Après-midis de Topologie - archive

Gabriel Angelini-Knoll

Matthew Feller

Tasos Moulinos

Sophie d’Espalungue: Boardman Vogt tensor product of operads in Cat and interchange of En structures

Iterated monoidal categories were introduced by Baltaneau, Fiedorowitzch, Schwänzl and Vogt as a categorical equivalent for iterated loop spaces and defined as algebras over operads $M_n$ with values in the category $Cat$ of categories. We construct a tensor product for operads in $Cat$ verifying $M_n \otimes M_m \cong M_{n+m}$. We equip the category of operads in $Cat$ with a model structure such that the tensor product preserves cofibrant objects, providing explicit models for homotopy n-fold monoidal structures.

15h30-16h30 Guglielmo Nocera: The E3-structure on the spherical category of a reductive group

Let G be a reductive group over the complex numbers, e.g. GLn,C. There is a monoidal triangulated/dg/∞-category Sph(G), called the spherical category of G, which plays an important role in the Geometric Langlands program. For example, its behaviour provides important constraints in the formulation of the Geometric Langlands Conjecture. This ∞-category is not symmetric monoidal, but it admits a t-structure whose heart is symmetric monoidal: more precisely, the heart is monoidal-equivalent to a category of representations of a group (the Langlands dual of G) with its (symmetric monoidal) tensor product. In this talk, I will present how to upgrade the existing E1-monoidal structure on Sph(G) to an E3-monoidal one, which formally recovers the symmetric monoidal structure of the heart. The construction implements ideas of Jacob Lurie and uses a strongly topologically flavoured presentation of Sph(G), namely as an ∞-category of constructible sheaves over a stratified space. If time permits, I will also briefly explain its connection to ongoing work of Campbell–Raskin, which allows to show that our E3-structure agrees with the interpretation of Sph(G) as the operadic E2-center of a derived ∞-category of representations induced by results of Bezrukavnikov–Finkelberg and Ben-Zvi–Francis–Nadler. Part of this work is joint with Morena Porzio.

Victor Pecanha Brittes: Localisations of the model structure for 2-quasi-categories

2-quasi-categories are a model for (infinity,2)-categories introduced by Ara. They are the fibrant objects of a model structure on the category of Θ_2-sets, where Θ_2 is a 2-dimensional analog of the simplex category Δ. In this talk, we will explain how we can localise this model structure to obtain models for the homotopy theory of spaces, 2-categories and homotopy 2-types.

Ran Azouri: Motivic invariants and singularities of schemes

The structure of the motivic stable homotopy category produces invariants for schemes of a quadratic nature, such as an Euler characteristic defined in quadratic forms, and Euler classes defined in cohomology theories that contain quadratic information. We will present some results on those invariants, refining classical formulas in integers. Then we turn to invariants related to degenerations using motivic nearby cycles, concluding in a quadratic conductor formula.

Victoria Callet: Topological modeling for music classification

Persistent homology is a computational tool created at the end of the 20th century for applied algebraic topology. The main idea is to understand the topological structure of a starting object by progressive approximations: for this we use simplicial theory and more precisely simplicial complex and homology, which we will quickly recall at the beginning. In practice, we extract a point cloud from our initial object and transform it into a filtered simplicial complex using a method called the Vietoris-Rips filtration. Persistent homology then encodes the evolution of homology classes and more precisely their lifetime during the newly created filtration: we will represent all this information on a family of graphs called barcodes, from which we will be able to analyze and compare further starting objects. We call this process Topological Data Analysis. As an illustration, we will see how this process can be applied to the classification of musical styles using the Discrete Fourier Transform.

Sylvain Douteau Weighted distances and topologies on Ran spaces

Given a Riemanian manifold M, its Ran space is a stratified space containing all finite configurations of points in M. Its strata, which are configuration spaces of some given cardinality, have been extensively studied, and the space itself is central in factorization homology. Yet, there is some ambiguity on the topology that one should consider on Ran(M). On one hand, there is the weak-topology, obtained by considering Ran(M) as the colimit of its truncation to configurations of bounded cardinal. While extremely natural, this topology is hard to describe. On the other hand, Ran(M) inherits a metric from M, called the Hausdorff distance, which induces another, coarser, topology. In this talk, I will present new topologies on Ran(M), induced from weighted distances which I will introduce. They interpolate between the weak topology and the Hausdorff topology, and collectively allow for a more explicit description of the weak topology.

