \(G\); dies sind Gruppen mit u.~A. der Eigenschaft, dass jeder Nachfolger von \(G\) ein Quotient von \(K\) ist. Wir stellen einen neuen Algorithmus hierfür vor, welcher wesentlich effizienter arbeitet als der bisher eingesetzte naive Algorithmus.

We present a type theory in which the user can directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on a model of type theory in cubical sets with connections. The identity type is defined as a type of paths and all properties of intensional equality are provable, with the usual definitional equalities. The fact that the user can directly manipulate n-dimensional cubes enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, the system also supports transforming any isomorphism into an equality and some higher inductive types like the circle and suspensions.

This is joint work with Cyril Cohen, Thierry Coquand, and Anders Mörtberg.

Stable fiber bundles are the nonautonomous analog of stable manifolds and these objects provide valuable information on the underlying dynamics. We propose an algorithm for their approximation that is based on computing zero contours of a particular operator.

The resulting program applies to a wide class of models, including noninvertible and nonautonomous discrete time systems. Precise error estimates are provided and fiber bundles are computed for several examples.

Maximality Principle (MP) states that for any first-order sentence \(\phi\) in the language of set theory, if it is forced by a set forcing that \(\phi\) is true in any further set generic extension, then \(\phi\) must be true. MP was proposed by Shanon and its basic theory was developed by Hamkins. In this talk, we will discuss several variants of maximality principles and their relations with forcing axioms, bounded forcing axioms, and large cardinals. This is joint work with Nam Trang.

Dynamical systems of the reaction-diffusion type with small noise have been instrumental to explain basic features of the dynamics of paleo-climate data. For instance, a spectral analysis of Greenland ice time series performed at the end of the 1990s representing average temperatures during the last ice age suggest an \(\alpha-\)stable noise component with an \(\alpha\sim 1.75.\) On the other hand, strong memory effects in the dynamics of global average temperatures are attributed to the global cryosphere. We model the time series as a dynamical system perturbed by \(\alpha\)-stable and fractional Gaussian noise, and develop an efficient testing method for the best fitting \(\alpha\) resp. Hurst coefficient. The method is based on the observed power variations of the residuals of the time series. Their asymptotic behavior in case of \(\alpha\)-stable noise is described by \(\frac{\alpha}{p}\)-stable processes, while in the fractional Gaussian case normal asymptotic behavior is observed for suitably renormalized approximations of the quadratic variation.

(joint work with J. Gairing, C. Hein, C. Tudor)

This talk aims to introduce an infinite dimensional version of the classical moment problem, namely the full moment problem on nuclear spaces, and to explore certain instances of this problem. Given a nuclear space \(X\), the question addressed is whether an infinite sequence of functions \(m_n\), s.t. each \(m_n\) is an element of the \(n\)-th symmetric tensor product of the topological dual \(X'\), is actually the sequence of moment functions of a finite non-negative Borel measure supported on a given subset \(K\) of \(X'\).

I present a recent joint work with Tobias Kuna and Aldo Rota about the case in which \(K\) is a generic closed basic semi-algebraic subset of the space of generalized functions on \(\mathbb{R}^d\). Our approach combines a classical result about the analogue of the Hamburger moment problem on nuclear spaces with some techniques recently developed for the moment problem on basic semi-algebraic sets of \(\mathbb{R}^d\). In this way, we derive a complete characterization of the support \(K\) of the realizing measure in terms of its moment functions. As concrete examples, I show how to apply our theorem to the set of all Radon measures, the set of all sub-probabilities, the set of all simple point configurations.

Starting from this result, I will also sketch some new directions that I am currently investigating in relation to the infinite dimensional moment problem.

A non-empty intersection of closed balls of unit radius in a finite-dimensional normed space is called a ball convex body. We discuss representations of ball convex bodies that can be seen as analogues of representations of classical convex bodies from inside (unions of simplices) and outside (intersections of half-spaces). The situation turns out to be more convenient if the underlying norm is strictly convex.

An application concerns the representation of diametrically maximal bodies. These are sets whose diameters increase as soon as one adds at least one point.

Model Theory studies the interplay of combinatorial properties of first-order theories and structural (often: algebraic) properties of their models. A particularly 'nice' class of theories are stable theories, and a well-known open conjecture in Model Theory is that every infinite field whose theory is stable (in the language of rings) is already separably closed. The stable fields conjecture is strongly related to a conjecture by Shelah on fields with a so-called NIP theory. NIP is a property of a theory implied by stability and which is currently extensively studied in the context of 'neo-stability'.

In this talk, I will explain a theorem (proven in joint work with Pierre Simon and Erik Walsberg) which asserts a special case of the Shelah conjecture.

Model Theory studies the interplay of combinatorial properties of first-order theories and structural (often: algebraic) properties of their models. A particularly 'nice' class of theories are stable theories, and a well-known open conjecture in Model Theory is that every infinite field whose theory is stable (in the language of rings) is already separably closed. The stable fields conjecture is strongly related to a conjecture by Shelah on fields with a so-called NIP theory. NIP is a property of a theory implied by stability and which is currently extensively studied in the context of 'neo-stability'.

In this talk, I will explain a theorem (proven in joint work with Pierre Simon and Erik Walsberg) which asserts a special case of the Shelah conjecture.

CATO ist eine in Java geschriebene Oberfläche, die die Eingaben für verschiedene CAS erleichtert und demzufolge in Mathematikvorlesungen für Ingenieure an Fachhochschule eingesetzt wird. Sie ist intuitiv benutzbar und lenkt daher beim unterstützendem Einsatz von der Mathematik nicht ab. Der Autor hat während der Verwendung von CATO mit einem CAS die Rückmeldungen und Verbesserungsvorschläge der Studenten gesammelt und sie in der Version 1.2 umgesetzt. Sie ließen sich problemlos in die Konzepte von CATO integrieren: Sei es das Zusammenstellen von häufig verwendeten Befehlen zu eigenen Paketen, verbesserte Hinweise bei der Führung zur Befehlsauswahl oder die Erklärung eigener Umschreibungen für einparametrige Befehle. Der Autor wird in seinem Vortrag CATO vorführen, die verschiedenen Prinzipien ansprechen und die Neuerungen erläutern. Selbstverständlich gilt für CATO immer noch, bei der Befehlseingabe ist es für den Benutzer nicht ersichtlich, welches CA-System er anspricht, die Eingabe bei CATO ist immer unabhängig von dem jeweils angebundenem System.

In the article "An Implicitization Challenge for Binary Factor Analysis" Tobis, Cueto and Yu computed the Newton polytope of the defining equation of a certain statistical model. The model is obtained by marginalising two variables in the undirected graphical model of the complete bipartite graph \(K_{2,4}\). The result, a polytope in 16-dimensional space with 17.214.912 vertices, was computed with "ray shooting" and the beneath-beyond method for vertex-facet conversion of polytopes. In this talk I present a more efficient method, namely an output sensitive geometric algorithm for recovering tropical polynomials from their tropical hypersurfaces. This algorithm arose in the study of tropical resultants.

This is joint work with Josephine Yu.

In an indirect Gaussian sequence space model lower and upper bounds are derived for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value \(\theta^\circ\) that generates the data. While this establishes posterior consistency, however, the concentration rate depends on both \(\theta^\circ\) and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each \(\theta^\circ\) separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive. This means that, given a parameter \(\theta^\circ\) or a class of parameters, respectively, the posterior distribution contracts at the oracle rate or at the minimax rate over the class. Notably, the hierarchical prior does not depend neither on \(\theta^\circ\) nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.

We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus \(g\), our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with \(g\) interior lattice points. It has dimension \(2g+1\) unless \(g \leq 3\) or \(g = 7\). We compute these spaces explicitly for \(g \leq 5\). The computations are based on TOPCOM and polymake. Joint work with Sarah Brodsky, Ralph Morrison and Bernd Sturmfels.

Recently, a numerical scheme for the dynamic programming problem has been proposed which is based on approximations by radial basis functions in combination with a least squares projection type approach. In this talk, we extend this method by adaptively choosing the basis functions' centers. We show convergence of this scheme for vanishing fill distance and present several numerical examples.

We study high-dimensional sample covariance matrices based on independent random vectors with missing coordinates. The presence of missing observations is common in modern applications such as climate studies or gene expression micro-arrays. A weak approximation on the spectral distribution in the "large dimension \(d\) and large sample size \(n\)" asymptotics is derived for possibly different observation probabilities in the coordinates. The spectral distribution turns out to be strongly influenced by the missingness mechanism. In the null case under the missing at random scenario where each component is observed with the same probability \(p\), the limiting spectral distribution is a Mar\v{c}enko-Pastur law shifted by \((1-p)/p\) to the left. As \(d/n\rightarrow y\in(0,1)\), the almost sure convergence of the extremal eigenvalues to the respective boundary points of the support of the limiting spectral distribution is proved, which are explicitly given in terms of \(y\) and \(p\). Eventually, the sample covariance matrix is positive definite if \(p\) is larger than \[1-\left(1-\sqrt{y}\right)^2,\] whereas this is not true any longer if \(p\) is smaller than this quantity.

We show that the coefficient of the Schur functor \(S^\lambda\) in the decomposition of the plethysm \(S^\mu(S^k)\) into irreducibles is the solution to a lattice point counting problem. Consequently, for each fixed \(\mu\) the solution to this problem is a piecewise quasi-polynomial in \((\lambda,k)\). We show how to use computer algebra to determine this function explicitly when \(\mu\) is a partition of \(4\) or \(5\). We also discuss asymptotics of the resulting piecewise quasi-polynomials. This is joint work with Mateusz Michałek.

Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal admitting no irreducible decomposition into binomial ideals, thus answering a question of Eisenbud and Sturmfels [1996].

I will talk about some algorithmic problems in polycyclic groups and analyze the complexity of them.

Given relatively slow fluid velocities, Navier-Stokes equations depend on a small Mach number, which renders the development of stable high-order numerical methods quite challenging.

In this talk we present recent developments on creating a high-order DG scheme coupled to an IMEX time discretization.

We show that a compact complex surface which admits a conformally Kähler metric \(g\) of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if \(g\) is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which \(g\) becomes a multiple of the Fubini-Study metric.

1. M. Kalafat, C. Koca, Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature. Geom. Dedicata 174 (2015), 401–408.

In the theory of vector lattices, ideals, disjointness and bands are well-investigated notions and a fundamental tool for the study of operators on vector lattices. Spaces of operators between vector lattices have a natural partial order, but are not vector lattices, in general. We introduce the above notions in partially ordered vector spaces and present their properties in pre-Riesz spaces, which cover all Archimedean directed partially ordered vector spaces. Moreover, disjointness preserving and band preserving operators on pre-Riesz spaces are discussed.

Christensen (Mathematical Finance 24, 2014, 156-172) has introduced an efficient numerical approach for obtaining upper bounds of American option prices in diffusion models. It relies on approximating the value of the option by European options with a larger payoff. In this talk we discuss the question whether or to what extent the value of an American option actually coincides on the continuation region with that of a properly chosen European payoff. In analytical terms this boils down to the question whether the harmonic function solving a free boundary problem can be extended to a harmonic function on the whole space.

For a monomial ideal in a polynomial ring the Taylor resolution is the best understood free resolution. It has been used in many proofs and already been generalized in many different directions: For example the Scarf complex gives a complex that arises in the same fashion. The minimal resolution of the monomial ideal then sits between the Taylor resolution and the Scarf complex. We want to generalize the construction for toric rings, i.e. semigroup rings with the semigroup coming from a cone intersected with a lattice. The resulting complex is not necessarily free, but applying the construction repeatedly we can get a free resolution up to any desired point. Additionally we can derive some results on the support of \(\text{Ext}\) modules of divisorial monomial ideals.

In recent years, the interplay between Markov jump processes and integrodifferential operators was the subject of many research activities. Several results have been obtained for solutions to nonlocal equations where the integrodifferential operator is of a (fractional) differentiability order less than 2. Scaling properties are essentially used in these approaches. In the talk we explain the use of scaling in these works and address limit cases where standard scaling fails. The objects that we study are closely related to geometric stable processes.

The talk is based on joint works with A. Mimica.

Over 50 years ago Polya stated the following problem. Given a plane convex set K find the shortest curve that bisects it into two pieces of equal area. Is it true that this curve is never longer than the diameter of a disk of the same area? Under the additional assumption that K is centrosymmetric (i.e. \(K=-K\)) he gave a simple proof that this is indeed the case. Without this assumption the proof is much harder, and I report on a joint paper with L.Esposito, V.Ferone, C.Nitsch and C.Trombetti containing the proof. It is remarkable that the answer to Polya's question is negative if only straight cuts are allowed. In that case N.Fusco and A.Pratelli were able to show that the Auerbach triangle and not the disc provides "the longest shortest cut". A related result states that a cylindrical bar of specific weight 0.5 does not need to have circular cross section to float in a metastable way in any horizontal orientation.

We present algorithms to compute the automorphism group of integral, finitely generated algebras that are graded by a finitely generated abelian group. We apply our methods to compute automorphism groups of Mori dream spaces. As an example, we compute automorphism groups of Fano varieties in the computer algebra system Singular.

In 2003 Cheon and Jun invented a polynomial-time algorithm, which solves the braid group-based Diffie-Hellman Conjugacy Problem (BDHCP). The algorithm was presented in their article ``A polynomial time algorithm for the braid Diffie-Hellman conjugacy problem'' - In Advances in Cryptology - CRYPTO 2003 Springer. The algorithm makes use of the Lawrence-Krammer representation which is an injective group homomorphism between the braid group \(B_n\) and the general linear group of degree \(\binom{n}{2}\) over the two-variable Laurent-ring \(\mathbb{Z}[t^{\pm1},q^{\pm1}]\). The algorithm searches for a matrix \(\mathcal{A} \in \text{GL}_{\binom{n}{2}}(\mathbb{Z}[t]/(p,f))\) (\(p\) (large) prime, \(f\) an irreducible polynomial (of large degree in \(t\))) which is a solution to several matrix equations. A part of the structure of the possible solutions \(\mathcal{A}\) is already known by Cheon and Jun. During our research on the algorithm we were able to find the complete structure, which improves the run-time complexity (\(\mathcal{O} (\ell^3n^{13,2}\log n)\)) of the algorithm by a constant factor.

Since the powers in the complexity are large, a reduction by a constant factor is not of much use. So our next aim is a clever implementation of the algorithm using interpolation methods and the Chinese remainder theorem, in order to do the calculations over small finite fields instead of doing them over the large field \(\mathbb{Z}[t]/(p,f)\). If we can reduce the run-time of the algorithm enough, then also the knowledge about the structure of the possible solutions should be of greater use.

Considering the dynamics near codimension one homoclinic cycles in flows that are equivariant under the action of the group \(D_n\) one finds an open problem in case that \(n\) is a multiple of 4. We present an explicit construction of families of \(D_{4m}\)-symmetric polynomial vectorfields in \(\mathbb{R}^4\) possessing such a codimension one homoclinic cycle. Based on this example we discuss problems that occur in bifurcation analysis of these cycles.

We present a notion of good-deal hedging, that corresponds to good-deal valuation and is defined by a uniform supermartingale property for the tracking errors of hedging strategies. No-good-deal restrictions are defined in terms of constraints on the Girsanov kernels of pricing measures, and good-deal valuations and hedges are derived from backward stochastic differential equations. Under model uncertainty about the market prices of risk of hedging assets, a robust approach leads to a reduction or even elimination of a speculative component in good-deal hedging, which is shown to be equivalent to global risk-minimization if uncertainty is sufficiently large.

Ordinary (bosonic) classical field theory consists of a "field" bundle on a spacetime manifold, a variational PDE on the field sections, its space of solutions (the "phase space", an infinite dimensional manifold), and the algebra of smooth functions ("observables") on the phase space, with an induced Poisson bracket. Fermionic field theory is defined analogously, except that the fibers of the field bundle are allowed to be supermanifolds instead of ordinary manifolds. In the physics literature, fermionic field theories are usually treated in an essentially algebraic way, at the level of the super-Poisson algebra of observables, with its interpretation as the algebra of functions on a phase space supermanifold lost. I will discuss how a modern, functorial formulation of supergeometry allows us to describe the fermionic phase space as a geometric object and to apply tools from analysis and PDE theory to answer some questions about fermionic theories that were difficult to study or even formulate in the algebraic treatment.

A real is called "infinitely often equal (ioe)" iff it coincides with every ground model real infinitely often. In joint work with Giorgio Laguzzi, we analysed the \(\sigma\)-ideal and the forcing partial order naturally related to ioe reals. Does such a forcing add Cohen reals? By unpublished work of Goldstern and Shelah, we know that some conditions do; but it is open whether all conditions do. I will present some results that could provide an answer. If there are conditions forcing that no Cohen reals are added, then this would provide an alternative solution to Fremlin's problem "can we add ioe reals without adding Cohen reals", recently solved by Zapletal.

Continuous-time ARMA\((p,q)\) (CARMA\((p,q)\)) processes are the continuous-time analog of the well-known ARMA\((p,q)\) processes. They have attracted interest over the last years. Methods to estimate the parameters of a CARMA process require an identifiable parametrization. Such an identifiable parametrization particularly requires the degree p of the autoregressive polynomial to be fixed. Thus, the degree p has to be known for parameter estimation. When this is not the case information criteria can be used to estimate \(p\) as well as \(q\). In this talk we investigate information criteria for CARMA processes based on quasi maximum likelihood estimation. Therefore, we first derive the asymptotic properties of quasi maximum likelihood estimators for CARMA processes in a misspecified parameter space. Then, we present necessary and sufficient conditions for information criteria to be strongly and weakly consistent, respectively. In particular, we study the well-known Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as special cases. The results can be generalized to multivariate CARMA processes as well. The talk is based on joint work with Vicky Fasen.

Given a positive integer \(t\), a set \(K \subseteq \mathbb{R}^d\) and a real multisequence \(s = \{ s_{\gamma_1, \ldots, \gamma_d} \}_{0 \leq \gamma_1+\ldots+\gamma_d \leq m}\) we will formulate new moment matrix conditions for \(s\) have a \(K\)-representing measure \(\sigma= \sum_{q=1}^t \varrho_q \delta_{w_q}\) with \(t\) atoms, i.e., \[s_{\gamma_1, \ldots, \gamma_d} = \int_{\mathbb{R}^d} x_1^{\gamma_1} \cdots x_d^{\gamma_d} d\sigma(x_1, \ldots, x_d) \quad {\rm for} \quad 0 \leq \gamma_1+\ldots + \gamma_d \leq m\] and \[w_1, \ldots, w_t \in K.\] Using these conditions, we will establish new minimal inside cubature rules for planar measures in \(\mathbb{R}^2\) and also pose a solution to the subnormal completion problem in \(d\) variables, i.e., given a collection of positive numbers \(\mathcal{C} = \{ \alpha_{(\gamma}^{(1)}, \ldots, \alpha_{\gamma}^{(d)}) \}_{0 \leq |\gamma| \leq m}\) we wish to determine whether or not \(\mathcal{C}\) gives rise to a \(d\)-variable subnormal weighted shift operator whose initial weights are given by \(\mathcal{C}\). We will also highlight recent results for a full moment problem in a countably infinite number of variables and briefly discuss an application to stochastic processes.

This talk is partially based on joint work with Daniel Alpay and Palle Jorgensen.

When searching for finite unit norm tight frames (FUNTFs) of \(M\) vectors in \(\mathbb{F}^N\) which yield robust representations, one is concerned with finding frames consisting of frame vectors which are in some sense as spread apart as possible. Algebraic spread and geometric spread are the two most commonly used measures of spread. A frame with optimal algebraic spread is called full spark and is such that any subcollection of \(N\) frame vectors is a basis for \(\mathbb{F}^N\). A Grassmannian frame is a FUNTF which satisfies the Grassmannian packing problem; that is, the frame vectors are optimally geometrically spread given fixed \(M\) and \(N\). A particular example of a Grassmannian frame is an equiangular frame, which is such that the absolute value of all inner products of distinct vectors is equal. The relationship between these two types of optimal spread is complicated. The folk knowledge for many years was that equiangular frames were full spark; however, this is now known not to hold for an infinite class of equiangular frames. The exact relationship between these types of spread will be further explored in this talk, as well as Plücker coordinates and mutual coherence, which are measures of how much a frame misses being optimally algebraically or geometrically spread.

Air is a compressible medium. Yet, experience shows that sound waves play a negligible role in the vast majority of meteorologically relevant atmospheric processes. Nevertheless, the family of sound-proof flow models, which correspond to the incompressible or zero-Mach number approximations in engineering fluid mechanics, has met with severe scepticism from a large fraction of the meteorological community since they were first introduce many decades ago.

In this lecture I will elucidate reasons for this scepticism, explain that a thorough analysis of nearly sound-free atmospheric flows involves a non-standard asymptotic three-scale problem, discuss formal estimates of the range of validity of available sound-proof models, and describe ongoing research aiming at an associated rigorous proof.

We present families of homogeneous supergravity backgrounds. We will describe in detail the field ingredients and discuss the free parameters. Furthermore, we will discuss some properties of their moduli spaces.

Let \(V\) be a subspace of an octonion division algebra over a field \(F\). We investigate the group generated by all left multiplications by non-trivial elements of \(V\). Using the autotopism group and the principle of triality, we get conclusive results for subspaces of sufficiently high rank which either contain the identity or are contained in the space of pure octonions. This yields a new approach to some of the isomorphisms between classical groups of low rank, in particular in the anisotropic case.

