EPFL Topology Seminar and Reading Group 2023 Fall

 Location: Real World 

Program

Reading group : Homotopy theory in algebraic geometry and vice-versa

On one hand, homotopy theoretical tools are becoming essential in algebraic geometry. On the other, algebro-geometrical ideas can provide intuition to some new developments in homotopical/higher algebra. The goal of this reading group is to understand the profound links between these two disciplines. 

Abstracts

Right-angled Artin groups, minors and de Verdière invariant (Ramón Flores)

In the last years, thorough research has been conducted in order to understand graph properties in terms of group properties of the associated right-angled Artin group (RAAG). These properties should be intrinsic, in the sense that they should not depend on a concrete system of generators of the group. In this talk we will show how to approach some graph properties (as for example planarity or outerplanarity) using as input different bases of the cohomology of the RAAG and the good behaviour of the Colin de Verdière invariant with respect to minors.

The Balmer spectrum of functor categories (Greg Arone)

One of the significant advances in equivariant stable homotopy theory in recent years occurred in the study of the spectrum of tensor triangulated ideals, a.k.a the Balmer spectrum, of compact G-spectra. This spectrum is now well understood for many groups G, in particular for abelian G.

The category of n-excisive functors from Spectra to Spectra is closed symmetric monoidal under Day convolution. As a stable monoidal category, it has many formal similarities to the category of G-spectra. For example, compact objects are dualizable. It therefore seems natural to apply the techniques of tensor-triangulated geometry to the study of the category of functors. In this talk we will describe the Balmer spectrum of the category of n-excisive functors. In the process, we describe the analogue of the Burnside ring for excisive functors, which is π_0 of the endomorphism ring of the identity. The result also requires calculating the Tate blueshift for the symmetric group with respect to the family of non-transitive subgroups. Joint with Tobias Barthel, Drew Heard and Beren Sanders.

Stratified homotopy theory: From the homotopy hypothesis to TDA

Roughly, the homotopy hypothesis – due to Grothendieck – states that the homotopy theory of spaces should be the same as the homotopy theory of infinity groupoids. Ayala, Francis and Rosenblyum conjectured a stratified analogue of this principle: The homotopy theory of (topological) stratified spaces should be the same as the homotopy theory of layered infinity categories – such infinity categories in which every endomorphism is an isomorphism. We are going to present a formal interpretation of this statement. Namely, we identify a category of stratified spaces – which include most classical examples such as Whitney stratified spaces and PL pseudo-manifolds – for which we prove that their localization at stratified homotopy equivalences is equivalent to layered infinity categories (even through a Quillen equivalence). We will then investigate how to apply these insights to investigating singular spaces in topological data analysis. Namely, we describe a stratified analogue of the filtered Cech-complex – the persistent stratified homotopy type – and show that is has most of the desirable properties of its non-stratified classical counterpart. These methods can be used to detect, investigate and classify singular shapes in non-stratified data.

Enhanced stable categories for infinite groups and applications (Juan Omar Gómez)

Knotted families from graspers (Danica Kosanović)

Lois ternaires formelles et cobordisme symplectique (Jean Fasel)

Brane action for coherent ∞-operads and string topology (Hugo Pourcelot)