**Title:**** **New model structures on simplicial sets**Place:** Room Seminar C3b

**Date:**Friday July 5th, 12h-13h

**Abstract:** In the way Kan complexes and quasi-categories model up-to-homotopy groupoids and categories, can we find model structures on simplicial sets which give up-to-homotopy versions of more general objects? We investigate this question, with the particular motivating example of 2-Segal sets. Cisinski's work on model structures in presheaf categories provides abstract blueprints for these new model structures, but turning these blueprints into a satisfying description is a nontrivial task. As a first step, we describe the minimal model structure on simplicial sets arising from Cisinski's theory.

**Speaker: **Marc Stephan (MPI, Bonn)**Title:**** **A multiplicative spectral sequence for free p-group actions**Place:** Room Seminar C3b

**Date:**Friday May 24th, 12h-13h

**Abstract:**Carlsson conjectured that if a finite CW complex admits a free action by an elementary abelian p-group G of rank n, then the sum of its mod-p Betti numbers is at least 2^n. In 2017, Iyengar and Walker constructed equivariant chain complexes that are counterexamples to an algebraic version of Carlsson’s conjecture. This raised the question if these chain complexes can be realized topologically by free G-spaces to produce counterexamples to Carlsson’s conjecture. In this talk, I will explain multiplicative properties of the spectral sequence obtained by filtering the mod-p cochains of a space with a free p-group action by powers of the augmentation ideal and deduce that the counterexamples can not be realized topologically. This is joint work with Henrik Rüping.

**Speaker: **Sune Precht Reeh (BGSMath-UAB)**Title:**** **A formula for p-completion by the way of the Segal conjecture**Place:** Room Seminar C3b

**Date:**Friday May 10, 10h-11h

**Abstract:**A variant of the Segal conjecture (theorem by Carlsson) gives a correspondence between homotopy classes of stable maps from BG to BH and the module of (G,H)-bisets that are H-free and where the module is completed with respect to the augmentation ideal I(G) in the Burnside ring of G. The details of this correspondence change depending on whether you add a disjoint basepoint to BG, BH, or both, and it is also not a priori clear what algebraic consequences the I(G)-adic completion has for the module of (G,H)-bisets.

Separately, we have the functor of p-completion for spaces or spectra. We can apply p-completion to each classifying space BG, and according to the Martino-Priddy conjecture (theorem by Oliver) the p-completed classifying space depends only on the saturated fusion system F_p(G) of G at the prime p.

Saturated fusion systems also have modules of bisets, and so it is not unreasonable to ask how p-completion interacts with the Segal conjecture: Suppose we are given a (G,H)-biset, we can interpret the biset as a stable map from BG to BH. Apply p-completion to get a stable map from BF_p(G) to BF_p(H). By the Segal conjecture for fusion systems, that stable map corresponds to an (F_p(G), F_p(H))-biset -- up to p-adic completion. Which (F_p(G),F_p(H))-biset do we get?

This innocent question was the starting point for a joint paper with Tomer Schlank and Nathaniel Stapleton, and in my talk I will give an overview of all the categories involved and how they fit together with functors. If time permits, we will even see how p-completion and fusion systems can help us understand the I(G)-adic completion for any finite group -- and I suppose we might even consider that "a formula for the Segal conjecture by way of p-completion".