Heteroclinic cycles connecting equilibria with different saddle indices, where one of the heteroclinic orbits is transverse and isolated, are referred to as T-points. In this talk we study the unfolding of a symmetric saddle-node of T-points in the context of reversible systems. We assume that the leading eigenvalues of the equilibria are real. We focus on the existence of shift dynamics and its creation or annihilation, respectively.

Magnetic Particle Imaging is a tomographic medical imaging technique that allows reconstructing the spatial distribution of magnetic nanoparticles. The relation between the measured voltage signal and the particle distribution is described by a linear system of equations. Due to the complex dynamic behavior of the magnetic nanoparticles the corresponding system matrix is not accurately known. For this reason the system matrix is usually measured column-by-column using a tedious calibration procedure involving a small delta sample and a robot. This measurement can last several days. In the talk it will be shown that the calibration measurement can be significantly accelerated using compressed sensing. Further the actual image reconstruction will be discussed. This involves regularization techniques due to the ill-conditioned system matrix. Using matrix compression in combination with iterative solvers the reconstruction time can be significantly reduced so that online reconstruction becomes feasible.

Magnetic Particle Imaging (MPI) is a tomographic medical imaging technique that uses iron-oxide based tracers in order to follow their spatial distribution when e.g. flowing through the cardiovascular system. In this talk the basic principles of MPI are introduced. Starting with the physical foundations involving static and dynamic magnetic fields responsible for spatial encoding we give an overview of the MPI signal chain. The later can be described by a forward model mathematically relating the particle concentration to the measurement signal that is detected in receive coils. In order to determine the particle concentration the inverse problem has to be solved. Due to the ill-conditioning of the MPI system matrix one has to apply regularization techniques that are described in this talk.

We apply the methods of nonstandard analysis to algebraic number theory and extend the results of G. L. Cherlin and others on nonstandard number fields. The ideal structure of the ring of integers of internal number fields is much richer than that of Dedekind domains. The ideals can be classified using filters on a lattice of internal ideals. The results are particularly interesting for external maximal and prime ideals. We determine various valuation rings and their residue class fields. The usual completion of a number field and the rings of adeles can be described as a subquotient of the enlarged field. Furthermore, we show that the nonstandard extension of the algebraic closure of a number field can be used to define \(\mathbb{C}_p\) and a spherical completion.

Mathematical vernacular -- the everyday language we use to communicate about mathematics is characterized by a special vocabulary. If we want to support humans with mathematical documents, we need a resource that captures the terminological, linguistic, and ontological aspects of the mathematical vocabulary. In the SMGloM project and system, we aim to do just this. We present the glossary system prototype, the content organization, and the envisioned community aspects.

Isometric Lie group actions on Riemannian manifolds are called polar if there is a submanifold, called section, which meets all orbits of the group action and meets them orthogonally at any intersection point; they are called hyperpolar in the special case where the section is flat.

I will talk about a result on hyperpolar actions on reducible compact symmetric spaces and a recent classification of infinitesimally polar (i.e. all slice representations are polar) actions on compact rank one symmetric spaces.

The latter is joint work with Claudio Gorodski.

The decomposition of the state space of a dynamical system into metastable or almost-invariant sets is important for understanding macroscopic behavior. This concept is well understood for autonomous dynamical systems, and has recently been generalized to non-autonomous systems via the notion of coherent sets. We elaborate here on the theory of coherent sets in continuous time for periodically-driven flows and describe a numerical method to find periodic families of coherent sets without trajectory integration.

Mimetic discretization methods for integrating the dynamical equations of Atmosphere and Ocean models on unstructured grids have recently gained much attraction. We review mimetic methods in view of their applications in numerical geophysical fluid dynamics. As a specific example we describe ICON-O a new general circulation model of the global ocean. ICON-O based on the Ocean Primitive Equations: the incompressible Navier-Stokes Equations on the sphere, in vector invariant form with a free surface plus the hydrostatic and the Boussinesq approximation.

The model solves the ocean primitive equations on a triangular icosahedral grid. The models dynamical core as well as its subgrid scale closure use a coherent discretization that is based on a mimetic discretization approach. We describe the mimetic disretization and some of its properties. A sequence of simulations is presented that range from idealized process studies to long-term global ocean simulations.

A recent breakthrough was the discovery that the spherical polynomials have an Almansi type representation. Hence, one may follow the framework of the multidimensional moment problem developed in previous works for the euclidean space: http://arxiv.org/abs/math/0509380 and http://arxiv.org/abs/0802.0023 We formulate the full moment problem on the sphere \(S^n\), and define also the truncated moment problem. We provide a solution for the truncated pseudo-positive moment problem. As a by-product we discover new cubature formulas on the sphere. The remarkable thing is that the new cubature method satisfies the classical criterion for weak* convergence of Osgood, Vitali, Lebesgue, Polya, and Banach.

This is a joint work with Hermann Render.

A non-parametric method to bootstrap locally stationary processes will be proposed, which combines a time domain wild bootstrap approach with a non-parametric frequency domain approach. The method generates pseudo time series which mimic (asymptotically) correct, the local second- and to the necessary extent the fourth-order moment structure of the underlying process. Thus it can be applied to approximate the distribution of several statistics that are based on observations of the locally stationary process. We prove a bootstrap central limit theorem for a general class of statistics that can be expressed as functionals of the preperiodogram, the latter being a useful tool for inferring properties of locally stationary processes. Some simulations and a real data example shed light on the finite sample properties and illustrate the ability of the bootstrap method proposed.

The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the fundamental results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project.

In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas, independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function \(f : N -> N\) such that every digraph of directed tree-width at least \(f(k)\) contains a directed grid of order \(k\). In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. In 2014 we finally managed to prove the conjecture in full generality.

In this talk we will give an introduction to directed tree width and present the main ideas of the proof of the directed grid theorem. We will also present some algorithmic application of this result to routing problems on digraphs as well as to Erdős-Pósa problems for directed graphs.

This is joint work with Ken-ichi Kawarabayashi, National Institute of Informatics, Tokyo.

Bei Seitenkanalangriffen versucht man nicht den mathematischen Algorithmus eines Kryptosystems sondern seine Implementierung in Soft- und Hardware anzugreifen. Eine wichtige Klasse solcher Angriffe sind Fehlerangriffe, bei denen durch gezielte physikalische Manipulationen (z.B. Manipulation der Spannungsversorgung, Beschuss mit Lasern oder anderen Strahlungen) bestimmte Fehlberechnungen induziert werden. Bei algebraischen Fehlerangriffen wird die erhaltene Information in polynomiale Gleichungen über endlichen Körpern umgewandelt und dann versucht, diese mit symbolischen Verfahren zu lösen. Wir führen verschiedene Arten und Beispiele algebraischer Fehlerangriffe vor und diskutieren auch mögliche Verteidigungsstrategien.

The feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the control a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examples are presented illustrating the stabilizing effect of the feedback control.

This is joint work with Sergio S. Rodrigues.

In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by Hk, k=1, where H denotes the mean curvature. We use a level set formulation of this flow and discretize the regularized level set equation with finite elements. In a previous paper we proved an a priori estimate for the approximation error between the finite element solution and the solution of the original level set equation. We obtained an upper bound for this error which is polynomial in the discretization parameter and the reciprocal regularization parameter. The aim of the present paper is the numerical study of the behavior of the evolution and the numerical verification of certain convergence rates. We restrict the consideration to the case that the level set function depends on two variables, i.e. the moving hypersurfaces are curves. Furthermore, we confirm for specific initial curves and different values of k that the flow improves the isoperimetrical deficit.

In this talk we will give some recent results about atmospheric flows which were obtained within the DFG Schwerpunktprogramm 1276 "MetStroem". In particular we will consider the numerical simulation of the gravity waves and some Benchmark problems which were designed within this DFG Schwerpunktprogramm. The simulation is based on the compressible Navier Stokes equation and for the numerical discretization we use discontinuous Galerkin methods within the DUNE context. Furthermore we will discuss the numerical simulation of surface flows modelled by shallow water equations including the wetting and drying process. Also in this case the numerical approximation is based on discontinuous Galerkin methods.

High order discontinuous Galerkin methods have emerged as ideal candidates for next-generation solvers of convection-dominated problems as they combine high accuracy with robustness. Hybridized discontinuous Galerkin discretizations (HDG) are special representatives mainly targeting implicit time stepping schemes with a considerably reduced the number of unknowns in the final linear systems: By suitable numerical traces, all element-related velocity degrees of freedom can be statically condensed into contributions for the velocities on the mesh skeleton only, while also offering favorable convergence behavior. In this talk we present our progress on taking HDG methods from a concept mostly used in theoretical works towards the practical application in large eddy simulation of incompressible turbulent flow. This includes the efficient implementation on high-performance parallel computers as well as iterative solvers for the final system in trace velocities and discontinuous pressures using block-triangular preconditioners. We will show comparisons of our new HDG solvers with established methods based on finite elements. Finally, turbulent-specific methodologies such as scale separation and subgrid scale models in the HDG context will be presented.

We shall present an abstract version of the Banach-Mazur-Choquet game, where two players alternately choose finitely generated models, building an increasing sequence whose union typically is a ``random" or ``generic" countably generated model. It turns out that various universal structures with high level of homogeneity (in particular, Fraisse limits) can be explained by the existence of a winning strategy of one of the players. We shall discuss applications of this approach beyond the standard model theory.

One of many definitions gives the rank of an \(m \times n\) matrix \(M\) as the smallest natural number \(r\) such that \(M\) can be factorized as \(AB\), where \(A\) and \(B\) are \(m \times r\) and \(r \times n\) matrices respectively. In many applications, we are interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank and are closely related to mixture models in statistics. Another rank I will consider in my talk is the positive semidefinite (psd) rank.

Both nonnegative and psd rank have geometric characterizations using nested polytopes. I will explain how to use these characterizations to derive a semialgebraic description of the set of matrices of nonnegative/psd rank at most \(r\) in some small cases, and to study boundaries of this set. The talk is based on joint work with Rob H. Eggermont, Emil Horobet, Elina Robeva, Richard Z. Robinson, and Bernd Sturmfels.

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achievea closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one moment, a suitable coarse-grained quantity computablefrom the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure. In this talk, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.

In most countries, trading gains have to be taxed. The modeling is complicated by the rule that gains on assets are taxed when assets are sold and not when gains actually occur. This means that an investor can influence the timing of her tax payments, i.e., she holds a timing option. In this talk, it is shown how the tax payment stream can be constructed beyond trading strategies of finite variation.

In addition, we analyze Constant Proportion Portfolio Insurance (CPPI) strategies in models with capital gains taxes and an Itô asset price process. CPPI strategies invest a constant fraction of some cushion in a risky asset (or index). For a fraction bigger than one, this leads to a superlinear participation in upward price movements while guaranteeing a given part of the invested capital, even if the cushion gets completely lost. It turns out that the associated tax payment stream is of finite variation if the fraction is bigger or equal to one and of infinite variation otherwise.

(Parts of the talk are based on joint work with Björn Ulbricht)

We are interested in the effect of Gaussian white noise on fast-slow dynamical systems with one fast and two slow variables. In the absence of noise, these systems can display mixed-mode oscillations, which are oscillation patterns in which small- amplitude and large-amplitude oscillations alternate. In particular, the system we study is not in gradient-form and contains several generic, but non-hyperbolic, singular points which generate the oscillations. The effect of weak noise can be quantified by analyzing the continuous-space, discrete-time Markov chain describing the returns of sample paths to a cross-section. The main result yields estimates of sample paths to deviate from the deterministic solution. The result implies estimates on transition probabilities of the discrete-time Markov chain on the finite state state space of mixed-mode patterns.

We give an overview of several variants of the Burrows-Wheeler Transform (BWT). This includes the sort transform (ST), bijective versions of the BWT and the ST, and generalizations by using permutations on the alphabet. The setting of the BWT with permutations (BWTP) is as follows. Let \(G\) be a group acting on an ordered alphabet \(\Sigma\). We write \(a^g\) for the letter obtained by applying the element \(g \in G\) to \(a \in \Sigma\). For \(u = a_1 \cdots a_n\) we let \(u^g = a_1^g \cdots a_n^g\) be the homomorphic extension to words \(u \in \Sigma^*\). Let \(\tilde{u}\) denote the lexicographically minimal element in \(\{u^g \mid g \in G\}\). Let \((\tilde{v}_1,\ldots,\tilde{v}_n)\) be the sorted list of the conjugates \(v_i\) of \(u\). The BWT with permutations (BWTP) of \(u\) is \(\mathrm{BWTP}_G(u) = (w,i,g)\) where \(w\) is the sequence of the last letters in the sorted list of the words \(\tilde{v}_i\), the number \(i\) is an index with \(\tilde{u} = \tilde{v}_i\), and \(g \in G\) satisfies \(\tilde{u} = u^g\).

In this talk we discuss the distribution of extreme events for dynamical systems for different classes of observables. In the last fifteen years the classical extreme value theory for stochastic processes has been extended to dynamical systems. Extreme value theory is concerned with either the asymptotical distribution of running maxima or the asymptotic of over threshold events for large thresholds and the relation between these two. We will review the aforementioned developments. Finally, we will discuss the behaviour of high dimensional chaotic systems for observables which do not have their maximal value in the interior of the attractor.

The latter is based on a joint work with Valerio Lucarini, Davide Faranda and Jeroen Wouters.

In Einstein-Maxwell theory a number of theorems hold for black holes. For instance, black holes are uniquely specified by their global charges; a static horizon implies a spherically symmetric spacetime as well as vanishing total angular momentum. However, these theorems do not generalize to theories with other types of fields. Yang-Mills fields violate uniqueness and allow for static non-spherically symmetric black hole spacetimes. The presence of a dilaton yields stationary black holes with static horizons as well as counterrotating black holes. Complex scalar fields produce hair on rotating black holes.

An orbit polytope is the convex hull of an orbit under a finite group \(G \leq \operatorname{GL}(d,\mathbb{R})\). We consider the possible affine symmetry groups of orbit polytopes. For every group, there is an open and dense set of ``generic points'' such that the orbit polytopes of generic points have conjugated affine symmetry groups and are minimal in a certain sense. For some groups \(G\), the affine symmetry group of every orbit polytope is strictly larger than \(G\), but for most groups, this is not the case. The affine symmetry group of a generic orbit polytope can be computed from the character of the group \(G\).

We also show that every abstract group that is isomorphic to the full euclidian symmetry group of an orbit polytope, is also isomorphic to the full affine symmetry group of an orbit polytope, with exactly three exceptions: the elementary abelian groups of orders \(4\), \(8\) and \(16\). This answers a question of Babai (1977).

This is joint work with Erik Friese.

(Based on a joint work with Sy Friedman.) We try to shed light on the non-trivial question of generalizing the random forcing for \(2^\kappa\), with \(\kappa\) uncountable. We answer a question posed by Shelah, asking whether one can construct a tree-like forcing adding new subsets of \(\kappa\), which is simultaneously \({<}\kappa\)-closed, \(\kappa^\kappa\)-bounding and satisfies \(\kappa^+\)-c.c., for \(\kappa\) inaccessible. We further investigate some properties of this forcing, comparing it with Shelah's one for \(\kappa\) weakly compact.

We derive a thermodynamically consistent diffuse interface model for tumour growth with chemotaxis and active transport. We couple a Cahn-Hilliard-Darcy system for a two component mixture of healthy cells and tumour cells, and a reaction diffusion equation for a nutrient. Specific choices of the fluxes allow us to include the effects of chemotaxis and active transport. Via a formally matched asymptotic analysis, we recover some of the recent sharp interface models studied for tumour growth. If time permitting, we discuss some recent results regarding the well-posedness of the Cahn-Hilliard nutrient subsystem without fluid flow.

In the first part of the talk we give an introduction to Fomin-Zelevinsky's theory of cluster algebras. We see that cluster algebras of finite type are classified by finite type root systems. Especially, the mutation rule defines a periodic sequence for every root system of rank \(2\). Next we study Fomin-Zelevinsky's mutation rule in the context of non-crystallographic root systems. In particular, we construct an almost periodic sequence of real numbers for every non-crystallographic root system of rank \(2\) and describe matrix mutation classes in rank \(3\).

A set \(K\subseteq\mathbb{R}^{d}\) is called lattice-convex if \(K\) is the intersection of \(\mathbb{Z}^d\) and a convex subset of \(\mathbb{R}^d\). A tiling is a partition of \(\mathbb{Z}^{d}\) into identical translated copies of some finite, lattice-convex set \(T\). In this talk we consider lattice tilings, i.e. the translation vectors form a sublattice \(\mathbb{L}\) of \(\mathbb{Z^{d}}\).

We study subsets of the form \(S\oplus T\) of lattice tilings \((\mathbb{L},T)\), where \(S\subseteq\mathbb{L}\) is finite and full-dimensional. We present a necessary and a sufficient condition for the existence of lattice-convex subsets of this form and show that these are `rare' in the plane if one additionally requires \(S\) to be not centrally symmetric. We apply our results to answer a problem in the the realm of discrete covariograms.

This is joint work with Gennadiy Averkov.

This is a survey talk about results on `chaotic' or complicated motion in the infinite-dimensional semiflows generated by delay equations. The focus is on analytically proven results, but numerical simulations will also be mentioned. Examples start from 1977 and include nonlinearities such as smoothed step functions, negative and also sine-like feedback, and state-dependent delay.

The proofs combine traditional ideas, like transverse homoclinic intersections described by Poincaré and homoclinic behavior as analyzed by L.P. Shilnikov for ODEs, with more recent approaches, e.g. fixed-point index methods for the proof of symbolic dynamics.

In this talk I will introduce the notion of group actions on moment graphs and apply it to construct a category of modules exhibiting a periodic behaviour. The periodic patterns arising in our category had been already studied by Lusztig and appear -or are expected to appear- in representation theory of affine Kac-Moody algebras, Lie algebras in positive characteristic, quantum groups at a root of unity, ...

This is joint work with Peter Fiebig.

In this talk we will consider the long-term behaviour of solutions to the Dirichlet-problem of a degenerate parabolic equation with a nonlinear and nonlocal contribution of the gradient, which arises in the context of evolutionary game dynamics. We will identify the (nonzero) \(\mathrm{W}^{1,2}\)-limit of solutions to \[ u_t = u\Delta u + u \int_{\Omega} |\nabla u|^2 \] in a bounded smooth domain under Dirichlet boundary conditions and prove convergence, if the initial data satisfy \(\int u_0 = 1\). (This long-term behaviour is significantly different from cases where the initial mass is either smaller or larger than 1.)

Tablets, smartphones as well as cardiograms and many other electronic devices make use of liquid crystals - a matter which shares properties of both conventional fluids and solid crystals. Liquid crystals are ubiquitous in our everyday life and, consequently, in the focus of ongoing research. In this talk, we study the equation of motion for nematic liquid crystals as a system of coupled nonlinear evolution equations. We present results on existence of generalized solutions to the Ericksen-Leslie model under general assumptions on the free energy potential. The method of proof relies on a suitable approximation scheme. Finally, we discuss possible relaxations of the model and an adaptation of the method of proof to non-nematic liquid crystals.

The Drinfeld and Heisenberg double are fundamental constructions in the theory of Quantum groups. The relation between the two constructions can be used to obtain a categorical action of modules of the former on modules of the latter. In a special case, this action can be restricted to give an action on representations of rational Cherednik algebras (where the parameter \(t\) is zero), using embeddings of Bazlov-Berenstein. In this case, the braided Drinfeld doubles of generalizations of the Fomin-Kirillov algebras (for any complex reflection group) are acting.

Bordismenringe erhält man, wenn man auf einer Menge von Mannigfaltigkeiten zwei Objekte miteinander identifiziert, die sich nur um einen Rand einer höher dimensionalen Mannigfaltigkeit unterscheiden. Charakteristische Zahlen spielen bei ihrer Berechnung eine wichtige Rolle. Zum Beispiel bestimmen Stiefel-Whitney Zahlen und K-theoretische Pontryagin-Zahlen die Bordismenklassen von orientierten und Spin-Mannigfaltigkeiten. Allerdings stimmt das nicht mehr, wenn man Strukturen betrachtet, die näher an einer Trivialisierung des Tangentialbündels liegen. In dem Vortrag werden neue charakteristische Zahlen für String-Mannigfaltigkeiten vorgestellt, die in der Kohomologietheorie der topologischen Modulformen liegen.

We consider models for random networks that are exchangeable in the sense that the distribution \(P_\mathcal{N}\) of the network is invariant under relabeling of the nodeset \(\mathcal{N}\). In particular we study exchangeable random networks that are extendable in the sense that \(P_\mathcal{N}\) is the marginal distribution of larger exchangeable random network with nodeset \(\mathcal{N}'\).

The set \(\mathcal{P}_\mathcal{N}\) of exchangeable distributions is a polytope and corners represent distributions that are uniform over isomorphism classes. Similarly the set of extendable distributions \(\mathcal{P}_\mathcal{N}^{\mathcal{N}'}\) is a polytope sitting inside \(\mathcal{P}_\mathcal{N}\). When the size of \(\mathcal{N}'\) tends to infinity, the limit \(\mathcal{P}_\mathcal{N}^{\infty}\) is a simplex where the extreme points are dissociated, so that presence of non-incident edges are independent, hence satisfying the bidirected Markov property w.r.t.\ the line graph of the complete graph on \(\mathcal{N}\).

