Mini-cours

Maria Yakerson (Sorbonne Université)
Motivic obstruction theory

In topology, obstruction theory is a powerful tool that allows one to tackle lifting and extension problems. In particular, obstruction theory helps to classify vector bundles in terms of algebraic invariants. More concretely, it often provides an answer to the question whether a vector bundle can be decomposed as a direct sum with a trivial subbundle. Motivic homotopy theory is a modern research area that combines homotopy theory with algebraic geometry. The main motto is that we apply, when possible, methods from homotopy theory to gain new information about objects in algebraic geometry. In particular, there is a motivic version of obstruction theory, developed by F. Morel, A. Asok, J. Fasel et al., which allows us to approach splitting problems for algebraic vector bundles on smooth affine varieties. In this lecture series, we will see the main instruments of motivic obstruction theory which stem from topological ideas, and we will learn how these ideas give applications in algebra and algebraic geometry.

Exposés invités

David Chataur (Amiens)
Stratified Homotopy Theory and Intersection Cohomology

In the 1980s, after having introduced Intersection Cohomology, M. Goresky and R. MacPherson proposed a number of problems and conjectures regarding homotopical foundations for this cohomological theory. Such an homotopical enhancement will have some potential applications to the study of singular sopaces in geometry. In this talk I will give a survey of such an enhancement based on a simplicial approach to Intersection Cohomology and on stratified homotopy theory. In particular I will discuss a "motivic like" approach to the subject developped in collaboration with S. Douteau.

Bérénice Delcroix-Oger (Montpellier)
Poset topology, Koszul duality and new criteria for shellability

Every (finite) poset can be associated with a simplicial complex called its order complex. A poset is said to be Cohen-Macaulay if this simplicial complex is homotopic to a wedge of spheres. Determining whether a poset is Cohen-Macaulay is a very rich question, which Björner and Wachs partially answered in the 1990s by presenting various combinatorial criteria called shellability. In the 2000s, Vallette drew the link between Koszulness of operads and Cohen-Macaulayness, which was further extended in a joint work with Joan Millès and Eric Hoffbeck to a link between PBW basis for operads and a certain kind of shellability. Finally, in a recent work with Matthieu Josuat-Vergès and Lucas Randazzo and a work in progress with Bishal Deb and Matthieu Josuat-Vergès, we introduced a new way to prove the shellability of a poset. In this talk, we will present these criteria for shellability, enhance the link between shellability and Koszulness of operads and present a new way to show the shellability of some posets. This presentation is based on collaborations with Eric Hoffbeck, Joan Millès, Bishal Deb, Matthieu Josuat-Vergès and Lucas Randazzo.

Jean Fasel (Grenoble)
Classification of vector bundles on real affine algebraic varieties: does topology help?

In this talk, I will report on recent work on the classification of vector bundles on smooth affine real varieties. I will mainly discuss the problem of splitting a free factor of rank one for vector bundles of corank 1, a real analogue of the so-called Murthy’s conjecture over algebraically closed fields. The proofs are based on motivic obstruction theory, and more precisely on interactions between the relevant algebraic obstruction groups and their analogues in topology. This is joint work with A. Asok and S. Lerbet.

Christine Lescop (Grenoble)
Sur des invariants topologiques qui comptent des graphes dans des variétés

On va décrire des manières de compter des graphes dans des variétés, en suivant des idées de Gauss, Witten, Kontsevich, Kuperberg et Thurston et d'autres pour produire des invariants topologiques exploitables. On commencera par présenter des exemples simples de tels invariants avant d'introduire une classe caractéristique dûe à Kontsevich et Watanabe qui compte des configurations du graphe complet à 4 sommets dans des fibrés lisses en boules de dimension 4 trivialisés le long des bords des fibres. Cette classe caractéristique a permis à Watanabe d'exhiber en 2018 un fibré lisse E_W non trivial en boules D^4 de dimension 4 au-dessus de la sphère S^2, dont le bord est le fibré trivial S^2 x ∂D^4. L'existence d'un tel fibré implique que le pi_1 du groupe de difféomorphismes préservant l'orientation de la sphère de dimension 4 n'est pas isomorphe au pi_1 du groupe orthogonal SO(5), contrairement à une prédiction souvent appelée « la conjecture de Smale en dimension 4 ». On terminera l'exposé en esquissant la démonstration de Watanabe de la réfutation de cette conjecture.