14h-15h : Mikala Janssen : Partial and unstable algebraic K-theory

We compare Yuan’s partial algebraic K-theory with the models for unstable algebraic K-theory given by the reductive Borel-Serre categories. Partial algebraic K-theory is a “non-group completed” version of algebraic K-theory defined in terms of a universal property whereas the reductive Borel-Serre categories are hands-on 1-categories that assemble to a monoidal category. It turns out that they are equivalent as E_1-spaces.This is work in progress, joint with Dustin Clausen.

15h15-16h15 : Erik Lindell : Stable cohomology of and the IA-automorphism group

The automorphism group of the free group is an object that for a long time has been of interest in group theory and in low-dimensional topology, where it appears as a kind of mapping class group of a finite graph. In the two recent decades, a lot of progress has been made in understanding the cohomology of this group. In particular, there have been several results about the stable cohomology, i.e. the part where n is sufficiently large compared to the cohomological degree, where it becomes independent of n. In this talk, I will describe recent results about the stable cohomology with certain “twisted” coefficients, and work-in-progress about how it can be applied to study the cohomology of the IA-automorphism group, i.e. the subgroup of automorphisms acting trivially on the abelianization of . This group is analogous to the Torelli subgroup of the mapping class group of a surface and very little is generally known about its cohomology.

16h30-17h30 : Félix Loubaton : L’univalence lax pour les -catégories

Dans cette présentation, je formulerai l’univalence lax pour les $\omega$-catégories. J’expliquerai ensuite comment ce résultat nous permet d’exprimer un lien fort entre la construction de Grothendieck pour les -catégories et les lax-colimites.

Ismaïl Razack: Hochschild cohomology, Batalin-Vilkovisky algebras and operads

The Hochschild cohomology of a differential graded algebra (DGA) is naturally endowed with a Gerstenhaber algebra structure. This structure can be enhanced into a Batalin-Vilkovisky algebra (BV algebra) when the DGA verifies some form of symmetry. For instance, Luc Menichi showed that HH*(C*(M)), the Hochschild cohomology of the cochain complex of a smooth, compact, simply connected manifold M, is a BV-algebra. The goal of this talk is to give a new proof of this result using operad theory (Barratt-Eccles operad) and without assuming that M is simply connected. We’ll also explain why we get a similar result if we replace a manifold by a pseudomanifold.

Ana Sopena Gilboy: Pluripotential operadic calculus

For complex manifolds, there exists a refined notion of weak equivalence related to both Dolbeault and anti-Dolbeault cohomology. This class of weak equivalences naturally defines a stronger formality notion. In particular, satisfying the ddbar-Lemma property does not imply formality in this new sense. The goal of this talk is to introduce a novel operadic framework designed to understand this homotopical situation. I will present pluripotential A-infinity algebras as well as a homotopy transfer theorem based on this strong notion of weak equivalence.

Lukas Waas: The topological stratified homotopy hypothesis

Roughly speaking, 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 topological analogue of this principle: The homotopy theory of (topological) stratified spaces should be the same as the homotopy theory of layered infinity-categories, i.e. such infinity-categories in which every endomorphism is an isomorphism. We are going to present a formal interpretation of this statement. Namely, we prove the existence of a simplicial semi-model structure for stratified spaces in which most geometrically relevant examples – such as Whitney stratified spaces and PL pseudomanifolds - are bifibrant. We then prove a Quillen equivalence (in terms of Lurie’s exit-path construction) of this model category with a left Bousfield localization of the Joyal model structure presenting layered infinity-categories. In particular, the following interpretation of the topological homotopy hypothesis follows: Lurie’s exit-path construction induces an equivalence between the localization of bifibrant stratified spaces at stratified homotopy equivalences and the homotopy theory of layered infinity-categories.

Dan Berwick-Evans (14h) : What is the homotopy type of quantum field theory?