We shall study the convex sets above and explain the connection to marginal binary independence models [4], de Finetti theorems for exchangeable arrays [1], [3], [2], and the theory of graphons and graph limits [5], as well the consequences for estimation of pararameters of random networks.

The lecture is based on joint work with Alessandro Rinaldo and Kayvan Sadeghi.

1. Aldous, D. (1981). Representations for partially exchangeable random variables. Journal of Multivariate Analysis, 11:581--598.
2. Aldous, D. (1985). Exchangeability and related topics. In Hennequin, P., editor, École d'Été de Probabilités de Saint--Flour XIII --- 1983, pages 1--198. Springer-Verlag, Heidelberg. Lecture Notes in Mathematics 1117.
3. Diaconis, P. and Freedman, D. (1981). On the statistics of vision: the Julesz conjecture. Journal of Mathematical Psychology, 24:112--138.
4. Drton, M. and Richardson, T.~S. (2008). Binary models for marginal independence. Journal of the Royal Statistical Society Series B, 70(2):287--309.
5. Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B, 96(6):933--957.

High-dimensional statistics is the basis for analyzing large and complex data sets that are generated by cutting-edge technologies in genetics, neuroscience, astronomy, and many other fields. However, Lasso, Ridge Regression, Graphical Lasso, and other standard methods in high-dimensional statistics depend on tuning parameters that are difficult to calibrate in practice. In this talk, I will illustrate the role of tuning parameters in modern data analysis. Moreover, I present two novel approaches to establish accurate calibration of these parameters. My first approach is based on a novel testing scheme that is inspired by Lepski's idea for bandwidth selection in non-parametric statistics. This approach provides tuning parameter calibration for estimation and prediction with the Lasso and other standard methods and is to date the only way to ensure high performance, fast computations, and optimal finite sample guarantees. My second approach is based on the minimization of an objective function that avoids tuning parameters altogether. This approach provides accurate variable selection in regression settings and, additionally, opens up new possibilities for the estimation of gene regulation networks, microbial ecosystems, and many other network structures.

Frame set problems in Gabor analysis ask the question for which sampling and modulation rates the corresponding time-frequencey shifts of a generating window allow for stable reproducing formulas of \(L^2\)-functions. In this talk we show that the frame set conjecture for B-splines of order two and greater is false. Our arguments are based on properties of the Zak transform (also known as the Bloch-Floquet transform and Weil-Brezin transform).

The study of wavelets in higher dimensions is generally restricted to the class of expanding dilations \(A\), i.e., all eigenvalues \(\lambda\) of \(A\) satisfy \(|\lambda|>1\). In contrast, much less attention has been devoted to the study of wavelets associated with general invertible dilations. In this talk we show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with the estimate on the number of lattice points inside dilates of balls by the powers of a dilation \(A \in GL_n(\mathbb{R})\). This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically.

We show that the lattice counting estimate holds for all dilations \(A\) with \(|\det A|\ne 1\) and for almost every lattice \(\Gamma\) with respect to invariant measure on the set of lattice. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and and almost every choice of a dilation \(A\) with respect to the Haar measure on \(GL_n(\mathbb{R})\).

This is joint work with Marcin Bownik.

I will first review Lusztig's quantum group of divided powers and how it leads to a non-semisimple "modular" category. This includes some own results, which remove restrictions on the order of the root of unity (e.g., even). Then I explain how one hopes to realize these categories from a uniformly constructed family of vertex algebras. I conclude by some examples.

We prove that the modular curvature of a conformal metric structure on the noncommutative torus \(T_\theta^2 (\theta\not\in\mathbb{Q})\) is invariant under Morita equivalence. More precisely, the curvature associated to a Hermitian structure on a Heisenberg bimodule \(E\) realizing the Morita equivalence between \(A_\theta = C(T_\theta)\) and \(A_{\theta'}\), with \(A_{\theta'}\) identified to the algebra of endomorphisms \(End_{A_\theta}(E)\), coincides with the intrinsic curvature of the conformal metric on \(T_\theta^2\) with corresponding Weyl factor. The main analytical tool is the extension of Connes? pseudodifferential calculus to Heisenberg modules, the novel technical aspect being that the entire computation is free of any computer assistance.

This is joint work with Henri Moscovici.

We consider cosmological models of Bianchi type. They yield spatially homogeneous, anisotropic solutions of the Einstein field equations. In particular, we are interested in the alpha-limit dynamics of the Bianchi model corresponding to the big-bang singular limit of the Einstein equations. Emphasis is on transient behaviour of solutions near the (backward) Bianchi attractor composed of the Kasner circle of equilibria and attached heteroclinic connections. The heteroclinic orbits in the Bianchi attractor form formal sequences of shift type. We prove the existence of unstable manifolds to heteroclinic sequences. This relates alpha-limit transients of cosmologies of Bianchi type to formal sequences of Kasner heteroclinics: a tumbling universe.

Qualitative properties of solutions to the evolution problem modeling microelectromechanical systems with general permittivity profile are investigated. The system couples a parabolic evolution problem for the displacement of a membrane with an elliptic free boundary value problem for the electric potential in the region between the membrane and a rigid ground plate. Conditions are specified which ensure the non-positivity of the membranes displacement. Moreover, assuming to have a non-positive displacement, it is shown that the solution develops a singularity after a finite time of existence.

Within the framework of variational modeling we derive a two-phase moving boundary problem that describes the motion of a semipermeable membrane separating two viscous liquids in a fixed container. The model includes the effects of osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short time. Moreover, we show that the manifold of steady states is locally exponentially attractive.

We consider the perturbed eigenvalue problem of the \(1\)-Laplace operator, which is formally given by the equation \begin{equation} -\mathrm{Div} \frac{Du}{|Du|} + f(x, u) = \lambda \Big(\frac{u}{|u|} +g(x,u)\Big)\tag{*} \end{equation} We stipulate certain growth, but not continuity assumptions on \(f\) and \(g\). This equation is highly singular and not well defined in the stated form. The associated variational problem is nonsmooth and non-convex. However, in the last years the unperturbed case (with \(f=g=0\)) has successfully been treated with methods of nonsmooth critical point theory, in particular eigenfunctions \(u\) are defined as critical points and eigenvalues \(\lambda\) as critical values of the associated variational problem. Since \(f\) and \(g\) break the homogeneity of the problem, this definition does not directly apply in the perturbed case. We will demonstrate, how to define reasonable solutions of \(\mathrm{(*)}\), prove the existence of a sequence of eigensolutions and provide a bifurcation result for the perturbed problem. In particular the eigenvalues of the unperturbed \(1\)-Laplace operator turn out to be bifurcation values of the eigenvalues of the perturbed problem.

I will report on some recent progess on spon models on random networks, such as the Curie-Weiss model or the Hopfield model on an Erdős-Renyi graph or more general graphs. This is based on joint work with Zakhar Kabluchko (Münster) and Franck Vermet (Brest).

Heteroclinic cycles and networks occur as prototypes for stop-and-go dynamics in a wide range of applications from geophysics to neurosciences. They consist of finitely many equilibria \(\xi_j\) and connecting trajectories \([\xi_j \rightarrow \xi_{j+1}] \subset W^u(\xi_j) \cap W^s(\xi_{j+1})\), and may be structurally stable in systems with symmetry. In this talk we consider simple heteroclinic networks in \(\mathbb{R}^4\)–constructed from simple, non-homoclinic, robust cycles. There are few ways by which such cycles can be joined to form a network, and we provide a complete list of these. Using the stability index from Podvigina and Ashwin (Nonlinearity 24, 887--929, 2011), we describe non-asymptotic stability properties of individual cycles and derive information about stability of the entire network as well as nearby dynamics. This strongly depends on the equivariance of the system–networks with seemingly identical geometry, but different symmetry groups, display very different stability configurations.

Inverse problems often suffer from ill-posedness, e.g. in the sense that the problems are underdetermined and/or the solution does not depend continuously on the given data. We study variational regularization methods, i.e. methods that minimize certain functionals. Noise models and prior information can be modelled via that approach. We touch upon regularizing properties of such functionals and also computational methods to solve the resulting convex minimization problems. The algorithms are build in a way that the scalable to large problems and produce mildly accurate minimizers quickly.

Sparse reconstruction aims to recover a sparse vector for underdetermined linear measurements. We introduce the notion of recoverable support (recoverable sign-pattern, to be precise) for such problems and analyze recoverable supports for the so-called basis pursuit method. Unlike many previous studies we do not consider any asymptotic regime but try to answer questions like "How many different recoverable supports can a matrix of a given size have?" or "How can one determine the largest recoverable support for a given matrix?". We show that answers to such questions can be given with the help of the geometry of certain convex polytopes.

We consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. We have used the Ruelle response theory on such a weakly coupled system to construct a surrogate dynamics, such that the expectation value of any observable agrees, up to second order in the coupling strength, to its expectation evaluated on the full dynamics. We show here that such surrogate dynamics agree up to second order to an expansion of the Mori-Zwanzig projected dynamics. This implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related, if one takes into account, in addition to the widely adopted stochastic forcing, the often neglected memory effects.

The climate is a complex, chaotic, non-equilibrium system featuring a limited horizon of predictability, variability on a vast range of temporal and spatial scales, instabilities resulting into energy transformations, and mixing and dissipative processes resulting into entropy production. Despite great progresses, we still do not have a complete theory of climate dynamics able to encompass instabilities, equilibration processes, and response to changing parameters of the system. We will outline some possible applications of the response theory developed by Ruelle for non-equilibrium statistical mechanical systems, showing how it allows for setting on firm ground and on a coherent framework concepts like climate sensitivity, climate response, and climate tipping points. We will show results for comprehensive global circulation models. The results are promising in terms of suggesting new ways for approaching the problem of climate change prediction and for using more efficiently the enormous amounts of data produced by modeling groups around the world.

V. Lucarini, R. Blender, C. Herbert, F. Ragone, S. Pascale, J. Wouters, Mathematical and Physical Ideas for Climate Science, Reviews of Geophysics 52, 809-859 (2014)

Motivated by a conjecture of Todorčević, we study strengthenings of the \(\kappa\)-chain conditions that are equivalent to the \(\kappa\)-chain condition in the case where \(\kappa\) is a weakly compact cardinal. We then use such properties to provide new characterisations of weakly compact cardinals. This is joint work in progress with Sean D. Cox (VCU Richmond).

We will present some recent results on the asymptotic preserving FV schemes for the shallow water and/or Euler equations.

We will show theoretically as well as by numerical experiments that the resulting methods yield consistent approximations with respect to a singular parameter. The main idea is to use a suitable splittting of the whole nonlinear problem into a linear singular operator (describing fast linear waves) and a nonlinear nonsingular one (describing slow nonlinear waves). Moreover, suitable approximation of the source terms will yield a well-balanced method uniformly with respect to the singular parameter, as well. The present work has been done in cooperation with G. Bispen, L. Yelash (University of Mainz).

Formally, the heat kernel corresponding to a Laplace-type operator on a Riemannian manifold can be written as an integral over the space of all paths that particles could take. One way to make this rigorous is by approximating the infinite-dimensional space of continuous paths by finite-dimensional paths of piece-wise geodesics. In the case that the manifold has a boundary, one has to take paths that reflect at the boundary; we discuss in particular, how different boundary conditions lead to different path integrals formulas.

This talk deals with a priori error estimates for space-time finite elements discretization of semilinear parabolic optimal control problems subject to inequality constraints on the state variable and its first derivative. These constraints are understood pointwise in time and averaged in space. Consideration of these constraints is motivated by industrial application in the steel and glass production, where stress averages are often considered in order to avoid material failure. Making use of the discontinuous Galerkin method for the time discretization and of standard conforming finite elements for the space discretization, we derive convergence rates as temporal and spatial mesh size tends to zero.

Die Diskussion um das forschende Lernen im Mathematikunterricht hat sich in den letzten Jahren national und international belebt. Forschendes Lernen und Projektlernen ist vermutlich der Arbeitsweise von Mathematikern aus der Perspektive der Unterrichtsformen am ähnlichsten. Obwohl Hattie dem forschenden Lernen eine Effektstärke von nur 0.38 zuschreibt und es sich damit im hinteren Drittel der erfolgreichen Unterrichtsmethoden befindet, halten wir forschendes Lernen für eine Chance, das Betreiben von Mathematik authentisch zu erleben.

Im Vortrag wird über das Projekt Mathe.Forscher berichtet, welches seit fast 5 Jahren an mehr als 20 Schulen bundesweit durchgeführt wird. Es wird aufgezeigt, wie sich Schüler aber auch Lehrerinnen und Lehrer auf das forschende und Projektlernen einlassen und welchen Nutzen Sie daraus ziehen. Die Schwierigkeiten bei diesem Unterrichtsansatz sollen aber nicht ausgespart werden.

I will report on work in progress with Håkon Gylterud and Erik Palmgren: a formalisation, in Coq, of the basic algebraic semantics of dependent type theory. Specifically, we aim to show the initiality of the syntactic category with attributes, for a small-ish dependent type theory. To minimise meta-theoretic assumptions, we use a setoid-based notion of categories with attributes, rather than categories or pre-categories in the sense of HoTT; however, due to the interaction between the set(oid)s of types and the (E-)category of contexts, a HoTT-like worldview inevitably pervades the work.

A related question is whether Homotopy Type Theory can eat itself - that is, whether one can construct in HoTT an interpretation function from some small fragment of its syntax into the actual universe of types. This has proven difficult, perhaps surprisingly so. The present project does not attempt to do this, but it perhaps sheds some light on the difficulties that arise.

In the talk I will present global bifurcation results for semilinear elliptic boundary value problems on annuli which are of Ambrosetti-Brézis-Cerami type, i.e. where the nonlinearity is sublinear near zero and superlinear near infinity and looks like \(\lambda u^q+u^p\) for \(1\lt q\lt 2\lt p\). It is proved that there are infinitely many global continua of nodal solutions emanating from the trivial solution.

The 7 geometries in the last row and the last column of the Freudenthal-Tits Magic Square are all connected with (octonion) Cayley-Dickson algebras. We discuss some features of these connections.

Let \(T\) be a (compact) torus that acts on a topological space \(X\). A polynomial assignment is a map \(A\) that assigns to any \(x\in X\) a polynomial function \(A(x): {\rm Lie}(T_x) \to {\mathbb R}\), where \(T_x\) is the stabilizer of \(x\); \(A\) is required to be \(T\)-invariant and satisfy a certain compatibility condition involving fixed points of various subtori of \(T\). The space of all such assignments is an algebra over the polynomial ring of \({\rm Lie}(T)\), the so-called assignment algebra. This notion was introduced by Ginzburg, Guillemin, and Karshon in 1999. For smooth actions on manifolds, connections with the \(T\)-equivariant cohomology algebra were established recently by Guillemin, Sabatini, and Zara (2014). I will explain that the same relationship between assignments and equivariant cohomology exists in the topological setting. I will also discuss assignment versions of the Chang-Skjelbred lemma and the Goresky-Kottwitz-MacPherson presentation.

This is a report on joint work with Oliver Goertsches (LMU Munich).

We survey some recent results obtained with A. Hurtado, V. Gimeno, and V. Palmer, concerning comparison geometric aspects of the Dirichlet spectrum and the mean exit time moment spectrum for extrinsic balls in minimal submanifolds. Possible extensions to foam structures and to similar comparison geometric results in Finsler spaces will also be discussed.

This talk was supposed to be given by Murray A. Marshall who suddenly passed away on the 1st of May 2015.

It is explained how a locally convex (lc) topology \(\tau\) on a real vector space \(V\) extends naturally to a locally multiplicatively convex (lmc) topology \(\overline{\tau}\) on the symmetric algebra \(S(V)\). This allows application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of \(\overline{\tau}\)-continuous linear functionals \(L: S(V)\rightarrow \mathbb{R}\) satisfying \(L(\sum S(V)^{2d}) \subseteq [0,\infty)\) (more generally, of \(\overline{\tau}\)-continuous linear functionals \(L: S(V)\rightarrow \mathbb{R}\) satisfying \(L(M) \subseteq [0,\infty)\) for some \(2d\)-power module \(M\) of \(S(V)\)) as integrals with respect to uniquely determined Radon measures \(\mu\) supported by special sorts of closed balls in the dual space of \(V\). The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and Šifrin. It is more general because \(V\) can be any locally convex topological space (not just a separable nuclear space), the result holds for arbitrary \(2d\)-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that \(L : S(V) \rightarrow \mathbb{R}\) is \(\overline{\tau}\)-continuous (not just that \(L\) is continuous on the homogeneous parts of degree \(k\) of \(S(V)\), for each \(k\ge 0\)).

This is a joint work with Mehdi Ghasemi, Salma Kuhlmann and Murray Marshall.

The basic equivalence between k-means clustering for the Euclidean metric and orthogonal non-negative matrix factorization (NMF) has attracted substantial interest in the last three years. We extend this approach to regularized NMF-methods with general p-norms and determine the equivalent k-means clustering algorithms. This gives rise to some algorithmically accessible non-standard k-means variants. We than apply this to MALDI Imaging data, which is a particular complex case of hyperspectral imaging data.

The Muskat problem is a moving boundary problem describing the evolution of two immiscible layers of Newtonian fluids with different densities and viscosities, for example oil and water, in a porous medium under the influence of surface tension effects and/or gravity. This problem has been studied in the last two decades by many mathematicians, the methods employed being various.

We show that in the absence of surface tension effects the Rayleigh-Taylor sign condition identifies a domain of parabolicity for the Muskat problem. When allowing for surface tension effects, the Muskat problem is of parabolic type for general initial and boundary data. The parabolicity property is used to establish the well-posed of the problem and to study the stability properties of equilibria.

Based on joint papers with Joachim Escher, Anca Matioc, Christoph Walker.

A permanent magnet machine is a generator where the excitation field is provided by a permanent magnet instead of a coil. The center of the generator, the rotor, contains the magnet. Our goal is to find the minimum volume of the rotor which guarantees a desired electromotive force. We apply model order reduction to speed up the optimization. We use snapshot POD to construct the reduced order model.

We present an optimal Berry-Esseen type theorem for approximating expectations of smooth functions (like \(f(x)= (1/6)|x|^3\)) of a standardized sum of i.i.d. random variables by corresponding expectations with respect to standardized symmetric binomial laws. Comparing the latter expectations to standard normal ones yields, as a corollary, a partial improvement of a Theorem of Tyurin (2009).

This is joint work with Irina Shevtsova.

Well-partial-orders play an important role in logic, mathematics and computer science. There are the essential ingredient of famous theorems like Higman's lemma and Kruskal's theorem. The maximal order type of a well-partial-order characterizes that order's proof-theoretical strength. Moreover, in many natural cases, the maximal order type of a well-partial-order can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent examples are Friedman's well-partial-orders on trees with finitely may labels with the so-called gap-embeddability relation. Friedman introduced these well-partial-orders in 1985 to obtain an independence result for the strongest theory of the Big Five in reverse mathematics. It was for a long time unknown if such a natural independence result even existed.

In this talk we discuss a conjecture of Weiermann about the connection between maximal order types of specific well-partial-orders, each ordered by a certain gap-embeddability relation, and ordinal notation systems based on the well-known collapsing functions \(\vartheta_i\). This conjecture yields a representation of for example the big Veblen number and the Howard-Bachmann number in terms of rooted trees. Furthermore, it implies an exact classification of Friedman's well-partial-orders in terms of maximal order types and ordinal notation systems.

According to the no-hair theorem, the Kerr-Newman black hole solution represents the most general asymptotically flat, stationary (electro-) vacuum black hole solution in general relativity. The talk shows how this solution can indeed be constructed as the unique solution to the corresponding boundary value problem of the axially symmetric Einstein-Maxwell equations in a straightforward manner.

A pseudo-Hermitian manifold in the classical sense is a pseudo-Riemannian manifold with an almost complex structure compatible with the metric. We study a more general setting with an almost hypercomplex and an almost quaternionic structure. It turns out that such manifolds with index \(4\) are already Hyperkähler or quaternionic Kähler manifolds if they are in addition homogeneous and have an irreducible isotropy group.

We start with an introduction to the field of functional data and principal components. Then we provide an overview on the literature on statistical topics in functional data analysis. In particular, we consider the model of functional linear regression and show that it is asymptotically equivalent to a white noise inverse problem. Furthermore we discuss asymptotic minimax results in nonparametric regression and classification for functional data.

Variational theories of ferromagnetism accommodates a variety of topologically non-trivial field configurations (domain wall, vortices, skyrmions). We shall discuss the effective dynamics of these particle-like structures based on localization principles for Landau-Lifshitz-Gilbert equations.

This talk is about doubly nonlinear evolution equations of the form \(\frac{d}{dt} Bu + Au = f\), where \(A,B\) are nonlinear operators and \(B\) does not admit a potential. A particular case are systems of doubly nonlinear reaction-diffusion equations \begin{equation*} \frac{\partial v}{\partial t} - \mathrm{div}\left(a(\nabla u)\right) = f \end{equation*} where \(u\) is vector-valued and the operator \(Au = - \mathrm{div}\left(a(\nabla u)\right)\) may be degenerate or singular. We discuss existence and further properties of solutions on the one hand for static relations \(v = b(u)\) between \(u\) and \(v\) which are nonpotential, i.e., \(b\) is not the derivative of a function \(\phi_b\), and on the other hand for additional dynamic equations for \(u\) determining the relation between \(u\) and \(v\), which are closely related to thermodynamics like, e.g., \(\frac{\partial u}{\partial t} = \frac{1}{\varepsilon} (v-b(u))\) with a relaxation time \(\varepsilon>0\).