Félix Loubaton (Bonn)
Exact double ∞-categories

It is a basic observation that equivalence relations on sets correspond bijectively to surjective maps. More generally, one can define equivalence relations in any category and ask whether this correspondence still holds. Categories with this property are usually called exact. This notion of exactness has been extended to ∞-categories by Lurie, Toën, and Vezzosi, where equivalence relations are replaced by internal groupoids. In this talk, I will present ongoing work with Jaco Ruit on a further generalization of exactness in the context of double ∞-categories, a two-dimensional generalization of ∞-categories. In this setting, internal groupoids are replaced by monads. As part of a collaboration with Fernando Abellán and Hugo Pourcelot, I will then explain how this framework gives rise to a notion of double sheaf, together with examples coming from geometry.

Wolfgang Pitsch (Barcelone)
Integral Homology Spheres: a tale of two filtrations

In this talk, I will explore the relationship between the Johnson filtration of the Torelli group of an orientable surface and the theory of finite-type invariants of 3-dimensional homology spheres. Finite-type invariants are known to be closely connected to the descending central series filtration of the Torelli group, which induces a genuine filtration on the set of integral homology spheres. Until recently, the analogous situation for the Johnson filtration remained unclear, as the first four stages of this filtration do not yield proper subsets of the integral homology spheres. After reviewing the general framework, I will explain how the study of trivial 2-cocycles on subgroups of the Johnson filtration leads to the construction of a finite-type invariant that indeed detects a proper stage of the Johnson filtration. The results presented are joint work with R. Riba.

Exposés sur proposition

Clovis Chabertier (Nantes et IMJ)
Invariants de Postnikov en homotopie rationnelle

Des modèles algébriques pour les types d'homotopie rationnels ont été proposés par Quillen en 1969 puis Sullivan en 1977. Le modèle de Quillen, utilisant les algèbres de Lie, permet de décrire les types d'homotopie rationnels simplement connexes, tandis que le modèle de Sullivan, utilisant les algèbres commutatives, décrit les types d’homotopie finis et nilpotents. Plus récemment, Buijs, Félix, Murillo et Tanré d’une part et Robert-Nicoud et Vallette d’autre part, ont étendu le modèle de Quillen au cas non-simplement connexe. Dans cet exposé, on présentera une théorie des invariants de Postnikov pour les algèbres de Lie qui est compatible avec le foncteur d’intégration de Robert-Nicoud et Vallette. On en déduira une intégration de la cohomologie de Chevalley-Eilenberg en la cohomologie d’espaces à coefficients locaux. Si le temps le permet, on exposera la motivation initiale pour construire une telle théorie des invariants de Postnikov, à savoir, une conjecture proposé par Félix et Tanré, qui relie les modèles en homotopie rationnelle et le modèle de Loday pour les n-types d’homotopie, i.e. les Cat^n-groupes. Si le temps le permet également, nous tenterons d’expliquer en quoi une théorie des invariants de Postnikov peut permettre d’axiomatiser les foncteurs d’intégrations.

Virgile Constantin (EPFL)
Higher Covering Maps, Deck Transformations, and Yoneda in an ∞-Topos

A classical and elementary result in topology identifies the group of deck transformations of a normal covering map (with discrete fibers) with a quotient of the fundamental group. In this talk, I will explain how to internalize this result in a fixed ∞-topos E, obtaining an isomorphism of group objects in E. Even better, for coverings with (n-1)-truncated fibers, we obtain (under an additional hypothesis) an analogous isomorphism of n-group objects between the deck transformation n-group and a suitable quotient of the fundamental n-group. The key input is an internal form of the Yoneda lemma (and embedding); I will present and motivate its formulation, and highlight its central role in the proof. As a direct corollary, we obtain a uniqueness result for quotients of higher groups. The talk is designed to be accessible: no prior knowledge of ∞-topoi is necessary.

Théo Deturck (Lille)
Some Ext-groups computations in the category of quantum polynomial functors

The q-Schur algebras are deformations of the classical Schur algebra and are linked with the representation theory of the finite general linear groups and of the quantum general linear groups. To study modules over this algebra, we can work in the category of quantum polynomial functors of Hong and Yacobi, which is a deformation with one parameter q of the more well-know category of strict polynomial functors of Friedlander and Suslin. When q is a root of unity, we can obtain a quantum polynomial functor from a strict polynomial functor via a process that we call the quantum Frobenius twist. The goal of this talk is to show that in many cases we can compute Ext-groups between quantum polynomial functors obtained via the quantum Frobenius twist using our wider knowledge of Ext-groups in the category of strict polynomial functors.

