**Research Seminar on Algebraic Topology, Summer Term 2022**

In the summer term 2022 we will meet every second Thursday at 16:15h in the gaps of the colloquium in mathematical physics (07/04, 21/04, 05/05, 19/05, 09/06, 23/06, 07/07). Most talks will be on zoom, local talks might be in 241. We will have a mixture of local and invited talks. If you are interested in participating, then please register for the mailing list of the seminar by sending "subscribe" to topologieseminar.math-request@lists.uni-hamburg.de

**Schedule:**

**09/11/20, 16:15h: Eva Höning, The Brun spectral sequence for topological Hochschild homology**: In this talk we will generalize a spectral sequence of Brun for the computation of topological Hochschild homology and we will discuss different applications. In particular, we will study the generalized Brun spectral sequence for the topological Hochschild homology of connective complex K-theory, whereby we recover a result of Ausoni. Furthermore, we will apply the spectral sequence to the topological Hochschild homology of the algebraic K-theory of finite fields.**16/11/20, 17:15h: Julian Holstein, Categorical Koszul Duality**: The algebraic analogue of the loop space construction of topological spaces is Adams' cobar construction. Together with the bar construction it induces a Koszul duality between algebras and coalgebras, providing an equivalence of suitable homotopy theories of augmented differential graded algebras and differential graded conilpotent coalgebras. Interesting things happen as one generalises this result, in particular dropping the augmentation on the dg algebra side corresponds to introducing a curvature term on the coalgebra side. I will review this story and then talk about joint work with Andrey Lazarev, in which we generalise this to a categorical Koszul duality and find a category of coalgebras Quillen equivalent to differential graded categories. I will show that this construction is closely related to the coherent nerve construction from simplicial categories to quasi-categories (i.e. from one flavour of infinity categories to another flavour of infinity categories).**23/11/20, 17:15h: Timothée Moreau, Exact model structures**: The interaction between exact structures and model structures give the theory of exact model structure. With very general assumptions, there is a one-to-one correspondance with these structure on morphisms with structures on objects. We discuss of how to induce exact model structures on chain complexes without generator. And we present some results about mixed exact model structures.**07/12/20, 17:00h: Manuel Rivera, The simplicial cocommutative coalgebra of chains on a space determines the fundamental group and much more**: Rational homotopy theory tells us that simply connected spaces, up to rational homotopy equivalence, may be classified algebraically by means of rational cocommutative coalgebras (Quillen) or in the finite type case by rational dg commutative algebras (Sullivan). Goerss and Mandell proved versions of these results for fields of arbitrary characteristic by means of simplicial cocommutative coalgebras and E-infinity algebras, respectively. The algebraic structures in these settings are considered up to quasi-isomorphism. In this talk, I will describe how to extend some of these results to spaces with arbitrary fundamental group. The key new observation is that the homotopy cocommutative coalgebraic structure of the chains on a space determines the fundamental group in complete generality. The corresponding algebraic notion of weak equivalence between simplicial cocommutative coalgebras can be understood as certain localization of Koszul duality. This is obtained by thinking of a homotopy type as an infinity category localized at all arrows and translating this to the algebraic setting of derived localizations as developed by Chuang, Holstein, and Lazarev. The end goal of this program is to completely understand homotopy types in terms of algebraic "chain level" structure.This talk will be (relatively) self contained but it will not be a repetition of my talk at the OATS on November 23. I will discuss new results and ideas as well as more details.

The work presented is part of different collaborations (some on going) with M. Zeinalian, F. Wierstra, E. Minichello, and G. Raptis.**14/12/20, 17:00h: Peter Haine, Revisiting classical splitting results**: Wickelgren and Williams used Morel's unstable \(\mathbb{A}^1\)-connectivity Theorem to deduce that the James Splitting \(\Sigma\Omega\Sigma X \simeq \Sigma X \vee \Sigma(X \wedge X) \vee \ldots \) holds for \(\mathbb{A}^1\)-connected motivic spaces over a perfect field. In this talk we'll explain why a more fundamental splitting \(\Sigma\Omega\Sigma X \simeq \Sigma X \vee(X \wedge \Sigma\Omega X)\) holds for motivic spaces over an arbitrary base scheme. In fact, a modern take on a classical proof of the James Splitting shows that a this more fundamental splitting holds in any \(\infty\)-category where all of the relevant terms are defined and pushout squares remain pushouts after basechange along an arbitrary morphism. This gives new contexts for this splitting, including profinite homotopy theory and (equivariant) motivic homotopy theory. We'll explain why some other classical splitting results hold at this level of generality and give some new descriptions of motivic spaces constructed from \(\mathbb{P}^1 \setminus \{0,1,\infty\}\). This is joint work with Sanath Devalapurkar.**04/01/21, 17:00h: Lukas Brantner, Purely inseparable Galois theory:**An algebraic extension of fields \(F/K\) of characteristic \(p\) is purely inseparable if for each \(x\) in \(F\), some power \(x^{p^n}\) belongs to \(K\). Using homotopical methods, we construct a Galois correspondence for finite purely inseparable field extensions \(F/K\), generalising a classical result of Jacobson for extensions of exponent one (where \(x^p\) belongs to \(K\) for all \(x\) in \(F\). This talk is based on joint work with Joe Waldron.**18/01/21, 16:15h, Birgit Richter, Detecting and describing ramification for structured ring spectra:**Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by the spectrum of Kähler differentials. For rings of integers in an extension of number fields it is important to distinguish between tame and wild ramification. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the norm map. We take the latter fact and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. In the talk I will discuss several examples in the context of topological K-theory and modular forms. This talk is based on joint work with Eva Höning.**25/01/21, 16:15h, Severin Bunk, Universal Symmetries of Gerbes and a Smooth Model for the String Group:**Gerbes are geometric objects describing the third integer cohomology of a manifold and the B-field in string theory; they can essentially be understood as bundles of categories whose fibre is equivalent to the category of vector spaces. I will start by explaining gerbes and their categorical features. The main topic of this talk will then be the study of symmetries of gerbes in a universal manner. We will see that these symmetries are completely encoded in a certain extension of smooth 2-groups. In the last part I will survey how this construction can be used to provide a new smooth model for the string group, via a theory of principal bundles and group extensions in \(\infty\)-topoi.**08/02/21, 16:00h: Dominic Culver, Towards tmf co-operations at the prime 3**In this talk I will discuss current work in progress on computing the ring of cooperations for the connective spectrum of topological modular forms. I will begin by explaining some history and what motivates this computation. I will then discuss an approach to the problem and, time permitting, delve into some of the computations.**15/02/21, 17:00h: Najib Idrissi, Configuration spaces of surfaces**Framed configuration spaces of a surface form a right module over the framed little disks operad. This rich algebraic structure has important consequences, for example for the computations of manifold calculus or factorization homology. Determining the homotopy type of this operadic right module remains however a difficult task. In this talk, I will explain how to compute the rational homotopy type for oriented compact surfaces. The end result is finite-dimensional purely combinatorial model. The proof involves several ingredients: Kontsevich's formality, Tamarkin's formality, and the cyclic formality of the framed little disks operad. (Joint work with Ricardo Campos and Thomas Willwacher.)