Literature.

• E. Di Benedetto, R.E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal. 12 (1981), 731--751.
• J. Merker, M. Krüger, On a variational principle in thermodynamics, Continuum Mechanics and Thermodynamics 25 (2013), 779-793.
• J. Merker, A. Matas, On doubly nonlinear evolution equations with nonpotential or dynamic relation between the state variables, preprint

We define \(\sigma\)-centred subforcings of the Blass-Shelah forcing order. Along a countable support iteration of length \(\aleph_2\) we construct an increasing system of maximal centred sets of pure conditions. Their projection to the set of subsets of \(\omega\) gives a non-rapid \(P_{\aleph_2}\)-point.

Guarded recursion is a form of recursion where the recursion variable is only allowed to appear guarded by a time step. The notion of time step is encoded in type theory by a modal type operator. Guarded recursion allows one to solve guarded variants of otherwise unsolvable type equations, and these have proved useful for modelling programming languages with advanced features inside type theory. Guarded recursion can also be used for constructing guarded variants of coinductive types, such as streams, and these can be used when constructing and reasoning about elements of coinductive types. In particular they can be used to encode the notion of productivity in types.

In the talk I will outline the use of guarded recursion in type theory and show how guarded recursion can be proved consistent with the univalence principles by constructing a presheaf model.

Lyapunov functions are functions whose orbital derivative is negative. They play a significant role in the stability analysis of equilibrium points of non-linear systems, since sublevel sets of a Lyapunov function are subsets of the domain of attraction of an equilibrium. However, constructing such functions is a challenging task. One of the numerical methods to construct Lyapunov functions is the RBF (Radial Basis Function) method.

We present two verification estimates to check the negativity of the orbital derivative of Lyapunov functions constructed by this method. The proposed estimates specify the density of the grid points in a given compact set, where we have to examine the sign of the constructed Lyapunov function which will in turn indicate the efficiency of the function.

We present and analyze cryptographic protocols which are based on Nielsen transformations.

One important driver of mathematical progress is the discovery of specific commonalities between structures, that allow for translating certain results in one to possibly novel results in the other. In fact, many theories - especially in the Bourbaki approach - are the result of extracting common principles of certain classes of structures, yielding e.g. the well-known hierarchical collection of basic algebraic theories such as groups, lattices, rings or vector spaces, all connected with each other by different translations, inclusions, reframings, etc. Consequently, it is a quite natural approach to see these as nodes in a theory graph and to look for useful operations extending it in an intuitive way; thus potentially giving rise to new interesting mathematical theories.

We will discuss specifically the concept of theory intersections along partial views (as a formal analogue to the process described above) and their current implementations in MMT (a module system for mathematical theories). By theory intersection, we mean: given two theories S and T, we would like to find "interesting" (for some adequate definition of the word) partial mappings v from S to T, yielding a corresponding common subtheory of v(S) and T. The actual question is consequently, how to find partial mappings between two given theories such, that the associated theory intersection becomes interesting.

In diesem Zeitschriftenprojekt bündeln sich vielfältige Motive unterschiedlicher Disziplinen sich in den 1920/30er Jahren systematisch der Mathematikgeschichtsforschung zuzuwenden und eine erste Institutionalisierung dieser Disziplin zu begründen. Dieses Vorhaben wurde ab 1926 von den Mathematikern Otto Toeplitz und Otto Neugebauer und dem Altphilologen Julius Stenzel vorbereitet, die erste Ausgabe erschien 1929 bei Springer. Die Mathematiker versprachen sich Orientierung in ihrer noch schwelenden philosophischen Grundlagendebatte und sie suchten für den immensen Studentenansturm der Zeit nach didaktischen Methoden für ihre Lehrtätigkeit. Otto Toeplitz entwickelte hierzu seine indirekt-genetische Methode in Abgrenzung zu Felix Kleins genetischer Methode. Von Seiten der Altphilologen hoffte man im Austausch mit Mathematikern auf Klärung der bislang umschifften dunklen Stellen in Platons Spätwerk.

Methodisch galt es über den Disziplinenrand hinaus zusammenzuarbeiten und in wechselseitigem Austausch von Mathematikern, Philosophen und Sprachwissenschaftlern, Klärung und neue Interpretationen zu erhalten und zudem jederzeit direkt an den historischen Quellen zu arbeiten und eigene Übersetzungen anzufertigen. Der politische Druck ließ das motivierte Projekt abrupt enden. Julius Stenzel war aus politischen Gründen 1933 strafversetzt worden und im Herbst 1935 verstorben, Otto Toeplitz erhielt ab Herbst 1935 Lehrverbot und wurde Ende 1938 als jüdischer Herausgeber denunziert, was letztlich Otto Neugebauer, der bereits 1934 emigriert war, dazu brachte, im Februar 1939 ebenfalls die Herausgeberschaft der Zeitschrift niederzulegen.

Throughout the literature, there are many equivalent descriptions of octonion algebras, in particular octonions are: (1) composition algebras, described by the Cayley-Dickson doubling process; (2) related to vector cross products, frequently discribed by the 'Fano plane mnemonic'; (3) related to a trilinear form in \(7\) dimensions; (4) constructible from root systems of type \(G_2\) as in the classification of semi-simple complex Lie algebras.

We will connect these constructions in a very concrete way, using an associated Clifford algebra and corresponding contraction operators. This point of view provides good insight into the structure of the Lie algebras and groups of type \(G_2\) (both split and anisotropic) while shunning long computations. In particular it helps us understand why certain exceptional behaviour arises in characteristics \(3\) (twisted Ree groups "\({}^2G_2\)") and \(7\).

It is an open problem to determine how many compatible symplectic forms a given Riemannian metric may admit. To understand this in low dimensions we study almost Hermitian \(4\)-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian \(4\)-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In this set-up we also fully describe almost Kähler \(4\)-manifolds.

Shallow water equations are a reasonable choice to describe the dynamics in the atmosphere or ocean. In this talk, we will show that different model order reduction techniques like Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM) can be applied to reduce the computational costs in further processing like data assimilation.

We report on joint work in progress with Akhil Mathew and Justin Noel. It permits to understand classical results like Quillen's F-isomorphism, Brauer induction and the Hopkins-Kuhn-Ravenel character theory on an equal footing and leads to new results.

In pioneering work, Thomason showed that Bott-inverted algebraic K-theory with finite coefficients admits étale descent. We present an entirely different proof of a slightly weaker result which relies crucially on the \(E_\infty\)-structure of algebraic K-theory and which generalizes to derived algebraic geometry. This is joint work with Akhil Mathew and Justin Noel.

Well quasi-orders are ubiquitous, through relational semantics, in many areas of philosophical logic such as provability, epistemic, temporal, and dynamic logics. The defining semantic conditions for such logics, given in terms of Noetherian or ancestral relations, cannot be expressed in a first-order language, but it is often possible to develop well-behaved proof systems. The talk will survey the results obtained (analyticity, semantic and syntactic completeness, decidability) and the current challenges.

We consider an optimal control problem governed by a phase-field fracture model. One challenge of this model problem is a non-differentiable irreversibility condition on the fracture growth, which we relax using a penalization approach. We then discuss existence of a solution to the penalized fracture model, existence of at least one solution for the regularized optimal control problem, as well as first order optimality conditions.

This is joint work with Thomas Wick and Winnifried Wollner

Motivated by the structure theory for sparse classes (see e.g. Sparsity, Springer 2012) we present several new WQO classes of graphs. The interpretation is a powerful tool for generating new WQO classes from old ones, we give several examples of such use.

This is a joint project with P. Ossona de Mendez.

Given a conic coisotropic submanifold \(B\) of a cotangent bundle of a manifold \(X\), Guillemin and Sternberg associated to it a certain algebra \(A\) of Fourier Integral Operators on \(X\). We will explain how the deformation of \(B\) relates to the full symbol calculus of the elements of \(A\) and show how to deduce an index theorem for Fredholm operators belonging to \(A\).

We present and analyze a new space-time parallel multigrid method for optimal control problems with parabolic equations as constraints. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time, and a finite element discretization in space. The key ingredient of the new algorithm is an inexact block Jacobi smoother. By using local Fourier mode analysis we determine asymptotically optimal smoothing parameters, a precise criterion for semi-coarsening in time or full coarsening, and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel, and show with numerical experiments its excellent strong and weak scalability properties.

Modeling of hydraulic fracturing (the injection of fluid at high pressures into the underground to create/widen fractures in the rock) requires the coupling of different physical processes, like rock deformation, fluid flow in the matrix and the open fracture. We present a 2d model that is described by the theory of poroelasticity and simulates the propagation of a single embedded fracture in a fully saturated, linear elastic, isotropic, porous material. Fluid flow within the matrix is given by Darcy's law and the open flow in the fracture is approximated by a parallel plate model. The used numerical method is the Extended Finite Element Method (XFEM). This way, no mesh-adapting step is needed when the geometry or location of the discontinuity changes. The coupling between the two domains is done via Lagrange multipliers. We discuss the implementation of the coupling during fracture growth and show examples of different fracture geometries and material properties.

We present the state of the art in the classification of terminal Q-factorial Fano threefolds that come with an effective action of a two-dimensional torus. To any such variety we associate its "anticanonical complex". The lattice points inside this complex control the discrepancies. This leads, amongst other, to simple characterizations of terminality and singularity. The explicit description of the anticanonical complex allows us to apply combinatorial and computational methods to the classification.

In 1978 R.M. Bryant and L.G. Kovács showed that for every subgroup \(H\) of the group \(\text{GL}(d,q)\) of invertible \(d\times d\) matrices with entries in a finite field of order \(p\), for a prime \(p\), there is a finite \(p\)-group \(P\) such that the automorphism group of \(P\) induces a subgroup isomorphic to \(H\) on a certain quotient of \(P\).

Their proof demonstrates the existence of such a \(p\)-group by considering sufficiently large quotients of a free group \(F\) on \(d\) generators.

In joint work with J. Bamberg, S. Glasby and L. Morgan we consider maximal subgroups of \(\text{GL}(d,p)\) for odd primes \(p\) and \(d\) at least \(4\). Guided by explicit computations of such groups in GAP and MAGMA we are able to prove the existence of the required \(p\)-groups and construct them algorithmically.

Implicit multistep/multistage approaches to a high-order accurate time integration seem well suited to be coupled with high-order DG space discretization of the unsteady compressible Navier-Stokes equa- tions. Implicit schemes, if accurate enough, can be very efficient and precise on highly-stretched grids even using large time step sizes. In this talk a short review of very accurate implicit methods is given and some issues of their implementation in the context of a matrix-free DG approach will be addressed. Furthermore, the influence of some physical (low Mach) and space discretization (anisotropic mesh) aspects on the performance of these schemes is discussed. The talk is completed by the presentation of recent numerical results of inviscid and viscous computations.

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the p-Laplacian converges, as \(p\) goes to \(\infty\), to a viscosity solution of a suitable eigenvalue problem for the \(\infty\)-Laplacian. We show among other things that the limiting eigenvalue is in fact the first nonzero eigenvalue, and derive a number consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.

This is a joint work with L.Esposito, B.Kawohl and C.Trombetti

Since the seminal work by Wadge in the 70s and 80s, a tradition has been established in descriptive set theory of using games to characterize certain important notions and classes of objects. Particular attention has been devoted to characterizing classes of functions in Baire space by games, with Wadge's game for continuous functions and Duparc's eraser game for the Baire class \(1\) functions as two important examples. In his PhD thesis, Brian Semmes introduced his tree game which characterizes the Borel measurable functions, and a restriction of the tree game which characterizes the Baire class \(2\) functions.

In this work, we show how to restrict Semmes's tree game in order to obtain games characterizing each finite Baire class, in a uniform way. The Wadge and eraser games are particular cases of our construction, but interestingly enough our construction for Baire class \(2\) gives a different -- though of course equivalent -- game than Semmes's.

The author would like to acknowledge that Alain Louveau and Brian Semmes proved the main result independently with a different proof.

Finite suspensions of Koras-Russell threefolds are contractible in \(A^1\)-homotopy theory.

For bounded symmetric domains in complex Euclidian Space there is a natural notion of areas of geodesic triangles; this is related to the Maslov index. In this lecture we shall explain this and also discuss some generalizations to other complex manifolds.

Let \(R\) be a regular local ring, \(K\) its field of fractions and \((V, \varphi)\) a quadratic space over \(R\). In the case of \(R\) containing a field of characteristic zero we show that if \((V, \varphi) \otimes_R K \) is isotropic over \(K\), then \((V, \varphi)\) is isotropic over \(R\). This solves the characteristic zero case of a question raised by J.-L. Colliot-Thélène in [C-T]. The proof is based on a moving lemma in algebraic cobordism of Levine-Morel.

Several approaches to treat thin-film type problems, i.e., degenerate fourth order parabolic equations, where the solution is only supported on part of the domain, will be reviewed. After discussing the effect of different degeneracies, an algorithm for this class of free boundary problems will be presented. The algorithm is used to compute solutions for different mobilities and for zero and non-zero contact angles in order to discuss the intricate behavior of the corresponding solutions (or their approximations) and as a validation for the algorithm, of course.

In causal inference, we often represent the causal structure of a data generating process with a directed graph. What do we use such causal models for? In many situations, we are interested in the system's behavior under a change of environment. Here, causal models become important because they are usually considered invariant under those changes. A causal prediction (which uses only direct causes of the target variable as predictors) remains valid even if we intervene on predictor variables or change the whole experimental setting. In this talk, we use data from different environments in order to estimate the causal structure and provide statistical guarantees. Reversely, we predict the models' behavior in different environments given the causal structure.

Driven by an overwhelming amount of applications numerical approximation of partial differential equations has established itself as one of the core areas in applied mathematics. During the last decades a trend for the solution of PDEs emerged, that focuses on employing systems from applied harmonic analysis for the adaptive solution of these equations. Most notably wavelet bases and also frames have been used and led for instance to provably optimal solvers for elliptic PDEs. Inspired by this success story also other systems with various advantages in different directions are currently beeing investigated in various discretization problems of PDEs. For instance, ridgelets where recently successfully used in the discretization of linear transport equations.

Maybe the most widely used anisotropic system today is that of shearlets, which admits optimally sparse representations of functions which are governed by discontinuities along smooth curves - so called cartoon like functions. This system constitutes a frame for \(L^2(\mathbb{R}^2)\). However, in order to apply such systems in adaptive discretization algorithms it is necessary to have a system on a bounded domain, which still yields a frame, is able to incorporate boundary conditions, and characterizes Sobolev spaces. Although there have been first approaches to construct shearlet systems for the solution of PDEs on bounded domains they fail to satisfy all the desiderata above. In this talk we will introduce a novel shearlet system that meets all the requirements mentioned above and admits optimal approximation rates for cartoon-like functions.

The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. A Bishop space is a pair \((X, F)\), where \(X\) is an inhabited set and \(F\), which is called a Bishop topology, or simply a topology, is a set of functions of type \(X \rightarrow R\) which includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop-continuous functions of type \(R \rightarrow R\). The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Since a morphism between two Bishop spaces and most of the topological notions related to Bishop spaces are defined in a function-theoretic way, and since all our proofs are within Bishop's informal system of constructive mathematics BISH, TBS is an approach to constructive point-function topology.

However remarkable the development of Homotopy Type Theory (HoTT) has been, we would like to provide a straightforward elementary counterpart of classical homotopy theory within BISH. We report on the current status of our reconstruction of basic homotopy theory within TBS. A similar study within formal topology was initiated by Palmgren in 2009. Since TBS is a function-theoretic approach to constructive topology, and since classical homotopy theory contains many function-theoretic concepts, it seems natural to try to develop such a reconstruction within TBS. If \((X, F)\) is a Bishop space, an \(F\)-path is a morphism from \([0, 1]\), endowed with the topology of the uniformly continuous functions, to \((X, F)\). In contrast to the "logical" character of paths in HoTT, not every Bishop space has the path joining property (PJP). We study the rich class of codense Bishop spaces, which generalizes the class of complete metric spaces in TBS, and we show that every codense Bishop space has the PJP. Also, the homotopy joining property holds for \((X,Y)\) when \(X\) is a Bishop space and \(Y\) is a codense Bishop space. Having these concepts as starting point, we translate some basic facts of the classical theory of the homotopy type into TBS.

We establish Balian-Low type theorems for Gabor spaces, that is, for spaces generated by a discrete set of time-frequency shifted copies of a single window function. Our results characterize windows that generate Gabor spaces which are invariant under time-frequency shifts that are not member of the space generating discrete set. Further, we observe that additional time-frequency invariance and good time-frequency decay of the window function are mutually exclusive properties. As generating sets, we consider symplectic lattices of rational density.

Joint work with Carlos Cabrelli and Ursula Molter.

During this talk we shall discuss the implementation of an analytic regularization scheme for time ordered products of quantum field theory on curved backgrounds. After discussing the general method and the ideas on which the method is based we shall present some concrete computations for analyzing interacting quantum field theory on cosmological spacetimes.

The classical Kronecker limit formula describes the derivative at \(s = 0\) of the non-holomorphic Eisenstein series for the modular group in terms of the Dedekind Delta function. In our talk, we will first recall how this formula can be used to compute the regularized determinant of the Laplacian on an elliptic curve. Then, we will discuss recent results for hyperbolic Riemann surfaces.

We consider some ideas to improve the well-known (inverse) FFT algorithm to compute a vector x from its Fourier transformed data. It is known that the FFT needs O(N log N) arithmetical operations. However, if the resulting vector x is a-priori known to be sparse, i.e., contains only a small number of non-zero components, the question arises, whether we can do this computation in an even faster way. In recent years, different sublinear algorithms for the sparse FFT have been proposed, most of them are randomized. We want to concentrate on deterministic sparse FFT algorithms and consider especially vectors with small support and sparse positive vectors. The talk is based on joint work with Katrin Wannenwetsch.

In a recent paper the authors have identified a new type of robust heteroclinic cycle for equivariant vector fields in \(\mathbb{R}^4\), which had been formerly mixed-up with the class of simple heteroclinic cycles. Simple cycles have well-known properties, in particular their asymptotic stability follows classical conditions on the eigenvalues of the Jacobian matrix evaluated at the equilibria which compose the cycle. In contrast pseudo-simple cycles appear to have different asymptotic behavior, being for example generically completely unstable if the symmetry group of the vector field belongs to \(\mathrm{SO}(4)\) and fragmentarily stable if the eigenvalues satisfy certain conditions and the symmetry group contains reflections in \(\mathbb{R}^4\). We present these objects and their properties.

A relational structure R is embeddable in a relational structure \(R'\) if \(R\) is isomorphic to an induced substructure of \(R'\). In the late forties, Fraisse, following the work of Cantor, Hausdorff and Sierpinski, pointed out the role of the quasi-ordering of embeddability and hereditary classes in the theory of relations. Recent years have seen a renewed interest for the study of hereditary classes particularly those made of finite structures. Many results have been obtained. Several are about their profile, i.e. the function \(\varphi_C\), the profile of a hereditary class \(C\), which counts for every integer \(n\) the number of members of \(C\) on \(n\) elements, counted up to an isomorphism. General counting results as well as precise results for graphs, tournaments, ordered graphs and permutations have been obtained, with a particular emphasis on jumps in the growth of the profile. I will illustrate the role of well quasi order in the classification of hereditary classes of small growth and conclude with several questions.

In the last decades a zoo of new symmetric periodic collision-less orbits for the n-body problem appeared in the literature as critical points of the Lagrangian action functional. Certainly one of the important features of such orbits, for a better understanding of the dynamics, is the knowledge of the Morse index as well as their linear (in)stability properties. A central device for computing this index is a Morse-type index theorem and a refined computation of the Maslov index. However, a key role in order to penetrate the intricate dynamics of this singular problem is represented by the collision orbits. In this talk, after a presentation of a new variational regularisation of the Lagrangian action functional, we will show how to define a suitable index theory for a special class of colliding trajectories. This is a joint work with V. Barutello, X. Hu and S. Terracini

We present a unified covariant multipolar framework for the description of test bodies in gravity. The framework covers a very large class of gravitational theories, and one can use it as a theoretical basis for systematic tests of gravity by means of extended deformable test bodies. The classes of theories covered range from simple generalizations of General Relativity, e.g. encompassing additional scalar fields, to theories with additional geometrical structures, which are needed for the description of microstructured matter. Furthermore, we discuss the impact of nonstandard couplings between matter and gravity on the resulting test body equations of motion.

Peridynamics is a nonlocal elasticity theory based on differences in the deformation instead of the deformation gradient. It is therefore suitable to describe long range forces as well as material failure. In this talk, we will give an introduction into the theory of peridynamics and consider its equation of motion as a nonlinear second order evolution equation. We present results on existence of weak and measure-valued solutions in the absence of any monotonicity assumption on the peridynamic operator. The method of proof also applies to other nonlocal partial differential equations.

The formalization and mechanical verification of mathematics in proof assistants is a growing trend in mathematics. However, current proof assistants employ incompatible logical foundations and libraries. This has the effect that all systems are mutually incompatible, and mathematical knowledge cannot be shared well across systems.

The MMT framework is a new approach aiming at overcoming this problem. MMT is a framework for representing logics, type theories, set theories, and similar languages in a uniform way. MMT achieves a high level of generality by systematically avoiding a commitment to a particular syntax or semantics. Instead, individual language features (e.g., function types, conjunction, etc.) are represented as reusable modules, which are composed into concrete languages. These modules can be declarative by specifying features as MMT theories or programmatic by providing individual rules as plugins.