Benachir El Allaoui (Sorbonne Paris Nord)
Vanishing of Ext-groups between representations of semi-additive categories

In this talk, I will show that the extension groups between linearisations of additive functors from a semi-additive category vanish. To prove it, I will relate these Ext-groups to the homology of simplicial inverse monoids. And I will show that for a simplicial inverse monoid, the group structure on his homotopy groups is in fact induced by his inverse monoid structure which allows us to compute his homology. The purpose of this result is to pursue the work of Djament-Gaujal to study the functor categories from a semi-additive category for which there are no polynomial functors, which is the case, for example, for the category of relations.

Colin Fourel (Strasbourg)
Flow categories as exit path categories

Given a Morse function on a closed smooth manifold and a Smale gradient-like vector field adapted to it, one can construct a topological category called the flow category associated with this data. Its objects are the critical points of the function, and its morphisms are the broken trajectories of the vector field connecting critical points. Like any topological category, a flow category models an ∞-category, and a theorem of Cohen–Jones–Segal states that the homotopy type underlying this ∞-category is that of the manifold itself. However, different choices of Morse–Smale pairs on the same manifold can give rise to non-equivalent ∞-categories. After making these ideas more precise, I will present a result I obtained, which asserts that the ∞-category associated with the flow category of a Morse–Smale pair is equivalent to the exit-path ∞-category associated with the stratification of the manifold by the ascending manifolds of the critical points.

Louis Hainaut (Chicago)
Croissance asymptotique de l'homologie des espaces de configurations sur les graphes

Les espaces de configurations sur des variétés topologiques ont fait et font toujours l'objet de nombreuses recherches. De nombreux résultats sont apparus dans la litérature, tant en ce qui concerne leur structure qu'en ce qui concerne des méthodes de calculer leur homologie. Par contraste, il existe peu de résultats lorsque l'espace de base n'est pas localement Euclidien. En collaboration avec Ben Knudsen et Nick Wawrykow, nous avons déterminé la vitesse de croissance asymptotique de l'homologie des espaces de configurations ordonnées sur n'importe quel graphe, d'une part au niveau des nombres de Betti, et d'autre part au niveau de la multiplicité des représentations irréductibles des groupes symmétriques. Dans cet exposé, je commencerai par rappeler certains résultats classiques concernant ces espaces de configurations sur des graphes, puis j'introduirai les structures algébriques nécessaires pour nos résultats et je présenterai les étapes principales de notre argument.

Yohan Mandin-Hublé (Marseille)
Perturbative invariants of combed 3-manifolds

In this talk, I will define an invariant of three-dimensional rational homology spheres equipped with a parallelization, following Witten, Kontsevich, and Kuperberg-Thurston. This invariant counts configurations of trivalent graphs in the given parallelized manifold. It can be corrected to obtain an invariant Z of the manifold using an invariant of the parallelization, which is a linear function of a Pontryaguin class. The invariant Z is a universal finite-type invariant of rational homology spheres known as the perturbative expansion of Chern-Simons theory. I will give a more flexible definition of the invariant Z using a nowhere vanishing vector field on the manifold instead of a parallelization.

Xiabing Ruan (Strasbourg)
A Computation of the Tamarkin-Tsygan Calculus 

The Tamarkin-Tsygan calculus is a collection of invariant and compatible structures of associative algebras. In algebraic geometry, where all the operations are commutative, the calculus posses concrete geometric interpretations. In non-commutative settings, however, the underlying geometric meanings remain mysterious. We compute explicitly the full Tamarkin-Tsygan calculus of a particular associative algebra, which serves as a counter-example of a conjecture of Polishchuk and Positselski relating to non-commutative geometry. Our computation shows that even for an algebra admiting an economical presentation, the operations in its Tamarkin-Tsygan calculus can behave intricately.

Noe Sotto (Paris Saclay)
Equivariant formal deformations and arithmetic deformations of homogeneous varieties

The theory of formal moduli problems studies infinitesimal deformations of algebro-geometric objects, such as varieties. Recent results have shown that all formal moduli problems are governed by kinds of derived Lie algebra (Lurie-Pridham for char 0, Brantner-Matthew for char p, Nuiten and Calaque-Grivaux over a base…). I will build on those results to study equivariant formal deformations with respect to the action of an algebraic group. Then, in the setting of deformations in mixed characteristic – i.e. the question of lifting varieties from a field of characteristic p to rings of higher characteristic – I will explain how the theory of equivariant formal deformations allows to show that a large class of homogeneous varieties have a supposedly rare behaviour: they admit no deformation to any ring of characteristic higher than p.