EPFL Topology Seminar and Reading Group 2023 Fall

 Location: Real World 

For questions about the seminar, please contact the organizer: Victor Roca i Lucio 
For question about the reading group, please contact the organizers: Léo Navarro Chafloque and Victor Roca i Lucio 
This semester, half of the talks will be seminar talks and half of them will be reading group talks by participant members.


Date Time Place Title Speaker
19.09.2023 14:00 CET MA C1 596 A panorama of derived geometry. (Reading group) Léo Navarro Chafloque
26.09.2023 14:00 CET MA B1 524 Right-angled Artin groups, minors and de Verdière invariant. (Seminar talk) Ramón Flores, Universidad de Sevilla
03.10.2023 14:00 CET MA B1 524 The Balmer spectrum of functor categories (Seminar talk) Greg Arone, Stockholm University
10.10.2023 14:00 CET MA B1 524 The cotangent complex (Reading group talk) Christina Kapatsori
17.10.2023 14:00 CET MA B1 524 Stratified homotopy theory: From the homotopy hypothesis to TDA (Seminar talk) Lukas Waas, Heidelberg University
24.10.2023     No event this week.  
31.10.2023 14:00 CET MA B1 524 Formal moduli problems Victor Roca i Lucio
07.11.2023 14:00 CET MA B1 524 Enhanced stable categories for infinite groups and applications (Seminar talk) Juan Omar Gómez, University of Bielefeld
14.11.2023 14:00 CET MA B1 524 Knotted families from graspers (Seminar talk) Danica Kosanović, ETH
21.11.2023     No event this week.  
28.11.2023 14:00 CET MA B1 524 Hilbert and Quot schemes (Reading group talk) Linus Rösler
05.12.2023 14:00 CET MA B1 524

Lois ternaires formelles et cobordisme symplectique

(Seminar talk)

Jean Fasel, Université de Grenoble
12.12.2023 14:00 CET MA B1 524 Spectral and derived approaches (Reading group talk) Sam Lavenir
19.12.2023 14:00 CET MA B1 524 Brane action for coherent ∞-operads and string topology Hugo Pourcelot, University of Florence

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. 




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)

In this talk we will introduce a homotopy-theoretic interpretation of the stable module category for (discrete) infinite groups over a field of positive characteristic. We present two applications of this approach: first, we provide a formula to describe the Picard group of the stable module category for groups with a finite dimensional cocompact model for the classifying space for proper actions; and second, we construct of a family of infinite degree separable commutative algebras giving a negative answer to an open question by P. Balmer.

Knotted families from graspers (Danica Kosanović)

For any smooth manifold M of dimension d≥4 we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into M, in every degree that is a multiple of d−3, and show that they are detected in the Taylor tower of Goodwillie and Weiss.

Lois ternaires formelles et cobordisme symplectique (Jean Fasel)

Le but de cet exposé est d’introduire la notion de loi ternaire formelle, dont le but est de comprendre le cobordisme symplectique, ou plus précisément de calculer une partie de ses groupes d’homotopie. Je commencerai par une brève introduction aux lois de groupes formelles et aux classes de Chern, et rappellerai le calcul des groupes d’homotopie du cobordisme « ordinaire » par Quillen. Je passerai ensuite aux classes de Borel et présenterai leurs propriétés élémentaires pour motiver la définition de loi ternaire formelle. Je calculerai ensuite ces lois dans certains cas particulier, les comparant en passant avec les 2-lois de groupes formelles à la Buchstaber. Je finirai par des questions ouvertes en lien avec le cobordisme symplectique.

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

I will explain the mechanism of brane actions, introduced by Toën in
2013. This construction takes a coherent ∞-operad P and yields a
canonical P-algebra via certain cospans of spaces. It was motivated by
the search for a categorification of Gromov–Witten invariants, and
later realized by Mann and Robalo (in genus 0).
After introducing a generalization of the construction to the colored
non-reduced situation, I will describe applications to string topology,
ie the study of free loop spaces. In particular, my plan is to sketch
how a fully fledged ∞-categorical bivariant theory, although not yet
available, would allow to construct chain level string topology
operations in large generality.