Despite this high degree of abstraction, it is possible to implement advanced algorithms generically at the MMT level. These include knowledge management algorithms (e.g, IDE, search, change management) as well as logical algorithms (e.g., parsing, type reconstruction, module system).

Thus, we can use MMT to obtain advanced implementations of logical languages at extremely low cost. Moreove, the resulting applications are very scalable and optimized for interoperability and knowledge sharing.

Tokamaks are a promising design of fusion reactors and the analysis of model equations poses a number of challenges. For a simplified two-fluid description of an idealized tokamak plasma some basic experimental findings can be recovered. We view high and low temperature components of the electron velocity distribution as miscible phases and account for nonlinear effect by drift only. Then the global attractor for large temperature difference is a confined, laminar steady state, which destabilises for low temperature difference and viscosity. Stable spatio-temporal oscillations in the form of travelling waves bifurcate and a number of secondary bifurcations off the laminar state occur. Numerically, we find secondary bifurcations along the branches, which yield complex dynamics and intricate bifurcation scenarios. This is joint work with Delyan Zhelyasov (L'Aquila) and Daniel Han-Kwan (Palaiseau).

As a simple model for phase separation, the Allen-Cahn equation possesses stable front-type interface solutions that are heteroclinic orbits in the planar spatial ODE. The coupling to a second reaction-diffusion equation is known to generate cusp singularities of fronts and to produce oscillatory front bifurcations. We focus on weak coupling to one or more linear equation, similar to the FitzHugh-Nagumo system, which allows for explicit analyses of the heteroclinic bifurcations and the PDE-stability. In particular, linear coupling to two equations produces a butterfly singularity and nonlinear coupling allows for the imbedding of arbitrary singularities in the dynamics of the front velocity. This is joint work with Martina Chirilus-Bruckner, Arjen Doelman and Peter van Heijster.

Given a \(7\)-manifold \(M\), it admits a \(G_2\)-structure if the structure group of its frame bundle can be reduced to the exceptional Lie group \(G_2\) or, equivalently, if it admits a stable \(3\)-form \(\varphi\) from which it is possible to define a Riemannian metric and a volume form on \(M\). If \(\varphi\) is not closed but is locally conformal equivalent to a stable closed \(3\)-form, the \(G_2\)-structure is said to be locally conformal calibrated and represents the \(G_2\)-analogue of locally conformal symplectic structures on even dimensional manifolds. In this talk, I will discuss a structure result for compact \(7\)-manifolds endowed with a locally conformal calibrated \(G_2\)-structure. In detail, after recalling some preliminary results, I will show that under some suitable and natural hypothesis on the \(3\)-form \(\varphi\), the \(7\)-manifold is fibered over the circle and each fiber is a \(6\)-manifold endowed with a coupled \(\mathrm{SU}(3)\)-structure \((\omega, \psi)\), that is, an half-flat \(\mathrm{SU}(3)\)-structure for which the exterior derivative of the Kähler form \(\omega\) is proportional to the real part of the complex volume form \(\psi\). I will conclude giving some explicit examples.

It is a long standing open problem whether the Thompson group \(F\) is amenable. In this talk I will give a brief introduction to the three Thompson groups \(F\), \(T\) and \(V\) and their \(C^*\)-algebras and Von Neumann algebras. Then I will discuss the paper "A computational approach to the Thompson group \(F\)", which I wrote in collaboration with Uffe Haagerup and Soren Haagerup. Here we estimate the norms of certain elements of the reduced \(C^*\)-algebra of \(F\), that suggest that \(F\) might not be amenable.

Reverse mathematics (RM) is a program that strives to classify the logico-existential strength of theorems of "ordinary" mathematics by means of set existence principles, mainly as they appear in subsystems of second order arithmetic. The graph minor theorem, GM, is arguably the most important theorem of graph theory. The strength of GM exceeds that of the standard classification systems of RM known as the "big five". An upper bound not too far removed from the biggest of the five was claimed in the literature but later rescinded. In this talk I shall survey the current knowledge about the strength of GM and other Kruskal-like principles, presenting lower and upper bounds.

Given an ordinary differential equation, a set of initial states, and a set of states considered to be unsafe, a barrier is positively invariant set that contains the set of initial states but does not contain any unsafe state. Hence the existence of a barrier certifies that no unsafe state is reachable from an initial state. Classical techniques for computing global information for dynamical systems are interval methods and set-oriented numerics. In the talk, we will present a method for computing barriers that tries to combine the advantages of both approaches, together with first numerical experiments.

Emil Artin und Helmut Hasse sind in demselben Jahr geboren, Artin am 3. März 1898 in Wien und Hasse am 25. August 1898 in Kassel. Der Briefwechsel reicht bis in das Jahr 1923 zurück; der frühe Briefwechsel ist vor allem mathematischen Themen gewidmet, man arbeitete auf denselben Gebieten. In diesem Beitrag jedoch wird der Schwerpunkt auf die Nachkriegszeit gelegt. Der Briefwechsel wurde 1953 wieder aufgenommen. Artin, der 1938 Hamburg hatte verlassen müssen, wirkte seit 1946 in Princeton. Erste Kontakte Artins zu Hamburg nach dem Krieg gab es bereits seit dem Jahre 1946. Helmut Hasse bekleidete seit 1950 eine Professur an der Universität Hamburg; seinen Bemühungen, war es schließlich zu verdanken, dass Artin 1958 nach Hamburg zurückkehrte und am 1. Oktober 1958 an der Universität wieder eine Professur übernahm. Diese war für ihn neu geschaffen worden. Artin gehörte damit zu den ganz wenigen Professoren, die wieder nach Deutschland zurückkehrten. Der Briefwechsel zwischen den beiden Mathematikern macht deutlich, wie sich zwischen 1953 und 1958 eine über die wissenschaftlichen Belange hinausgehende Freundschaft entwickelte. Diese führte dazu, dass aus dem Lieber Herr Hasse/Lieber Herr Artin schließlich ein Lieber Hasse/Lieber Artin wurde und man zum Du überging. In Hamburg erlebten sowohl Artin wie auch Hasse eine sehr fruchtbare Zeit; leider verstarb Artin bereits 20. Dezember 1962, Hasse am 26. Dezember 1979. Die BMGN-Bibliothek in Hamburg verfügt über zahlreiche Mitschriften von Artins Vorlesungen. Und Hasse betreute in Hamburg 10 Doktorarbeiten.

Riemannian manifolds with holonomy \(G_2\) are an active research topic in differential geometry that has applications in theoretical physics, too. Most of the known non-compact, complete \(G_2\)-manifolds are of cohomogeneity one; i.e., they admit an isometric action whose generic orbits have codimension one. We study a special class of \(\mathrm{SU}(2)^2\)-invariant metrics of that kind and investigate if they have infinitesimal deformations that do not change the holonomy. This question is equivalent to an eigenvalue problem for a differential operator on the orbit \(\mathrm{SU}(2)^2\). With help of the Peter-Weyl theorem our problem can be simplified even further. The spectrum of our operator coincides with the eigenvalues of an infinite series of matrices. Since we are interested only in the lowest eigenvalues, our question can be answered by numerical methods.

In recent work, we found a new proof of a conjecture due to Erdős and Sós stating that large quasirandom \(3\)-uniform hypergraphs that are weakly quasirandom with density greater than \(\frac{1}{4}\) contain four vertices spanning at least three hyperedges. This was proved earlier by Glebov, Kral, and Volec with the help of flag algebras and computers. The new proof is based on the hypergraph regularity method and gave rise to further developments in this field that are surveyed in this talk.

Coxeter elements play an important role in the theory of finite Coxeter groups. One can generalize their definition and call an element \(c\) in a Coxeter group \(W\) "Coxeter element" if \(c\) is maximally regular in the sense of Springer theory. In this talk, I will show that an element \(c\) in \(W\) is a Coxeter element in this generalized sense if and only if there exists a simple system of reflections such that \(c\) is the product of the generators in this simple system.

If time permits, I will also provide analogous statements for Shephard groups and for complex reflection groups, and I will also show that this general definition yields a simple transitive action of the Galois group of the field of definition on the set of conjugacy classes of Coxeter elements.

(Joint work with Heiko von der Mosel and Henryk Gerlach.) In order to investigate the elastic behavior of knotted loops of springy wire, we minimize the classic bending energy regularized by ropelength, i.e., the quotient of length over thickness, in order to penalize self-intersection. Our main objective is to characterize the limit configurations of energy minimizers as the regularization parameter tends to zero, which will be referred to as elastic knots.

The elastic unknot turns out to be the round circle. In all non-trivial knot classes where the natural lower bound \((4\pi)^2\) for the bending energy is sharp, any elastic knot is shown to belong to the one-parameter family of tangential pairs of identical circles, where the parameter is the angle in between the circles ranging from \(0\) to \(\pi\).

Finally, for every odd \(b > 1\) and the respective class of \((2,b)\)-torus knots (containing the trefoil) we obtain a complete picture showing that the respective elastic \((2,b)\)-torus knot is the twice covered circle.

The causal perturbation theory approach to renormalization, based on the seminal paper of Epstein and Glaser from 1973, is a mathematically rigorous framework which allows to study foundations of perturbative QFT. In this talk I will explain how BV algebras arise naturally in this construction. The physical motivation is the study of gauge theories. Mathematically, the construction which I present allows to obtain interesting examples of BV algebras from a class of differential Gerstenhaber algebras.

We review the hyperbolic circle problem and explain some recent results concerning the error term. We also explain how these results relate to L-functions of certain automorphic forms.

We consider a diffuse interface model for tumor growth proposed by Hawkins-Daarud, van der Zee and Oden. This model consists of the Cahn-Hilliard equation for the tumour cell fraction nonlinearly coupled with a reaction-diusion equation for the nutrient-rich extracellular water volume fraction. We shall first present a result on the existence of a weak solution, then we show that the weak solution is unique and continuously depends on the initial data. Furthermore, we shall give a result on the existence of a strong solution that allows to show that any weak solution regularizes in nite time. The last results will be on the existence of the global attractor in a phase space characterized by an a priori bounded energy and on some rigorous asymptotics.

Joint works with P. Colli, S. Frigeri, M. Grasselli, G. Gilardi, J. Sprekels.

In dimension three, the Shilnikov model of a homoclinic cycle to a saddle-focus is one of the most famous and rich examples in the dynamical systems theory, in which a simple configuration generates a very complicated behaviour around the neighbourhood of the cycle. Concerning the study of chaos arising from the presence of rotating nodes, the next big challenge is the study of cycles involving a bifocus in dimension four. In a homoclinic network associated to a non-resonant hyperbolic bifocus, we prove that the rotation combined with a non-degeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, creates switching behaviour. Trajectories that realize switching lie on suspended hyperbolic horseshoes that accumulate on the network. This is a joint work with Santiago Ibanez from Oviedo University (Spain).

We present a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these can even be zero, i.e. we prove uniqueness of solutions to continuity equations. Also new existence results are proved. Applications to proving well-posedness of Fokker-Planck-Kolmogorov equations associated with SPDEs and of continuity equations associated with PDE are discussed.

This is joint work with Vladimir Bogachev, Giuseppe Da Prato and Stanislav Shaposhnikov

In joint work with Markus Spitzweck and Paul Arne Østvar, we study the spectral sequence based on Voevodsky's slice filtration. This filtration on the stable homotopy category of motivic spectra over a field F measures the amount of Tate suspensions which are necessary to construct a given motivic spectrum. Work of Levine and Voevodsky shows that the slices of the motivic sphere spectrum are determined by the second page of the topological Adams-Novikov spectral sequence. We use this information to compute the zeroth and the first stable motivic homotopy groups of spheres over fields of characteristic zero. This supplies an independent proof of Morel's identification of the zeroth stable stem with the graded Milnor-Witt K-theory of the base field. The first stable stem is described as an extension of a Milnor K-theory group modulo \(24\) and the image of the unit map for hermitian K-theory. An important ingredient are convergence results for the slice spectral sequence of cellular motivic spectra of finite type.

Waldhausen's algebraic K-theory machine can be applied to suitable categories appearing in motivic homotopy theory. The talk will discuss some properties of the resulting homotopy types. For example, the path components will be related to the Grothendieck ring of varieties.

Chemotaxis describes a biological phenomenon of self organization and pattern forming of cell populations caused by chemical substances. It can be modelled by a two component reaction diffusion system in which the equations are coupled by a quasilinear cross-diffusion term. In this talk, we consider an optimal control problem with Neumann boundary control for the chemoattractant.

This is joint work with Hendrik Feldhordt and Michael Winkler

Im Kontext des Lernens von Mathematik im Schulunterricht spielen mentale Repräsentationen für den Aufbau adäquater Grundvorstellungen bei Schülerinnen und Schülern eine zentrale Rolle. In diesem Vortrag wird anhand des Modells von Concept Image - Concept Definition das herausfordernde Wechselspiel zwischen formalen Aspekten der mathematischen Theorie und individuell unterschiedlichen Vorstellungen der Lernenden am Beispiel von Visualisierungen im Oberstufenunterricht aufgezeigt. Eine Schlüsselfrage ist dabei, inwiefern Lehrerinnen und Lehrer in ihrer fachlichen und fachdidaktischen Ausbildung für diese Problematik sensibilisiert werden können, um umfassende begriffliche Vorstellungen bei den Schülerinnen und Schülern zu fördern und Fehlvorstellungen entgegenzuwirken.

I will start with a result on \(U_q(\mathfrak{sl}_n)\)-representations: I will give an explicit resolution of every simple \(U_q(\mathfrak{sl}_n)\)-module in terms of tensor powers of the fundamental representations.

Then I will recall the construction of Queffelec and Rose of the \(\mathfrak{sl}_n\)-link homology and explain how to use the first result to obtain an homology theory which decategorify on the colored \(\mathfrak{sl}_n\)-invariant for framed links.

Symmetries are abundant in PDEs from applications. In this talk we show how symmetries can be used to approximate similarity solutions to PDEs by direct long-time simulations. The basic approach is to separate the behavior of the solution into two parts, one describing the evolution of the solutions shape and the other describing the movement of the solution in a symmetry-group, that is related to the problem. This approach leads to a partial differential algebraic equation that has to be solved numerically.

As a simple example we consider the viscous and inviscid Burgers' equation and generalizations of it to more than one spatial dimension. In this case the symmetry includes spatial shifts and scalings of space and time and can be described by (non-)abelian Lie-groups.

\(D\)-modules are algebraic counterparts of systems of linear differential equations and play an important role in algebraic geometry, singularity theory and representation theory. The foundations of the theory were laid by Kashiwara and others. Mebkhout described the formalism of Grothendieck's six operations for \(D\)-modules. One of them, the direct image, corresponds in some sense to integration of differential forms. In this talk we describe an algorithmic approach to the \(D\)-module direct image extending ideas of Oaku/Takayama and Walther.

A solid and distinctive online record of a scientist's research achievements is nowadays almost inevitable to advance one's academic career. This necessitates a reliable presentation of a scientist's research achievements, in particular publications. However, authorship identification is a nontrivial problem for various reasons: author related data contained in a publication may be incomplete (abbreviated or missing name parts) or incorrect (typos or transliteration problems), or a given author may publish under different names. Author disambiguation is a challenge that can be taken up only with a proper mixture of identification algorithms and manual curation. One approach is trying to grab pieces of information everywhere, from any Internet service providing author profiles (like ORCID, MGP, ResearchGate, Mathnet.ru, ...). Through a reliable matching of the corresponding author profile, one can fetch biographical or bibliographical information which, in return, allows to extend and refine the authorship disambiguation of the given author profiles. The talk gives an overview of the recent developments in zbMATH, in the directions of algorithmic and manual disambiguation of author profiles. In particular, we will present our 1-year-old graphical interface through which zbMATH users can have a direct impact on author profiles. We will also discuss which linkings to and collaboration with other services zbMATH uses for its authorship disambiguation process.

Based on the deterministic radial basis interpolation method and the sums of square decomposition we discuss the construction of Lyapunov functions for asymptotically stable equilibria in dynamical systems generated by random and stochastic differential equations.

We investigate optimal control problems for some reaction diffusion equations, where patterns of traveling wave fronts, impulses, spiral waves, and other phenomena appear. In particular, we discuss the consideration of pointwise state constraints. We derive first-order necessary optimality conditions for the associated control problem and present various numerical examples.

It is a classical result that for any fixed \(N\) and any prime \(p\), there are only finitely many congruence classes of eigenforms level \(N\) modulo \(p\), and these occur in weights bounded by a constant depending only on \(p\). The situation modulo higher prime powers is however unclear. In this talk, I will explain the problem, and state results on the existence of weight bounds recently obtained in a joint work with Ian Kiming and Gabor Wiese. I will also discuss computer experiments that may shed light on the nature of these bounds.

Let a torus act on a compact symplectic manifold with isolated fixed points. In this talk I will discuss about recent results concerning the classification of the topological invariants of such a manifold, including equations involving the Chern numbers of the manifold, depending on its minimal Chern number. This includes both the Hamiltonian and non-Hamiltonian case.

I will describe the results of a joint paper with K. Ishige and K. Nakagawa. In this paper we start the investigation of concavity properties of solutions to systems of PDE's in convex domains. In particular we prove that suitable powers of solutions to some weakly coupled elliptic and parabolic systems are concave.

We prove that on non-compact harmonic spaces, the Abel transform and its dual are topological isomorphisms. Relying on L. Schwartz's classical result on mean periodic functions we then derive that functions satisfying the mean value property for two generic radii must be harmonic. Moreover, functions with vanishing integrals over all spheres (or balls) of two generic radii must be identically zero.

The phenomenon of black hole radiation (Hawking, 1975), which is still only partially understood, provides an interesting connection between the behaviour of quantum matter and the classical geometry of a background spacetime.

For a simplified description at late times, where the black hole has settled down to a stationary state, one conjectures that the quantum matter is in a kind of ground state, which is well-behaved across the black hole horizon. Restricting this state to the region outside the black hole yields a thermal (KMS) state at the Hawking temperature (Hartle and Hawking, 1976; Israel, 1976).

The conjectured existence of the ground state was argued to be false for certain stationary black holes by Kay and Wald (1991). However, we will discuss some aspects of the first general existence proof (2015), which concerns free scalar fields in a class of static spacetimes with a bifurcate Killing horizon (including the Schwarzschild black holes). The proof combines detailed methods from geometry and analysis (local and global).

In a seminal paper published in 1952, Alan Turing proposed a mechanism for the development of spots and stripes on animal coats that relies on the spontaneous formation of spatially periodic patterns based on diffusion and reaction of chemicals. These ideas have been tremendously influential not only in morphogenesis but also in other areas of biology, chemistry, and physics. I will give an overview of Alan Turing's original idea, its mathematical manifestations, and its success in explaining many pattern-forming processes in nature. I will also discuss recent efforts to model stripe formation on zebrafish that involve Turing's mechanism.

In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding non-trivial homology classes of these compactifications is related to finding non-trivial string operations. However, the homology of these complexes is largely unknown.

In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams. In the genus zero case, I'll give a reinterpretation of it in terms of weighted partitions, give some computational results, connectivity results and some conjectures and open problems.

This talk is based on joint work with F. Lutz.

Skew Howe duality for the general linear group enables to describe the braiding of \(gl(m)\)-representations using a dual Lie algebra \(gl(k)\). This has proven to be extremely useful, in particular in the quantized setting, for the combinatorial study as well as for categorification purposes.

In the talk, I will present an extension of skew Howe duality in which the vector representation of the Lie superalgebra \(gl(m|n)\) and its dual appear at the same time. This relates finite-dimensional representations of \(gl(m|n)\) with infinite-dimensional representations of a dual Lie algebra \(gl(k)\). All of this has a natural quantized version and should hopefully be of great help for understanding categorification of \(gl(m|n)\)-representations.

(This is joint work with H. Queffelec)

In one variable, Prony's method is a well-known procedure to reconstruct sparse exponentials from integer data. The talk concerns its multivariate analog where ideal bases have to be computed from kernels of certain Hankel matrices. By means of homogeneous H-bases this can be done in a fairly fast and numerically quite stable way where it turns out that, though computationally more challenging, an increased number of variables stabilizes things significantly. This is a typical behavior of polynomials where numerical issues often depend in a moderate way from the total degree and not the number of coefficients.

In this talk we derive some basic properties of a certain mapping obtained via Lévy mixing. Using this, we study the class of infinitely divisible distributions obtained by subordinating a Lévy process through a subordinator. We show that this class is closed under convolutions and it is in a bijection with the family of infinitely divisible distributions whose support is contained in \((0,\infty)\) (subordinators). In particular, we use our results to solve the so-called recovery problem for Lévy bases as well as moving average processes which are driven by subordinated Lévy processes.

This talk is based on a joint work with Almut Veraart.

When we consider surfaces of prescribed mean curvature \(H\) with a one-to-one orthogonal projection onto a plane, we have to study the nonparametric \(H\)-surface equation. Now the \(H\)-surfaces with a one-to-one central projection onto a plane lead to an intricate elliptic differential equation which is derived in §1; in the case \(H=0\) this p.d.e. has been invented by T.Radó. We establish the uniqueness of the Dirichlet problem for this H-surface equation in central projection in §2. Moreover, we develop an estimate for the maximal deviation of large \(H\)-surfaces from their boundary values, resembling an inequality by J.Serrin. In §3 we provide a Bernstein-type result for the case \(H=0\) with methods from the book on Minimal Surfaces by U. Dierkes, S. Hildebrandt, and F. Sauvigny; thus we can classify the entire solutions of the minimal surface equation in central projection. Furthermore, we solve the Dirichlet problem for \(H=0\) by a variational method. In §4 we construct solutions of the Dirichlet problem for nonvanishing H by the deformation method and an approximation. Finally, we investigate the boundary regularity under a suitable curvature restriction.

