Structured Ring Spectra - TNG

Barnes/Roitzheim The category of K(p)-local spectra is an important approximation to the stable homotopy category that is somewhat easier to study. When p=2 this category is rigid, that is, all of the higher homotopy information of K(2)-local spectra is contained in the triangulated structure of the homotopy category. For p=3 this is not true, as well as K(3)-local spectra there is the exotic model of Franke. The homotopy category of this exotic model has the same triangulated structure as K(3)-local spectra, but arises from a different homotopy theory.
Tilman Bauer The unstable Adams spectral sequence based on a generalized homology theory K can be used to compute the homotopy groups of the K-completion of a space. In practice, this can be of little help since the E^2-term of this spectral sequence consists of nonabelian derived functors, which are not easily computable. I will show how the Adams spectral sequence based on many nonconnective theories, including the Morava K-theories, can be linearized to make computations much more feasible. This requires a good understanding of what an "unstable comodule" should be in this setting.
Bob Bruner The Adams spectral sequence for the homotopy of a ring spectrum and the homological homotopy fixed point (and related) spectral sequence(s) have operations and differentials induced by the ring structure of the spectrum. We try to generalize this. Part of the talk will be a report on work of my student, Sean Tilson.
Ethan Devinatz I will use nilpotence technology to prove the following result: If X is a type n finite spectrum with an appropriately homotopy commutative ring spectrum structure, and f is a v_n self-map, then the cofiber of a sufficiently large iterate of f also has a homotopy commutative ring spectrum structure, with one less level of coherence.
Nora Ganter I will describe joint work with Arun Ram on the Weyl-Kac character formula and Schubert calculus in elliptic cohomology.
Teena Gerhardt In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups. In particular, we consider the K-theory of truncated polynomial algebras in several variables. Techniques from equivariant stable homotopy theory are often key to algebraic K-theory computations. In this case we use n-cubes of cyclotomic spectra to compute the topological cyclic homology, and hence K-theory, of the rings in question.
John Harper We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups - this result should be thought of as an algebras over operads in spectra analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology, and use this to define Quillen homology completion - in the sense of Bousfield-Kan - for algebras over operads in symmetric spectra. We prove that under appropriate connectivity conditions, the coaugmentation into Quillen homology completion is a weak equivalence - in particular, such algebras over operads can be recovered from their topological Quillen homology. The results described are joint with K. Hess and M. Ching.
Kathryn Hess (Joint work with Patrick Müller.) In this talk I'll describe a general homotopy-theoretic framework for studying descent and completion, and the dual notions of codescent and cocompletion, in model categories enriched over monoidal model categories. In particular, this framework is applicable to categories enriched over simplicial sets, chain complexes or spectra. I'll present general criteria, reminiscent of Mandell's theorem on E-algebra models of p-complete spaces, under which homotopic descent is satisfied. I'll also construct general descent spectral sequences, explain how to interpret them in terms of derived completion and homotopic descent and relate them to generalized Adams spectral sequences. To conclude I'll sketch a couple of applications.
Mike Hill It is well known that if R is a commutative ring spectrum and x is a homotopy element, then the localization R[x^{-1}] is a commutative R-algebra. In the equivariant context, this doesn't have to be true! In this talk, I'll give examples in which this is true and in which it is false, and I will discuss how one knows when a localization gives a commutative ring again. This work is all joint with Hopkins.
Niles Johnson This talk will introduce the notion of Azumaya object and Brauer group in general bicategorical settings. We will give a characterization of Azumaya objects generalizing that of Azumaya algebras over a commutative ring. For homotopical settings, we describe triangulated bicategories and explain how to extend the Azumaya characterization in that case. We will explain how this relates to the work of Toen and Baker-Richter, among others.
Daniel Ramras This talk will describe recent results about representation spaces of discrete groups, and related questions about characteristic classes of (families of) flat bundles. The proofs of these results make crucial use of ring and module structures on K-theory spectra associated to topologized categories of representations, and lead to some subtle questions about these structures.
Shoham Shamir In commutative algebra, complete intersections rings are the next best thing after regular rings. Such rings also have homological and homotopical characterizations, one given by Gulliksen in the 70's and another developed recently by Benson and Greenlees. It is therefore natural to try and adapt these definitions to the mod-p cochains algebra of a connected space, or rather to some equivalent commutative model of the cochains which mimics a commutative ring. In this joint work with John Greenlees and Dave Benson we show how two of these definitions turn out to be equivalent also for the mod-p cochains algebra of a space.
Vesna Stojanoska It has been observed that certain localizations of the spectrum of topological modular forms tmf are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves M_{ell}, yet is only true in the derived setting. When p is inverted, choice of level-p-structure for an elliptic curve provides a geometrically well-behaved cover of M_{ell}, which allows one to consider tmf as the homotopy fixed points of tmf(p), topological modular forms with level-p-structure, under a natural action by GL_2(Z/p). Specializing to p=2 or p=3 we obtain that as a result of Grothendieck-Serre duality, tmf(p) is self dual. The vanishing of the associated Tate spectra then makes tmf itself Anderson self-dual.
Markus Szymik I will discuss Brauer groups in the context of commutative S-algebras, as defined in joint work with Baker and Richter. If time permits, I will also mention some more geometric aspects of the theory such as Brauer spectra and derived Brauer-Severi schemes.
Sarah Whitehouse In various contexts, the ring of K-theory operations can be shown to map to the corresponding ring of operations for other cohomology theories, in such a way that the image of this map is precisely the centre of the target ring. I will discuss some results of this sort, both old and new, including joint work with Imma Galvez and with M-J Strong.