Program and abstracts

Abstract: Given a (nice enough) group, there is an isomorphism, due to Lück, relating the rationalised K-theory groups of its classifying space to a large product of cohomology groups, some with rational and some with p-adic coeffecients.

In joint work with Irakli Patchkoria, we identify a generalised cohomology theory capturing the p-adic part of this product. Working in Out(Fn), in ranks close to p we can fully compute this p-adic part and in this way produce an infinite family of odd-dimensional summands in the rationalised K-theory of Out(Fn).

I will discuss these results and the tools that go into them, which range from spherical group rings to the lemma that is not Burnside's, via results about centralisers in Out(Fn): I will try to explain how all these various ideas fit together!

Abstract: For an infinite discrete group G, proper equivariant G-spectra are defined in terms of the finite subgroups of G. In joint work with Tobias Barthel, Drew Heard, and Irakli Patchkoria, we study the Balmer spectrum of the category of dualizable proper G-spectra. We introduce a new separability property on infinite groups, and in the first part of this talk I will discuss how this property guarantees that said Balmer spectrum is given by a colimit of the Balmer spectra for the finite subgroups of G.

In the second part of this talk, I will present some examples of groups with and without this property, and relate it to other commonly studied properties of groups.

Abstract:

Abstract: Tate cohomology can be obtained via a stabilisation process for cohomology groups. Thus, it has been generalised from finite groups to all groups by several authors using different constructions. Therefore, a uniform theory is needed explaining why their approaches all lead to the same conclusions. The goal of the talk is to provide a rough sketch of such a uniform and general theory. After highlighting to which settings this theory applies, the talk concludes by stating two fundamental properties of the resulting generalisation of Tate cohomology.

Abstract: In joint work with Patchkoria, we introduce a family of generalizations, one for each natural number n, of the classical notion of orbifold Euler characteristic of a group as studied by Wall and Serre (which is the case n=0 of our family). These generalizations arise naturally in the study of the generalized (co)homology of infinite discrete groups, with coefficients in the spectra studied in chromatic homotopy theory (as opposed to just rational cohomology). Our work concerns two new aspects of duality between homology and cohomology of groups with finite classifying space for proper actions: we prove the vanishing of Farrell-Tate cohomology with T(n)-local (or K(n)-local) coefficients for such groups, and we construct a new duality functor on the category of proper G-spectra. We compute our generalized Euler characteristics in many examples, including many (but not all) mapping class groups and arithmetic groups.

Abstract: In this talk I will give an overview of the history of Walter Neumann's "Strengthened Hanna Neumann Conjecture" on intersections of subgroups in a free group.  I will then discuss an analogue for locally quasi-convex virtually compact special hyperbolic groups.  I will touch on aspects of the proof using methods from combinatorial topology, L^2-cohomology, and geometric group theory.  Based on joint work with Marco Linton.

Abstract:

Abstract: Tempered cohomology theories are a family of equivariant cohomology theories introduced by Lurie. This includes classical theories such as equivariant K-theory and the Morava E-theory of classifying spaces, but also more exotic theories such as equivariant TMF. In this talk I will give a background on these theories, and then discuss a modular interpretation for their geometric fixed points obtained in joint work with William Balderrama and Jack Davies. I will them explain how this recovers familiar calculations in equivariant K-theory, such as its derived defect base, as well as how it gives a geometric explanation of chromatic blueshift.

Abstract: We consider equivariant (co-)homology groups and aim add their computation.  They occur just in connection with  ordinary group (co-)homology, Borel (co-)homology or  the sources of the assembly maps in the Baum-Connes or Farrell-Jones Conjecture.  Rationally there are rather general formulas, provided that the coefficients carry a kind of Mackey structure. Integral computations are only possible in special but interesting situations and are usually based on good models for the classifying space of proper actions. These computations have applications to conjectures and problems in algebra, group (co-)homology, operator theory, and topology.

Abstract: Let G be a discrete group and R be a ring. The assembly map in K- (resp. L-) theory is a comparison map between G-equivariant homology with coefficients in algebraic K- (resp. L-) theory of R, and K- (resp. L-) theory of the group ring R[G] on the other side. The Farrell--Jones conjecture asserts that such maps are equivalences of spectra, and implies important conjectures in topology such as the Borel Conjecture in high dimension. I will present a categorification of the L-theoretic assembly map, namely a functor of Poincaré categories whose L-theory coincides with the classical assembly map and whose kernel admits explicit generators. As a corollary of such a presentation, I will recover Ranicki's splitting theorem for the L-theory of twisted Laurent polynomials.

Abstract: On a finite lattice, a lot of crucial homotopy-theoretic information can be encoded via transfer systems. These are a combinatorial notion first observed in equivariant homotopy theory. As an obvious advantage over more general model categories, the set-theoretic struggles related to the underlying base categories are, in the lattice setting, replaced with much more hands-on, manageable methods with numerous connections to established combinatorial structures.

Specifically, we will discuss how Bousfield localisation of model categories works on a finite lattice, using transfer systems. The talk will involve plenty of examples and not assume any background knowledge.