Elastoplastic deformations play a tremendous role in industrial forming. Many of these processes happen at non-isothermal conditions. Therefore, the optimization of such problems is of interest not only mathematically but also for applications.

In this talk we will present the analysis of the existence of a global solution of an optimal control problem governed by a thermovisco(elasto)plastic model. We will point out the difficulties arising from the nonlinear coupling of the heat equation with the mechanical part of the model. Finally, we will discuss first steps to show the directional differentiability of the control-to-state mapping and to obtain necessary optimality conditions.

The talk is based on joint work with Roland Herzog and Christian Meyer.

The complexity of coloring graphs without long induced paths is a notorious problem in algorithmic graph theory. An especially intriguing case is that of \(3\)-colorability. Here, the state of the art is our recent poly-time algorithm to solve the problem on graphs without induced paths on seven vertices, so-called \(P_7\)-free graphs.

So far, much less was known about certification in this context. We prove that there are \(24\) minimally non-\(3\)-colorable graphs in the class of \(P_6\)-free graphs, and give the complete list. In particular, we obtain a certifying algorithm for \(3\)-coloring graphs in this class.

We also show that our result is best possible, in the following sense. If \(H\) is a connected graph that is not an induced subgraph of \(P_6\), then there are infinitely many minimally non-\(3\)-colorable \(H\)-free graphs.

Joint work with Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Maya Stein, and Mingxian Zhong resp. Maria Chudnovsky, Jan Goedgebeur, and Mingxian Zhong

The string diagrammatic calculus is a well established tool for computations in linear algebra. A 3-dimensional topological field theory with defects can be regarded as a categorification of such a calculus: It encodes operations of various compositions, of generalized traces and center constructions.

As example we show how to compute the value of a cylinder with a defect line in a state-sum description of a Turaev-Viro theory. This produces known algebraic relations between Drinfeld centers of fusion categories. This is joint work with Christoph Schweigert and Jürgen Fuchs.

Quadratic alternative algebras are connected with the geometries of the Freudenthal-Tits Magic Square. Traditionally, there is a split and a nonsplit version of this square, and for the first column, these two version coincide. We discuss a possibility to introduce "degenerate" versions of the Magic Square that connect the split with the nonsplit version. This would turn the square into a triangular prism: the \(i\)-th column would admit \(2i - 1\) different versions, and hence the last column would correspond to \(7\) different non-associative but possibly degenerate Cayley-Dickson algebras. The corresponding geometries are projective remoteness planes.

Higson and Roe have used homological algebra over \(C^*\)-algebras to map the surgery exact sequence for smooth manifolds to an exact sequence of K-theory groups of \(C^*\)-algebras (the latter containing as particular case the Baum-Connes assembly map). Jointly with Paolo Piazza, we have developped an appropriate secondary large scale index theory to directly construct all the maps involved in terms of higher index theory of the signature operator. This allows in particular to extend the result to the topological category. We present this result. To obtain numerical results we show how one can systematically map further to cyclic homoloty groups to obtain numerical invariants.

Die Mathe-Wichtel stammen aus dem DMV-Schülerwettbewerb, bekannt als "Mathe im Advent". Weltweit nehmen inzwischen jährlich 150.000 Schüler/innen, 6.500 Lehrer/innen mit 10.000 Klassen und 5.000 Spaßspieler/innen daran teil. 2014 war "Mathe im Advent" ein BMBF-Projekt im Wissenschaftsjahr 2014—Die digitale Gesellschaft. 30.000.000 Seitenaufrufen im Advent (page impressions) und eine durchschnittlichen Verweildauer von 8 Minuten auf der Webseite bestätigten die Beliebtheit des digitalen Angebots.

Die humorvollen Aufgaben, Lösungen, mathematischen Exkursionen und Blicke über den Tellerrand dieses Wettbewerbs geben einen wunderbaren Einblick in die Vielfalt der Mathematik. Sie fördern den mathematischen Entdeckungsdrang und das kreative Weiterdenken auch in benachbarte Disziplinen auf spielerische Weise. So erweitern sie das in der Schule vermittelte Bild der Mathematik und begeistern selbst diejenigen, die mit ihr bisher auf Kriegsfuß standen. An dem Wettbewerb können Schüler/innen, Lehrer/innen der Grund- und Sekundarstufe I und Spaßspieler/innen jeden Alters teilnehmen. 2014 sind über die schönsten Aufgaben zwei Bücher im Springer Spektrum Verlag erschienen.

Die Referentin stellt den Weg zum erfolgreiche Konzept des Schülerwettbewerbs vor und analysiert die Stärken und Schwächen des Online-Angebots, die aus 600.000 Emails der Teilnehmer pro Saison gewonnen wurden. Sie erläutert zudem das didaktische Konzept, welches hinter den Aufgaben, Lösungen und Zusatzangeboten steckt.

Suppose that \(A\) is a Dedekind finite set in a model of set theory without choice. We consider the possible sizes which \(A\) might have in generic extensions. For instance, we show that it is consistent that the injective collapse of any Dedekind finite set \(A\) to an infinite cardinal preserves cardinals, and therefore \(A\) may be arbitrarily large in cardinal preserving extensions. This is a joint project with Asaf Karagila.

Viele mathematische Vorlesungen und Seminare an Hochschulen setzen das Verständnis mengentheoretischer Grundlagen voraus. Diese Aspekte werden im Schulunterricht meist nur auf einer intuitiven Ebene behandelt, so dass hier gegenüber den Anforderungen der Hochschule eine Lücke besteht. Das kann für Schulabsolventen zu einem Hindernis werden, die an Hochschulen und im wissenschaftlichen Bereich verwendeten mathematischen Formalismen nachzuvollziehen. Es soll die Fragestellung diskutiert werden, inwieweit die Grundlagen der Mengenlehre im Schulunterricht vermittelt werden können, um diese Lücke zu schließen.

Magnetic materials have the special property that they react to applied external fields in remarkable ways and have therefore many technological applications. They can not only be found in medical applications, but, for example, also in loud speakers and shock absorbers.

We propose a model for micromagnetic materials in the framework of complex fluids. The system of PDEs to model the flow of the material is derived in a continuum mechanical setting from variational principles including the least action principle and the maximum dissipation principle. We outline this process of modeling and the energetic variational approach. Moreover, we highlight the coupling between the elastic and the magnetic properties of the material.

The obtained model reflects most of the phenomena represented in micromagnetic fluids but, on the other hand, is rather complex. Therefore, we provide also a simplified version of the model that is amenable for analysis but applies only to some particular flow regimes. As an illustration, we will concentrate on the two dimensional case where an explicit ansatz for the solution of the magnetization can be found. With this ansatz we simplify the model even further and show existence of weak solutions.

This is joint work with Johannes Forster (Institute for Mathematics, University of Würzburg, Germany), Carlos García-Cervera (Mathematics Department, University of California, Santa Barbara, USA), and Chun Liu (Department of Mathematics, Penn State University, University Park, USA).

Well quasi-orders and Kruskal's Tree Theorem are nowadays being applied in several areas of computer science, a development which perhaps no-one expected in the seventies. I shall sketch what led me to study these topics then, but also who was there before me and laid vital foundations, and what their backgrounds and objectives were.

We consider spatially periodic pattern at the Eckhaus boundary. We explain the occurrence of a waiting time phenomenon for solutions which are pinned for \( x \to -\infty \) at an Eckhaus stable equilibrium and for \( x \to -\infty \) at the equilibrium at the Eckhaus boundary. Secondly, we prove the nonlinear diffusive stability of the equilibrium at the Eckhaus boundary.

Pyragas control is a widely used time-delayed feedback control for the stabilization of periodic orbits in dynamical systems. In this talk we investigate how we can use equivariance to eliminate restrictions of Pyragas control, both to select periodic orbits for stabilization by their spatio-temporal pattern and to render Pyragas control possible at all for those orbits. Another important aspect is the optimization of equivariant Pyragas control, i.e., to construct larger control regions. The ring of \(n\) identical Stuart-Landau oscillators coupled diffusively in a bidirectional ring serves as our model.

In der Mathematikdidaktik und -geschichte dominiert der Gegensatz von abstraktem mathematischen Inhalt versus seine Veranschaulichung und Visualisierung. Was die das Denken begleitenden mathematischen Bilder als mathematische ausmacht und von anderen Phantasien wesentlich unterscheidet, ist dabei nicht geklärt.

In dem Vortrag wird eine Theorie mathematischer Vorstellungsbildung vorgestellt, die aus der pädagogischen Praxis abstrahiert wurde, aber philosophisch begründet wird. Mit ihr ergibt sich eine enge und direkte Verbindung zwischen mathematischem Phantasiebild und gedanklichem Inhalt, ein neues Verständnis von dem Verhältnis der euklidischen Definitionen und Postulate in den Elementen und einige grundlegende Anregungen für den Mathematikunterricht.

Was glaubt ein Hochschullehrer noch vom Schulalltag zu wissen und was ein Schullehrer von den Anforderungen einer Hochschulausbildung? Wir stellen die Erfahrungen aus mehr als 10 Jahren einer gegenseitigen Kooperation vor und entwickeln ein Modell, um dieses Konzept auf der Basis der vorgestellten mathematischen Brückenkurse nachhaltig zu implementieren.

I present a general characterization of the obstructions for higher WZW-terms (higher gerbes with connection) defined on some higher (or derived) group stack \(G/H\) to have a parameterization over higher Cartan geometries locally modeled on \(G/H\). Applied to the canonical Kostant-Souriau line bundle the construction reproduces metaplectic pre-quantization. For the traditional degree-\(3\) WZW term it reproduces the Green-Schwarz anomaly; for the degree-\(7\) WZW term we get a Fivebrane-analog, for the degree-\(11\) term a Ninebrane-analog. Applied to the exceptional cocycles on extended super-Minkowski spacetimes the construction yields a forgetful innity-functor on globally dened (classical anomaly free) Green-Schwarz super \(p\)-brane sigma models propagating on higher super étale stacks, which sends these to \(G\)-structures on these super stacks, for \(G\) the higher Heisenberg group stack of the higher WZW term. Specically for the super-\(5\)-brane sigma-model this yields a forgetful innity-functor from its classical anomaly free backgrounds to super etale \(3\)-stacks satisfying the equations of motion of \(11\)-dimensional supergravity and satisfying a further topological constraint.

In 1991 Lawvere suggested a) that the future of category theory revolves around toposes equippped with an adjoint system of idempotent (co-)monads [1] and that b) this is formalization of what the ancients had called the "objective logic" [2]. While for 1-toposes this seems inconclusive, one finds [3] that internal to infinity-toposes equipped with such adjoint systems much of higher differential geometry and of modern physics has a succinct and useful synthetic formalization. But here the syntax of this internal language is modal homotopy type theory [4]. In this talk I survey the immensely rich semantics and the potential prospects of its full syntactic formalization, in the hope to motivate the type theory community to further look into this fascinating but under-explored aspect of their theory.

1. http://ncatlab.org/nlab/show/Some+Thoughts+on+the+Future+of+Category+Theory
2. http://ncatlab.org/nlab/show/Tools+for+the+advancement+of+objective+logic
3. http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos
4. http://ncatlab.org/schreiber/show/Quantum+gauge+field+theory+in+Cohesive+homotopy+type+theory

A recent joint result with Asger Törnquist is that both in the Sacks and the Miller extensions of L, there is a co-analytic maximal orthogonal family. Orthogonal families are but one example of what I will here call maximal independet sets. Many other examples were discussed in a classic paper of Miller: most notably, Hamel bases for the real numbers, 2-point sets, and maximal independent subssets of the natural numbers; aprés Miller, more examples have been studied. Unfortunately, the result of Törnquist-S wasn't general enough to apply to these latter examples of maximal independent families. In this talk, I shall generalize our previous result to Hamel bases and some of the other examples.

The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly nonlinear theories can be constructed using higher order expansion terms. While these methods undoubtedly have great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of the extension of classical stability analysis to a nonlinearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the nonlinear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient \(\Delta T\). We explore three settings of \(\Delta T\), representative of relatively weak turbulence, well-developed turbulence and intermediate conditions.

We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition for instability. Barotropic instability is present only for a few very unstable CLVs for large values of \(\Delta T\). Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following the classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmospheric dynamics.

Schubert, S., & Lucarini, V. (2015). Covariant Lyapunov vectors of a quasi-geostrophic baroclinic model: analysis of instabilities and feedbacks. to appear in: Quarterly Journal of the Royal Meteorological Society (doi:10.1002/qj.2588).

In this talk, we present recent results on efficient time integration for the hybridized discontinuous Galerkin method. As this method necessitates implicit time-stepping, it is desirable to keep the effort per time-step as low as possible, while retaining a given order of temporal accuracy. The class of multiderivative ODE integrators in general yields higher accuracy while retaining less internal stages in comparison to standard Runge-Kutta methods. Coupling those integrators to the HDG method locally increases the number of spatial unknowns. These, however, can be accounted for quite efficiently within this framework.

A random geometric graph is constructed by connecting two points of a Poisson process in a compact convex set whenever their distance does not exceed a prescribed distance. The aim of this talk is to investigate the asymptotic behaviour of the total edge length or, more general, sums of powers of the edge lengths of this random graph as the intensity of the underlying Poisson process is increased and the threshold for connecting points is adjusted. Depending on the interplay of these two parameters one obtains limit theorems where the limiting distribution can be Gaussian, compound Poisson or stable. This talk is based on joint work with Laurent Decreusefond, Matthias Reitzner and Christoph Thäle.

In an operator algebra furnished with an anti-linear involution it is possible to consider Lagrangian projections which specify KR-group elements. Pairing such projections with KR-cycles leads to index theorems which can be \(2\mathbb{Z}\)- or \(\mathbb{Z}_2\)-valued. The heart of the argument is based on methods from symplectic linear algebra and leads to a new type of Kramers degeneracy argument. Applications concern topological condensed matter systems.

Let \(C\) be a Tarski-style mono-conclusion consequence relation on a monoid \((S,*,1)\) that satisfies the counterpart of disjunction elimination, with \(*\) in place of disjunction. Let \(E\) be the multi-conclusion entailment relation, in the sense of Scott, that extends \(C\) and captures what it means that \(*\) and \(1\) are viewed as disjunction and absurdity, respectively. Then \(E\) and \(C\) have the same mono-conclusion sequents. In particular, we eliminate the Krull-Lindenbaum Lemma, the appropriate instance of Zorn's Lemma, from otherwise elementary proofs. Instances include that the theory of integral domains is conservative, for definite Horn clauses, over the theory of commutative reduced rings. Our approach appears to be somewhat more direct and/or more general than the related results obtained with dynamical algebra.

The classical approach to measure the descriptive complexity of a subset of a topological space is to use ordinals to estimate the number of iterations of suitable set-theoretic operations needed to obtain the set from the open sets. In contrast, ordinals are often not sufficient to measure the descriptive complexity of functions between topological spaces in an appropriate way. Some useful classifications of functions are obtained by employing natural well quasi-orders, in particular those arising from suitable embeddings between labeled trees.

In this talk, we survey recent results in the specified direction, considering some such classifications from descriptive set theory and automata theory.

We present a classification of compact homogeneous spaces with positive Euler characteristic admitting an almost complex structure and more generally a tangent bundle which is stably complex. We show that such a space is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure, or have stably trivial tangent bundle, or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures. The talk is based on joint work with Andrei Moroianu.

Cardinal characteristics such as unbounding and dominating number have been being studied for long time and have applications to real numbers, connections to topology on the Baire and Cantor spaces, regularity properties of subsets of reals and tree forcing notions. However, many of them are defined in a purely combinatorial way which allows natural generalizations to an arbitrary uncountable cardinal kappa. Inspired by the standard work systematically dealing with the classical countable case being done by A. Blass (Combinatorial Cardinal Characteristics of the Continuum) I would like to give an overview of the current state of the art for the generalized case, remarking the similarities and deviations from the countable case.

Tropical geometry provides polyhedral models of algebraic varieties over fields. Often invariants of the classical geometric objects are translated to the combinatorics of these models.

We will look at how a computational approach has aided in specific problems surrounding moduli spaces of tropical and classical curves and surfaces.

The adaptive grid method is presented for numerical modelling of impulse water waves generated by submarine landslides moving along irregular bottom profiles and consecutive run-up of the generated waves on the coast. A nonuniform submarine landslide moving on a nonuniform slope is modelled by a "quasi-deformed" rigid body [1]. The surface water waves are simulated using the hierarchy of mathematical models including the nonlinear dispersive shallow water model, the nonlinear shallow water model, and the potential flow model. The wave run-up on coast is modelled by the moving mesh approach with the allocation of waterfront. The formulas for waterfront position and velocity are obtained using the exact analytical solutions of the shallow water equations in the vicinity of waterfront [2]. The predictor-corrector scheme on moving grids is used [2]. The grid is constructed by the equidistribution method [3]. The numerical results are presented for several test problems.

Literature.

1. Shokina N., Aizinger V. Submitted to Environ. Earth Sci. (2015)
2. Shokina N. Proc. Appl. Math. Mech., Special Issue: 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Erlangen 2014. 14(1), 853-854 (2014)
3. Khakimzyanov G.S., Shokina N.Yu. Comput. Technol. Vol. 17(2), 79-98 (2012)

We offer a secure homomorphic encryption of elements of a finite nonabelian simple group. According to a result of Khamsemanan-Ostrovsky-Skeith, this entails construction of a secure fully homomorphic encryption scheme.

Bevor Karl Weierstraß 1885 den Beweis seines Approximationssatzes (WAS) in der reellen Funktionentheorie veröffentlichte, erörterte er in Briefwechseln mit seiner Schülerin Sofja Kovalevskaja, mit Paul du Bois-Reymond und mit Hermann Amandus Schwarz die Möglichkeit, Cantors neuen Begriff des Inhalts von Punktmengen für die Verallgemeinerung des im Beweise benutzten Riemannschen Integralbegriffs zu benutzen. Am Ende gelangte Weierstraß nur zum nichtadditiven oberen Darbouxschen Integral und verzichtete auf eine Publikation dieser Überlegungen. Der Vortrag wird auch einige weitergehende Bemerkungen über das Verhältnis von Weierstraß zu seinem ehemaligen Schüler Georg Cantor enthalten, zu einer Zeit, da beide zunehmenden Angriffen von Leopold Kronecker ausgesetzt waren.

Using classical hypergeometric identities, we compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. We deduce several explicit computations on the related Green function and Martin kernel.

Joint work with Christophe Profeta (Evry).

Motivated from art restoration, "inpainting" stands for methods for the reconstruction of damaged or missing parts of images. Such damage can be caused for example by degradation of the real image. On the basis of those data of the image which are not damaged, inpainting methods try to reconstruct the damaged parts in a suitable way. There exists a wide range of mathematical inpainting methods differing in the choice of the algorithm as well as in the space of the solutions, both strongly depending on the interpretation of what suitable means. Tensor product splines are, among other things, useful due to their simple structure and efficient implementability. In this talk we present an inpainting method that uses tensor product splines for the reconstruction. This is joint work with Tomas Sauer.

A graph \(G\) is said to be \(H(N, \Delta)\)-universal if it contains a copy of every graph on \(N\) vertices with maximum degree at most \(\Delta\). Determining the threshold for the property that a typical graph \(G \sim G(n,p)\) is \(H(N, \Delta)\)-universal is an intriguing question in the theory of random graphs, with two most common scenarios being \(N = n\) (spanning subgraphs) and \(N = (1 - \epsilon)n\) (almost-spanning subgraphs). A result of Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi shows that for \(N = (1 - \epsilon)n\) it suffices to take \(p \ge (\log n / n)^{1/\Delta}\). This was further improved by Dellamonica, Kohayakawa, Rödl and Rucińcki, who showed that for the same value of \(p\) one can take \(N\) to be as large as \(n\). On the other hand, the only known lower bound on the threshold for these two properties is of order \(n^{-2/(\Delta + 1)}\). It is worth noting that even for the simpler property of containing a single (arbitrary) spanning graph \(H \in H(n, \Delta)\), no better bound on \(p\) is known (result of Alon and Füredi).

We make a step towards closing this gap. In particular, we bypass a natural barrier of \(p \ge (log n / n)^{1 / \Delta}\) by showing that, in the case \(\Delta \ge 3\), a typical graph \(G \sim G(n, p)\) is \(H((1 - \epsilon)n, \Delta)\)-universal for \(p \ge n^{-1/(\Delta - 1)} \log^5 n\). This determines, up to the logarithmic factor, the asymptotic value of the threshold in case \(\Delta = 3\).

Joint work with David Conlon, Asaf Ferber and Rajko Nenadov.

Recently, the classical linearized metrizability has been understood for a large class of parabolic geometries.

This leads to the quest for subriemannian metric partial connections within the class of the Weyl structures on a given parabolic geometry. I will illustrate the procedure on some explicit examples like the Lie contact structures or quaternionic contact structures. The talk will reflect work in progress, joint with David M.J. Calderbank and Vladimir Soucek.

We discuss how motivic sheaves help our understanding of graded representation categories.

Software citations in literature are typically sparse and contain often not more than the name of the software. This makes the identification of software references in publications difficult. Up to now, the swMATH information service (http://www.swmath.org) on mathematical software uses heuristic methods for identifying software citations in publications. A standard for software citations would really increase the visibility and a positive identification of the software. In the first part of the talk, the situation for software references is analyzed which differs from that of publications. Then, some proposals for standardizing software citations are presented.

We outline the proof of the theorem stated in the title. As a corollary we obtain that the additivity of the Mycielski ideal is less or equal than the additiviy of the meager ideal. In forcing terminology this means that every reasonable amoeba for Silver forcing adds both Cohen and dominating reals. We also give some background of the open problem whether there is a Tukey reduction of the Mycielski to the null ideal, which, by Pawlikowski's theorem, would be a strengthening of our result.

Coquand's cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. We show that the underlying cube category is the opposite of the Lawvere theory of De Morgan algebras. The topos of cubical sets itself classifies the theory of "free De Morgan algebras". We will relate this to Johnstone's topological topos and the nerve construction. This provides us with a topos with an internal "interval". Using this interval we construct a model of type theory following van den Berg and Garner. We are currently investigating the precise relation with Coquand's model. We do not exclude that the interval can also be used to construct other models.

We will discuss abelian categories of Mixed Tate Motives over arithmetic base schemes satisfying the Beilinson-Soule vanishing Conjecture. They arise as heart of a t-structure on integral triangulated Tate Motives. This t-structure restricts to a t-structure on compact objects, giving rise to an abelian category of integral Tate Motives of finite type. Finally we address integral geometric fundamental groups whose representations model Tate Motives.

The (one-dimensional) KPZ equation is a stochastic PDE describing the motion of growing fronts, generated when a stable phase is in contact with a metastable one. While the equation has been around since 1986, only recently we start to better understand its mathematical structure. In particular, the KPZ equation is a beautiful example for an integrable stochastic system. There are many other models which, either expected, numerically supported, or proved, have the same statistical properties as the KPZ equation when both are viewed on large space-time scales. I will review the case of interacting diffusions. One can think of them as a collection of one-dimensional diffusions \(x_j(t), j \in \mathbb{Z}\), where diffusion with label \(j\) interacts with its left neighbor, \(x_{j-1}(t)\), and right neighbor, \(x_{j+1}(t)\). In general, these models are expected to be in the KPZ universality class. But for a very particular choice of the interaction the model turns out to be integrable and thus allows for a deeper analysis.

In this talk we focus on reliable a posteriori error estimation techniques for state-constrained optimal control problems, a particular emphasis will be laid on convergence results for a sequence of discrete solutions computed on adaptive grids without the use of maximum-norm error estimates.

Asymptotically efficient quantum algorithms that render some classical computational hardness assumptions invalid are widely known, and the availability of these algorithms motivates research in post-quantum cryptography. A more fine-grained resource analysis of such quantum algorithms is desirable in order to understand their cryptanalytic impact. How many qubits and how many gates of which type do we need to attack actually deployed schemes, and what is the depth of such a quantum circuit?

In this talk we consider the multivariate continuous-time GARCH(1,1) process driven by a Lévy process emphasising stationarity properties. The focus is on the volatility process which takes values in the positive semi-definite matrices.

In the univariate model existence and uniqueness of the stationary distribution as well as geometric ergodicity are well-understood, whereas for the multivariate model only an existence criterion is known as far as strict stationarity is concerned. We shall first review the multivariate COGARCH(1,1) model and its properties focussing on strict and weak stationarity. Thereafter, the main part of the talk is devoted to establishing sufficient conditions for geometric ergodicity and thereby for uniqueness of the stationary distribution and exponential strong mixing.

We follow a classical Markov/Feller process approach based on a Foster-Lyapunov drift condition on the generator. Apart from finding an appropriate test function for the drift criterion, the main challenge is to prove an appropriate irreducibility condition due to the degenerate structure of the jumps of the volatility process, which are all rank one matrices. We present a sufficient condition for irreducibility in the case of the driving Lévy process being compound Poisson.

Recently, a phase field model for surfactant and two or more fluids has been presented. It generalises a Cahn-Hilliard-Navier-Stokes system which is coupled to an advection-diffusion equation for a soluble surfactant to multiple phases. In the interfaces and triple junctions, local chemical equilibrum with regards to the surfactant has been assumed. Using matched asymptotic expansions the approach could be shown to converge to the desired sharp interface problem which is supported by numerical simulation results. In the talk, extensions of the model will be discussed with a focus on non-equilibrum conditions both in the interfaces and the triple junctions.

We combine methods of real algebraic geometry, linear system theory and harmonic analysis for the construction and for parameterization of classes of tight wavelet frames. Recent algebraic results guarantee that nonnegative trigonometric polynomials in two variables have a sum-of-squares decomposition. This result is useful in solving two matrix extension problems which occur in the construction of bivariate tight wavelet frames, namely the unitary and oblique extension principles. The masks of the wavelet frames are finite, if the constructions are based on the unitary extension principle, and infinite otherwise. For the oblique extension principle, a new and efficient method for the construction of tight wavelet frames with finite masks is presented. It uses the interpretation of certain rational trigonometric functions as transfer functions of a linear system.

This is joint work with M. Charina, M. Putinar and C. Scheiderer.

We present a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for the joint image reconstruction and segmentation in magnetic particle imaging.

We recall various ways of splitting fibrations and discuss the role equality of objects plays in there.

I will explain an analogue of the Atiyah-Patodi-Singer index theorem for the (Loretzian) Dirac operator on a globally hyperbolic spacetime with boundary. Since this operator is not elliptic this does not fall into the framework of classical index theory. I will also discuss some consequences of this theorem for quantum field theory on curved spacetimes. (joint work with C. Bär)

The continuum random tree (CRT) was constructed by David Aldous in the early nineties and plays a major role in the study of typical metric properties of large random trees and graphs. We show that the model "all unlabelled unrooted trees equally likely" admits the CRT as a scaling limit. This confirms a long-standing conjecture by Aldous and completes the now long list of families of random discrete objects converging towards the CRT. Our approach is based on the cycle-pointing method, a combinatorial technique developed by Bodirsky, Fusy, Kang and Vigerske.

In this talk we consider a currency exchange rate model given by a differential equation with state-dependent delay. After describing the basic assumptions and properties of the differential equation under consideration, we state a result about the existence of periodic orbits and briefly explain its proof. Afterwards we introduce a more general exchange rate equation with state-dependent delay and discuss the attempt to carry over the result about the existence of periodic orbits to this more general situation.

The study of Riemannian manifolds all of whose geodesics are closed is a classical subject with anewed interest branching out into the field of contact geometry. I will explain which results are known for the pseudo-Riemannian case, especially how the theorem of Waldsley on geodesic foliations generalises to the pseudo-Riemannian world. Further I will give counterexamples to possible generalisations and comment on the problem of determining the set of pseudo-Riemannian metrics all of whose geodesics are closed.

The talk is concerned with a damage model including two damage variables, a local and a non-local one, which are coupled through a penalty term in the free energy functional. After introducing the precise model, we prove existence and uniqueness for the viscous regularization thereof. Moreover, we rigorously study the limit for penalization parameter tending to infinity. It turns out that in the limit both damage variables coincide and satisfy a classical viscous damage.

Let \(\Omega \subset \mathbb{R}^{n}\) be a bounded domain and let \(G:% \bar{\Omega}\times \bar{\Omega}\rightarrow \left[ 0,\infty \right] \) be the Green function for \(-\Delta u=f\) in \(\Omega \) under zero Dirichlet boundary conditions. What can one say about the supremum of the so-called \(3G\) -function \[ G_3(x,y):=G\left( x,y\right) ^{-1}\int_{\Omega }G\left( x,z\right) G\left( z,y\right) dz? \] It is conjectured that for simply connected domains the essential supremum is reached at some (opposite) boundary points: \[ \underset{x,y\in \Omega }{\mathrm{ess~sup}}~G_3\left( x,y\right) = \underset{x,y\in \partial\Omega }{\mathrm{ess~sup}}~G_3\left( x,y\right). \] Indeed for \(\Omega\) being a ball, this conjecture has been confirmed. For \(n=2\) this conjecture remains open but in higher dimensions one may construct a domain \(\Omega \) that gives a counterexample. This \(3G\)-function is related with the expected lifetime of a certain conditioned Brownian motion. The main part will focus on joint work with Matthias Erven.

In this talk, I aim to give an overview of some known results and several open questions concerning geometric and topological properties of the moduli space of stable Higgs bundles of fixed rank and degree on a compact Riemannian surface. I shall in particular discuss its construction as the solution space of a certain system of nonlinear elliptic equations, called Hitchin's self-duality equations. A novel geometric compactification of the moduli space is presented by adding to its smooth part configurations which are singular in a finite number of points. We finally describe some aspects of the asymptotic geometry of a natural hyperkaehler metric moduli space is endowed with.

(Joint work with Rafe Mazzeo (Stanford), Hartmut Weiß (Kiel) and Frederik Witt (Münster)).

I will present the result that Thompson's group \(V\) is acyclic. This embeds into a general discussion of the symmetries of algebraic theories and their algebraic K-theory. Other examples and applications of this circle of ideas will be given insomuch as time permits.

To obtain a model with no lifting for Borel/Meager Shelah defined oracles, what it means for a forcing poset to be \(\bar M\)-cc for an oracle \(\bar M\), and described how these posets should be iterated. I am trying to define the corresponding notions in the generalized Baire space with the scope of generalizing Shelah's involved construction of a model with no lifting homomorphisms.

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this talk is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are discussed in the second part of the talk. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces.

This joint work with: Stephan Dahlke , Filippo De Mari, Ernesto De Vito, Sören Häuser , Gabriele Steidl.

The specific features of mathematical information have led to an ecosystem of specialized services: reviewing databases like zbMATH, collections like EuDML, repositories of mathematical objects like the OEIS, or libraries of formalized mathematics. While they well-designed to solve the typical problems they have been build for, the 21st century challenge of building a global information system for mathematics is to build connections between them, thereby enhancing the value of each system considerably. We outline some approaches which currently take shape to achieve this goal.

If \(G\) is a finite group, then \(G\) has finitely many irreducible finite dimensional representations, and each finite dimensional representation of \(G\) can be expressed uniquely as a direct sum of finitely many irreducible representations. Unfortunately, the basic theory of the infinite dimensional unitary representations of countably infinite groups is much less satisfactory. In particular, such a group typically has uncountably many irreducible infinite dimensional unitary representations. In this talk, I will consider questions such as:

• For which countably infinite groups \(G\) is it possible to classify its irreducible representations?
• What does it mean to classify an uncountable set of irreducible representations?

Along the way, we will see the representation theorists Mackey, Glimm and Efros making fundamental contributions to descriptive set theory, and the descriptive set theorists Kechris and Hjorth making fundamental contributions to representation theory.

Let \(G\) be a countable discrete group and let \(\text{Sub}_{G}\) be the compact space of subgroups \(H \leqslant G\). Then a probability measure \(\nu\) on \(\text{Sub}_{G}\) which is invariant under the conjugation action of \(G\) on \(\text{Sub}_{G}\) is called an invariant random subgroup. In this talk, I will discuss the invariant random subgroups of inductive limits of finite alternating groups.

Shadowing theory studies properties of approximate trajectories (pseudotrajectories). The main question of the shadowing theory is the following: when for any pseudotrajectory does there exist a close exact trajectory?

The main difference between the shadowing problem for vector fields and the similar problem for discrete time dynamical systems is related to the necessity of reparametrization of shadowing trajectories in the former case. In the modern theory of shadowing the most important types of allowed reparametrizations correspond to standard and oriented shadowing properties, investigated in 80's by Komuro and Thomas. In 1984 Komuro proved that those notions are equivalent for nonsingular vector fields and posed a question if those two notions are different in general . We provide an example showing that those shadowing properties are not equivalent. An example is a non-structurally stable \(4\)-dimensional vector field based on a special \(2\)-dimensional vector field whose trajectories look like spirals.

Eine Mathematiklehrerfortbildung, die als nachhaltiges Projekt intendiert wird und über Kompetenzveränderungen der Teilnehmer schließlich auch die Kompetenzfaktoren von Schüler/innen in den Klassen erreicht, ist mehr als nur eine gute Idee, die eines engagierten Kursleiter und durchaus begeisterte Kursteilnehmer bedarf.

Der Autor berichtet aus seiner Tätigkeit im Design von Fortbildungsveranstaltungen im Deutschen Zentrum für Lehrerfortbildung Mathematik (DZLM) und die hier zugrunde gelegten Prinzipien. Wirklich nachhaltige Veranstaltungen müssen systemische und letztlich auch politische Implikationen anstreben und realisierbar machen. In den beiden letzten Jahrzehnten haben sich international wichtige Paradigmenwechsel für die Professionalisierung von Lehrpersonen vollzogen, die in Deutschland dekliniert werden müssen.

Two Borel probability measures \(\nu\) and \(\mu\) on Cantor space are orthogonal if there is a Borel set which has measure \(1\) for \(\nu\), but measure \(0\) for \(\mu\). An orthogonal family of measures is a family of pairwise orthogonal measures; it is maximal if it is maximal under inclusion.

It can be shown that no analytic maximal orthogonal family (mof) exists (Preiss-Rataj), but if \(V=L\) then there is a \(\Pi^1_1\) (lightface coanalytic) mof (Fischer-T.). However, if we add a Cohen or Random real to \(L\), then there are no \(\Pi^1_1\) mofs (Fischer-Friedman-T.).

This motivated the question: Can a \(\Pi^1_1\) mof coexist with a non-constructible real? In this talk, we answer this by showing that there is a \(\Pi^1_1\) mof in the Sacks and Miller extensions of \(L\). By contrast, we also show that if we add a Mathias real to \(L\) then there are no \(\Pi^1_1\) mofs.

A \(2\)-factor in a graph is a set of disjoint cycles covering all vertices of the graph. A complete characterization of the maximal graphs without \(2\)-factors is presented. The proof is based on the general \(2\)-factor theorem. Also an easy proof of the theorem that any \((2r+1)\)-regular graph with at most \(2r\) bridges has a \(2\)-factor is given, and moreover all \((2r+1)\)-regular graphs with \(2r+1\) bridges without a \(2\)-factor are found. This generalizes Julius Petersen's famous theorem (1891) that any \(3\)-regular graph with at most two bridges has a \(1\)-factor, and in addition the Sylvester graphs. The results will be put into a historical context.

The first to obtain the general \(2\)-factor theorem was Hans-Boris Belck in 1949 in his Ph.D. thesis, written at the University of Frankfurt, when he was only 20 years old. In fact Belck obtained the general \(k\)-factor theorem, and he also presented the first purely graph theoretic proof of Tutte's \(1\)-factor theorem from 1947. In addition to his dissertation Belck published only one mathematical paper (in Crelle's Journal 1950), and thereafter he disappeared out of sight. The lecture will present biographical details about Belck. It is really a pity that he was not properly honoured in his lifetime for his remarkable mathematical achievements.

However, the terminology of Belck was unusual and his proofs not as easy to read as the very elegant theory of \(1\)-factors, presented in 1950 by T. Gallai (who obtained as a byproduct what we now call the Edmonds-Gallai Theorem). In 1951 Tutte obtained the general \(f\)-factor theorem as a generalization of the \(k\)-factor theorem, and in 1953 he showed how to reduce it in a simple way to his \(1\)-factor theorem.

The research behind this talk was initiated at the Department of Mathematics at London School of Economics in the fall of 2011, in collaboration with Jan van den Heuvel.

G.A. Dirac initiated and created several areas of modern graph theory, starting with his ph.d. thesis in 1951 on critical graphs. He was of Hungarian origin, but grew up in England, where he graduated with Richard Rado as supervisor. He obtained strong contacts with graph theory in Germany through his appointments in Hamburg and Ilmenau around 1960 and the personal relations that developed. These contacts flourished during his appointment at Aarhus University in Denmark from 1966 on. In the lecture an overview of Dirac's achievements in structural graph theory will be presented, together with an evaluation of the significance of his German relations.

In this talk I would like to present about the inverse problem of identifying the diffusion matrix in the Dirichlet problem for an elliptic partial differential equation of second-order, when a solution of the direct problem is imprecisely given by the observation data. The convex energy functional method with Tikhonov regularization is used to our identification problem. We analyze convergence rates of the method under a new source condition which is weaker than that of the theory of regularization for nonlinear ill-posed problems. In discrete case the finite element method is applied to strictly convex minimization problems for solving the identification problem. We investigate an error bound of the finite element approximation solutions. Furthermore, a gradient-projection algorithm is employed for finding minimizers of these minimization problems. The strong convergence of iterative solutions to that of the identification problem is shown without smooth assumption on the sought matrix. Finally, we present a numerical experiment which illustrates our theoretical results.

The optimal control of low-frequency electromagnetic fields is considered in a time-harmonic setting. For the state equation, a non-standard \(H\)-based formulation of the equations of electromagnetism is used for multiply connected conductors. The magnetic field \(H\) in the conductor is obtained from an elliptic equation with the \(\mathrm{curl}\,\sigma^{-1}\,\mathrm{curl}\)-operator, while the \(\mathrm{div}\,\mu\,\nabla\)-operator is set up for a potential in the isolator. Both equations are coupled by interface conditions. The control is the electrical current in the conducting domain. In particular, the problem of sparse optimal control is sketched.

This is joint work with Alberto Valli, University of Trento.

Classical Schur-Weyl duality says that the actions of \(\mathbb{C}[S_d]\) and \(\mathfrak{gl}_n\) on \(T=(\mathbb{C}^n)^{\otimes d}\) commute and generate each others commutant. In particular, one can recover \(\mathbb{C}[S_d]\) as \(\mathrm{End}_{\mathfrak{gl}_n}(T)\) by taking \(n\geq d\).

This is just the tip of the iceberg of a huge class of algebras called centralizer algebras. We discuss a general method to study their representation theory for the case where \(\mathfrak{gl}_n\) is replaced by \(\mathbf{U}_q(\mathfrak{g})\) and \(T\) is replaced by any \(\mathbf{U}_q=\mathbf{U}_q(\mathfrak{g})\)-tilting module. That is, we show that \(\mathrm{End}_{\mathbf{U}_q}(T)\) is equipped with a cellular basis.

As an application, we explain how Jantzen's sum formula can be used to check semi-simplicity criteria for these centralizer algebras.

Joint work with Henning Haahr Andersen and Catharina Stroppel.

We present a framework to numerically treat spatially distributed optimal control problems with an infinite time horizon, illustrating the approach by some examples. The basic idea is to consider the associated canonical systems in two steps. First we perform a bifurcation analysis of the steady state canonical system, yielding branches of patterned canonical steady states. In a second step we compute time dependent canonical system paths to steady states having the so called saddle point property. It turns out that often patterned canonical steady states are optimal.

Mice are countable sufficiently iterable models of set theory. Itay Neeman has shown that the existence of such mice with finitely many Woodin cardinals implies that projective determinacy holds. In fact he proved that the existence and \(\omega_1\)-iterability of \(M^{\#}_n(x)\) for all reals \(x\) implies that boldface \(\Pi^1_{n+1}\)-determinacy holds.

We prove the converse of this result, that means boldface \(\Pi^1_{n+1}\)-determinacy implies that \(M^{\#}_n(x)\) exists and is \(\omega_1\)-iterable for all reals \(x\). This level-wise connection between mice and projective determinacy is an old so far unpublished result by W. Hugh Woodin. As a consequence we can obtain the determinacy transfer theorem for all levels n. These results connect the areas of inner model theory and descriptive set theory, so we will give an overview of the relevant topics in both fields and briefly sketch a proof of the result mentioned above. The first goal is to show how to derive a model of set theory with Woodin cardinals from a determinacy hypothesis. The second goal is to prove that there is such a model which is iterable. For this part the odd and even levels of the projective hierarchy are treated differently.

This is joint work with Ralf Schindler and W. Hugh Woodin

Bei Würdigungen von Karl Weierstraß (1815-1897) anlässlich seines 200sten Geburtstags in diesem Jahr wird er zumeist als Analytiker gekennzeichnet. Allerdings gibt es in seinem Werk durchaus Beziehungen zur Algebra: So betonte er in seinem vielzitierten „Glaubensbekenntnis“, dass die Funktionentheorie „auf dem Fundamente algebraischer Wahrheiten aufgebaut werden muss“. Ebenso ist eine Normalform linearer Abbildungen nach ihm benannt. In dem Vortrag wird unter diesem Aspekt ein genauerer Blick auf sein mathematisches Werk geworfen, der Weierstraß in seinen algebraischen Arbeiten als erstaunlich „modern“ (im Sinne von van der Waerden bzw. Artin und Noether) zeigt, etwa bei seiner axiomatischen Charakterisierung der Determinante.

The Jones isomorphism relates Hochschild homology \(HH_{-\bullet}(S^*X)\) and cohomology of the free loop space \(H^\bullet(LX)\), for any simply connected space \(X\). This and its \(S^1\)-equivariant version have provided algebraic models for string topology. In work in progress, we use similar simplicial methods to explore the \(O(2)\)-equivariant case and give an isomorphism \(DH_{-\bullet}(S^*X)\cong H^\bullet_{O(2)}(LX)\), involving a flavour of dihedral homology.

In this talk I will present new upper bounds for the maximum density of translative packings of superspheres in three dimensions (unit balls for the \(l^p\)-norm). This will give some strong indications that the lattice packings experimentally found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proof of our new upper bounds is computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.

This is joint work with Maria Dostert, Cristobál Guzmán, and Fernando Mário de Oliveira Filho.

The predictability problem of the atmosphere and climate has been a central concern for decades since the discovery of the property of sensitivity to initial conditions in models based on the conservation laws for fluid flows. Nowadays, the short term predictability (up to 15 days) of the large-scale atmosphere is well understood. However new challenges are arising with the necessity to resolve convection-scale processes (short-time and kilometer scale) on one hand, and on the other hand, the necessity to produce decadal time-scale climate forecasts. In this talk, we review key results of the past analyses of the predictability of the atmosphere and the current challenges at both ends of the spatio-temporal spectrum associated with very short term high-resolution forecasts, and long-term climate predictions.

This talk considers scalar delay differential equations of the form \[ \dot{x}\left(t\right)=-ax\left(t\right)+f\left(x\left(t-1\right)\right), \] where \(a>0\) and \(f\) is a strictly increasing \(C^{1}\)-function. We say that a periodic solution has large amplitude if it oscillates about at least two unstable equilibria. We investigate what type of large-amplitude periodic solutions may exist at the same time when the number of unstable equilibria is arbitrarily large. We also discuss the geometrical properties of the unstable sets of certain large-amplitude periodic orbits oscillating about exactly two unstable equilibria.

Discovery of regularized theta lifts made by Harvey-Moore and Borcherds lead to many advances in physics and number theory. For a long time such regularized theta lifts where known only for dual reductive pairs \((SL_2, O(V))\), where \(V\) is a rational quadratic space. Recently, J. Bruinier has defined regularized theta lifts from \(SL_2\) to orthogonal groups over totally real fields. In this talk we will analyze CM values of such theta lifts.

In dimension one, moment problems are closely related to classical interpolationproblems for bounded analytic functions of the unit disc (or on the upper half plane). The analogues of these interpolation problems in higher dimension were considered largely intractable till the groundbreaking work of Agler in the late 1980s – early 1990s who discovered the (correct) relationship with multivariable operator theory (von Neumann inequality) and (as it became apparent only quite recently) certain sums of squares decompositions. I will review some of these developments, with a particular emphasis on the joint work with Joe Ball and Cora Sadosky that both provides a more constructive approach and shows a relation to the work of Geronimo and Woerdeman on the two-dimensional trigonometric moment problem.

This talk will be about the classification of right coideal subalgebras \(C\) of a quantum group with generic \(q\), where \(C\) has the additional property that all irreducible representations are \(1\)-dimensional and \(C\) is maximal with this property.

We call such a right coideal subalgebra a Borel subalgebra. This is due to a theorem of Sophus Lie stating that the Borel subalgebras of a semisimple Lie algebra have only \(1\)-dimensional representations and are maximal with this property. Borel subalgebra and subgroups are in the theory of algebraic groups, semisimple Lie algebras and representation theory the basic components of many standard constructions (flag varieties, spherical varieties, Verma modules and their irreducible quotient, etc.).

We shall see that indeed there are the so-called standard Borel subalgebras and their reflections which are parametrized by an element of the Weyl group. But there are more examples, already in \(Uq(sl2)\) appears a family of Weyl algebras generated by two elements. So the question arises, which other kinds of Borel subalgebras exist.

L. von Bortkiewicz (1868-1931) was an outstanding statistician, less known is his major contribution to the popularisation of statistiscs in the mid 1920s. He became the editor of a series of popular books on statistics ("Serie populärer statistischer Bücher") which were published by Rudolf Mosse publishing house (Rudolf Mosse Buchverlag Berlin) in Berlin between 1925 and 1929. The seven volumes "The world in figures" (Die Welt in Zahlen) became a role model of this kind of publications. In the talk we'll describe the collaboration between the couple Woytinsky (Wladimir S. Woytinsky (1885-1960) and Emma S. Woytinsky (1893-1968)) and L. von Bortkiewicz producing these volumes. Furthermore, we'll discuss the motivations of the latter to participate in this project.

Finite metric graphs are used to describe many phenomena in science, from phylogenetic trees in biology to Feynman diagrams in physics, so one would like to understand the spaces that parametrize such graphs. Techniques from geometric group theory have shown that the moduli space of all metric graphs with a fixed number of loops and marked points (or univalent vertices) is closely related to the group of automorphisms of a free group. Thus algebraic tools can be used to help understand the geometry and topology of these moduli spaces and, conversely, geometric tools can be used to help understand the algebraic structure of these automorphism groups. I will discuss what we know about these spaces and groups, and then show how to bootstrap information about moduli spaces of graphs with a small number of loops to obtain new information about moduli spaces of larger graphs. This involves using tools from various different areas of mathematics, including group theory, algebraic topology, representation theory and modular forms.

We investigate the numerical solution of a boundary control problem with elliptic partial differential equation by the hp-finite element method. We prove exponential convergence with respect to the number of unknowns for an a-priori chosen discretization. Here, we have to prove that derivatives of arbitrary order of the solution are in suitably chosen weighted Sobolev spaces. Numerical experiments confirm the theoretical findings.

We will explain the definition and properties of the noncommutative Maslov index for projective modules over \(C^\ast\)-algebras and its connection to recent work of Barge and Lannes, who studied a Maslov index over commutative rings. In particular we correct and make more precise a statement on the dependence of the choices contained in our previous work on the subject. All this generalizes classical work of Cappell, Lee and Miller. The proofs involve index theory over \(C^\ast\)-algebras.

I will discuss a variational existence theory for three-dimensional fully localised solitary water waves with weak surface tension. The water is modelled as a perfect fluid of finite depth, undergoing irrotational flow. The surface tension is assumed to be weak in the sense that \(0 \leq B \leq 1/3\), where \(B\) is the Bond number. A fully localised solitary wave is a travelling wave which decays to the undisturbed state of the water in every horizontal direction. Such solutions are constructed by minimising a certain nonlocal functional on its natural constraint. A key ingredient is a variational reduction method, which reduces the problem to a perturbation of the Davey-Stewartson equation.

We construct a semiflow of differentiable solution operators for an autonomous delay differential equation in the general case which covers time-invariant and state-dependent delay, bounded or unbounded. This semiflow lives on a submanifold of finite codimension in the Fréchet space of continuously differentiable maps on the nonpositive reals. The hypothesis on the functional defining the differential equation is continuous differentiability (in the sense of Michel and Bastiani) together with a mild extension property for the derivatives.

Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that functions of random variables are typically near their means. Of particular importance is the case where \(f(X)\) is a function of independent random variables \(X=(X_1, \ldots, X_n)\). Here the well-known bounded differences inequality (also called McDiarmid's or Hoeffding--Azuma inequality) establishes sharp concentration if the function \(f\) does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that \(|f(X)-f(X')| \leq c_k\) whenever \(X,X'\) differ only in \(X_k\). While this is easy to check, the main disadvantage is that it considers worst-case changes \(c_k\), which often makes the resulting bounds too weak to be useful.

In this talk we discuss a variant of the bounded differences inequality which can be used to establish concentration of functions \(f(X)\) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations as to why concentration should hold. Indeed, given an event \(\Gamma\) that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where \(\Gamma\) occurs. The point is that the resulting typical changes \(c_k\) are often much smaller than the worst case ones.

If time permits, we shall illustrate its application by considering the reverse \(H\)-free process, where \(H\) is \(2\)-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdős.

We study bifurcation from a branch of trivial solutions of semilinear systems of elliptic Dirichlet boundary value problems on star-shaped domains, where the bifurcation parameter is introduced by shrinking the domain. We associate to this bifurcation problem two curves of Lagrangian subspaces of a symplectic Hilbert space and construct a Maslov index, which, roughly speaking, counts the number of intersections of these curves. Our main result states that a non-vanishing Maslov index entails bifurcation. Our proof uses a generalised Morse index theorem for the linearised equations, which reduces for strongly elliptic equations to a theorem of Stephen Smale from the sixties. This is joint work with Alessandro Portaluri from the University of Turin.

A partition of a positive integer \(n\) is an additive decomposition of \(n\) into a sum where the summands are positive integers as well and the order of these summands does not matter. The partition function \(p(n)\) counts the number of all partitions of \(n\), and it is a traditional problem of constructive number theory to find an explicit \textit{finite} representation of \(p(n)\) (rather than infinite series expansions such as Rademacher's formula). Recently, Bruinier and Ono [2013] established the first finite representation of \(p(n)\) as a finite sum of algebraic numbers being singular moduli for a certain weak Maass form. It is the purpose of this talk to establish several alternatives which exhibit a spectral nature of the partition function. The results may be summarized as follows:

• The \(n\)th partition number \(p(n)\) can be expressed and computed by means of the \(n\)th power of a certain sparse Hessenberg matrix \(H\).
• \(p(n)\) admits an explicit representation as finite linear combination of powers of eigenvalues of \(H\). This spectral representation shows that \(p(n)\) can be regarded as a superposition of sinusoids which explains the oscillating behaviour of \(p(n)\).
• The above formulae yield several identities involving the divisor function or series of partition numbers.

References
J.H. Bruinier and K. Ono: Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Adv. Math. 246 (2013), 198-219
M. Weba: A finite spectral representation of the partition function, tentatively accepted for publication by the Ramanujan Journal.

Systemic risk refers to the risk that the financial system is susceptible to failures due to the characteristics of the system itself. The tremendous cost of this type of risk requires the design and implementation of tools for the efficient macroprudential regulation of financial institutions. We propose a novel approach to measuring systemic risk. Key to our construction is a rigorous derivation of systemic risk measures from the structure of the underlying system and the objectives of a financial regulator. The suggested systemic risk measures express systemic risk in terms of capital endowments of the financial firms. Their definition requires two ingredients: first, a random field that assigns to the capital allocations of the entities in the system a relevant stochastic outcome. The second ingredient is an acceptability criterion, i.e. a set of random variables that identifies those outcomes that are acceptable from the point of view of a regulatory authority. Systemic risk is measured by the set of allocations of additional capital that lead to acceptable outcomes. The resulting systemic risk measures are set-valued and can be studied using methods from set-valued convex analysis. At the same time, they can easily be applied to the regulation of financial institutions in practice. We explain the conceptual framework and the definition of systemic risk measures, provide an algorithm for their computation, and illustrate their application in numerical case studies. We apply our methodology to systemic risk aggregation as described in Chen, Iyengar & Moallemi (2013) and to network models as suggested in the seminal paper of Eisenberg & Noe (2001), see also Cifuentes, Shin & Ferrucci (2005), Rogers & Veraart (2013), and Awiszus & Weber (2015). This is joint work with Zachary G. Feinstein and Birgit Rudloff.

Orthogonally additive and order bounded (not necessary linear) operators between vector lattices \(E\) and \(F\) form a Dedekind complete vector lattice \(\mathcal{U}(E,F)\) provided \(F\) is Dedekind complete (1990). Those operators are called abstract Uryson operators and generalize the well known Uryson integral operators. In Archimedean vector lattices finite elements (as abstract analogon to continuous functions with compact support) have been introduced in 1972 and are actively studied in the last years. In the talk finite elements in \(\mathcal{U}(E,F)\) are dealt with. A description of the finite elements is given, in particular, for \(\mathcal{U}(\mathbb{R}^n,\mathbb{R}^m)\). Some cases are considered when rank one operators are finite elements in \(\mathcal{U}(E,F)\).

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume knowledge of the evolution equations. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but instead we have sampled data which is contaminated with noise. Our approach is to first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.

We consider axisymmetric solutions of the ElectroHydroDynamic equations in three dimensions. We analyze possible singularities and show in a certain regime convergence to a fluid cusp.

In den letzten Jahren gab es beträchtliche Fortschritte auf dem Gebiet der tomographischen Bildgebung in der Medizintechnik mittels der Methode MPI (Magnetic Particle Imaging). Da es sich um eine vergleichsweise junge Methode handelt sind die aktuellen Aktivitäten äußerst vielfältig. Im Gegensatz zu anderen Modalitäten wird zur Signalerzeugung ein Kontrastmittel zwingend benötigt. Die Leistungsfähigkeit dieser nanometergroßen Magnete beeinflusst in hohem Maße die Bildqualität und hat großen Einfluss auf das Design der Signalkette. Im Vortrag wird nach einer kurzen Einleitung gezeigt, wie sich mittels stochastischer Differentialgleichungen (Langevin-Gleichung) oder partieller Differentialgleichungen (Fokker-Planck-Gleichung) die zeitlichen Ummagnetisierungsprozesse (Néel und Brown-Rotation) modellieren lassen und welche Auswirkung das auf die Signalqualität hat.

In the talk I will outline the results of joint work with Aravind Asok and Stefan Kebekus in which we compared the \(A^1\)-homotopy and \(A^1\)-h-cobordism classification of projectivizations of rank two vector bundles on the projective plane. Over algebraically closed fields of characteristic not \(2\), the \(A^1\)-homotopy classification is "classical", everything is determined by characteristic classes. The \(A^1\)-h-cobordism classification of projective line bundles is somehow related to moduli spaces of rank two vector bundles, which leads to some subtleties. For topologically split bundles, \(A^1\)-h-cobordism and \(A^1\)-weak equivalence of bundles agree, while in the other cases the \(A^1\)-h-cobordism classification is still unknown.

Trading networks can be modeled as directed graphs: The vertices correspond to companies or countries, the arcs indicate the flow of trade goods within a given period of time.

The underlying trading markets can be organized in different ways. Some resemble production chains, where goods are iteratively sold from one group of companies to the next one (hierarchical market structure). Others have a group of companies in the center of the market, which sell their goods to several peripheral company groups (center-peripheral market structure). To classify a given market in this manner is hence interesting from both a scientific and a strategic viewpoint.

The market classification can be modeled as a combinatorial optimization problem. We express it as a nonlinear integer program, which is actually a generalization of well-known problems such as the Quadratic Assignment, Linear Ordering, and the Traveling Salesman Problem. An exact solver is presented which uses new linearization techniques and exploits the relations to the problem's well-known special cases. It is able to classify networks up to 10,000 times faster than comparable approaches from the literature.

The solver is applied to real-world trading network data. We present results for the recent trading between German photo agencies as well as for international trading data provided by the United Nations.

Elementary patterns of resemblance, which are finite structures of nested trees, were discovered by Timothy J. Carlson and constitute the basic levels of his general program on patterns of embeddings as an ultrafine-structural approach to contribute to Gödel's suggestion of using large cardinals to solve mathematical incompleteness. The particular class of pure patterns of order two is well-quasi ordered with respect to coverings, as was shown by Carlson. We show that this result is unprovable in the subsystem \(Pi^1_1\)-Comprehension with set induction of second-order number theory, which is the strongest system of the so-called Big Five in reverse mathematics.

Elementary patterns of resemblance, which are finite structures of nested trees, were discovered by Timothy J. Carlson and constitute the basic levels of his general program on patterns of embeddings as an ultrafine-structural approach to GoedelÂ’s idea of using large cardinals to solve mathematical incompleteness. In this talk I will explain the arithmetical analysis and computation of patterns of resemblance, starting from an overview of their basic structural properties. Recent results will be presented. See also my contribution to the minisymposium on well-quasi orders.

We consider models for the spatio-temporal evolution of populations of microorganisms, moving in an incopressible fluid, which are able to partially orient their motion along gradients of a chemical signal. According to modeling approaches accounting for the mutual interaction of the swimming cells and the surrounding fluid, we study parabolic chemotaxis systems coupled to the (Navier-)Stokes equations through transport and buoyancy-induced forces.

The presentation discusses mathematical challenges encountered even in the context of basic issues such as questions concerning global existence and boundedness, and attempts to illustrate this by reviewing some recent developments. A particular focus will be on strategies toward achieving a priori estimates which provide information sufficient not only for the construction of solutions, but also for some qualitative analysis.

We consider variants of the Keller-Segel system of chemotaxis which contain logistic-type source terms and thereby account for proliferation and death of cells. We briefly review results and open problems with regard to the fundamental question whether solutions exist globally in time or blow up. The primary focus will then be on the prototypical parabolic-elliptic system \[ u_t=\varepsilon u_{xx} - (uv_x)_x + ru - \mu u^2, \\ 0= v_{xx}-v+u,~~~~~~~~~~~~~~~~~~~~ \] in bounded real intervals. The corresponding Neumann initial-boundary value problem, though known to possess global bounded solutions for any reasonably smooth initial data, is shown to have the property that the so-called carrying capacity \(\frac{r}{\mu}\) can be exceeded dynamically to an arbitrary extent during evolution in an appropriate sense, provided that \(\mu<1\) and that \(\varepsilon>0\) is sufficiently small. To achieve this, an analysis of the hyperbolic-elliptic problem obtained on taking \(\varepsilon\to 0\) is carried out; indeed, it turns out that the latter limit problem possesses some solutions which blow up in finite time with respect to their spatial \(L^\infty\) norm. This result is in stark contrast to the case of the corresponding Fisher-type equation obtained upon dropping the term \(-(uv_x)_x\), and hence reflects a drastic peculiarity of destabilizing action due to chemotactic cross-diffusion, observable even in the simple spatially one-dimensional setting. Numerical simulations underline the challenge in the analytical derivation of this result by indicating that the phenomenon in question occurs at intermediate time scales only, and disappears in the large time asymptotics.

In this talk the numerical approximation of Neumann boundary control problems governed by linear elliptic partial differential equations is discussed. The naive strategy of using a full finite element discretization of the optimality system with piecewise linear finite elements for the state and dual state, and a piecewise constant approximation of the control yields only the convergence rate one for the discrete control in the \(L^2(\Gamma)\)-norm. This poor convergence behaviour can be improved with advanced strategies like postprocessing and variational discretization which allow an improvement up to the convergence rate two when the computational domain has a smooth boundary. However, on polyhedral domains the solution does not possess the required regularity as singularities can occur in the vicinity of edges and corners. For this reason local mesh refinement can be used to preserve the optimal convergence rates, and we study in detail how strong the refinement has to be, and in which cases optimal convergence is guaranteed even on quasi-uniform meshes.

The Galvin-Mycielski-Solovay theorem confirms a conjecture of Prikry saying that a set of reals is strong measure zero if and only if it can be translated away from each meager set. This connection gives rise to a variety of new "notions of smallness", among them the notion of strongly meager where meager is replaced by null in the translation characterization. In my talk, however, I will focus on another variant based on the ideal of Marczewski null sets which is connected to Sacks forcing. In order to further explore the situation, I will introduce the notion of Sacks dense ideals, that is, translation-invariant sigma-ideals dense in Sacks forcing. Moreover, I will discuss why the cofinality question for ideals such as the one of Marczewski null sets is related to this. (Partially joint work with Jörg Brendle and Yurii Khomskii.)

In this talk, we will consider a stationary model for fluid structure interaction of an incompressible fluid with an elastic structure using the ALE-framework. We will show, that under some suitable assumptions, in particular on the size of the problem data, the solution of the fluid structure interaction problem is differentiable with respect to the given data. We will discuss the essential necessity of the smallness assumption as well as the influence of the chosen continuation of the structure displacement with the help of numerical experiments.

One of the well-known features of the BRST formalism is that the physical space is obtained as the cohomology of the BRST differential. On the other hand, standard approaches to linear quantum fields on curved spacetime use a description of the classical phase space in terms of space-compact solutions of a hyperbolic differential operator. To make those two aspects work together, one needs a good understanding of the interplay of the BRST differential with the equations of motion. I will explain the resulting issues on the level of differential operators and give a rigorous definition of the classical phase space in the BRST formalism, proposed recently in a joint work with J. Zahn, then point towards new challenges in the construction of physical states for higher-spin fields.

I will discuss plane wave solutions and their stability in a one-dimensional complex cubic-quintic Ginzburg--Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced.

(Read More: http://epubs.siam.org/doi/abs/10.1137/130944643)

Variable annuities represent certain unit-linked life insurance products offering different types of protection commonly referred to as guaranteed minimum benefits (GMXBs). They are designed for the increasing demand of the customers for private pension provision. We propose a framework for the pricing of variable annuities with guaranteed minimum repayments at maturity and in case of the insured's death. If the policyholder prematurely surrenders this contract, his right of refund is restriced to the current value of the fund account reduced by the prevailing surrender fee. For the financial market and the mortality model an affine linear setting is chosen. For the surrender model a Cox process is deployed whose intensity is given by a deterministic function (s-curve) with stochastic inputs of the financial market. Hence, the policyholders' surrender behavior depends on the performance of the financial market and is stochastic. The presented pricing framework allows for an incorporation of the so-called interest-rate, moneyness, and emergency-fund hypothesis and is based on suitable closed-form approximations.

Weinstein conjectured in 1978 that any Reeb vector field on a closed contact manifold carries a periodic solution. The conjecture is far from being fully established. So-called "half-plugs" could be used to disprove the conjecture whose non-existence was conjectured by Hofer in 2012. In my talk I will explain how to construct such "half-plugs" disproving Hofer's conjecture based on a joint work with Hansjörg Geiges and Nena Röttgen.

Current processor architectures are able to perform many more floating point operations per time than typical used by numeric codes. Processors heavily use data and instruction parallelism at different levels, together with a deep memory hierarchy. Standard programming approaches cannot exploit this parallelism in total or adapt to the memory layout. However, many numerical algorithms on current systems tend to be memory bandwidth limited, which is a severe limitation. We will discuss cache aware algorithms, vectorization strategies and memory layouts for the case of Finite Differences stencil computations.