Spaces of quantum theories are the fundamental objects in several modern applications of algebraic topology to theoretical physics. In this talk, I will begin by explaining how twisted equivariant K-theory encodes the homotopy type of the space of (supersymmetric) quantum mechanical systems. Viewing quantum systems as 1-dimensional quantum field theories, generalizing these structures suggests a connection between 2-dimensional (supersymmetric) quantum field theories and twisted equivariant elliptic cohomology, building on ideas of Segal, Stolz and Teichner.

Sacha Ikonicoff (15h15) : Quillen Cohomology of Divided Power Algebras over an Operad

Divided power algebras are algebras equipped with additional monomial operations. They naturally arise in the context of positive characteristic, particularly in the study of simplicial algebras, crystalline cohomology, and deformation theory. An operad is an algebraic object that encodes operations: there is an operad for associative algebras, one for commutative algebras, for Lie algebras, Poisson algebras, and so on. Each operad produces a category of associated algebras, as well as a category of divided power algebras. The purpose of this talk is to demonstrate how Quillen cohomology generalizes to many categories of algebras using the notion of operad. We will introduce the concepts of modules and derivations, as well as an object representing modules - called the universal enveloping algebra - and an object representing derivations - called the module of Kähler differentials - which will allow us to construct an analogue of the cotangent complex. We will show how these concepts allow us to recover known cohomological theories on certain categories of algebras and provide new and somewhat exotic notions when applied to divided power algebras.

Pedro Magalhaes (16h45) : Formality of Kähler manifolds revisited

The interaction of Hodge structures with rational homotopy theory is a powerful tool to provide restrictions on the homotopy types of Kähler manifolds and of complex algebraic varieties. An example is the well-known result of Deligne, Griffiths, Morgan and Sullivan, stating that compact Kähler manifolds are formal. In the simply connected case, it implies, for instance, that the rational homotopy groups of such manifolds are a formal consequence of the cohomology. Despite this fact, the mixed Hodge structure on their rational homotopy groups is not, in general, a formal consequence of the Hodge structures on cohomology. To understand this phenomenon, we will introduce a stronger notion of formality which arises from studying homotopy theory in a category encoding the Hodge structures. We will also introduce obstructions to this strong formality, generalizing the classical ones, and study when are Kähler manifolds formal in this stronger sense.

Francesca Pratali : Rectification of operadic left fibrations

By a result of Heuts-Moerdijk, the oo-category of simplicial diagrams on the nerve of a discrete category A is equivalent to that of left fibrations over the nerve of A. This is an instance of the well known Grothendieck-Lurie straightening-unstraightening theorem. In this talk, we will explain how one can generalize this result to the operadic case. More specifically, by working with the dendroidal formalism we show how, given any discrete operad P, one can functorially rectify an operadic left fibration over the dendroidal nerve of P and obtain a simplicial algebra on P. After explaining how this extends an analogous functor for categories, we prove that it establishes an equivalence of oo-categories between operadic left fibrations over the nerve of P and simplicial P-algebras. A first step towards operadic straightening-unstraightening! If time permits, we will conclude the exposition by presenting possible future applications.

Birgit Richter : Involutive Hochschild homology and reflexive homology as equivariant Loday constructions

In joint work with Lindenstrauss and Zou we identified real topological Hochschild homology of a nice genuine commutative C_2 ring spectrum A with the C_2-Loday construction for the one-point compactification of the sign representation. The same C_2-Loday construction applied to the fixed Tambara functor associated to a commutative ring R with involution can be identified with a Tambara functor of a two-sided bar construction and this in turn relates in good cases to the involutive Hochschild homology and the reflexive homology of R. In work in progress our aim is to get rid of the assumption of commutativity. This is joint work with Ayelet Lindenstrauss.

Victor Carmona : AQFTs vs. Factorization Algebras: toward a higher comparison

Quantum Field Theory (QFT) is an exciting yet elusive domain within mathematics and physics. Despite the lack of rigorous foundations to support many advancements made by physicists, mathematicians have engaged in a fruitful endeavor to formalize QFTs. In current times, we find ourselves at a crossroads: while we still lack the comprehensive techniques and language to fully grasp QFT, numerous distinct axiomatic frameworks are attempting to capture its essence. A natural question arises: how are these approaches connected? This talk will focus on two such frameworks: Algebraic Quantum Field Theories (AQFTs) and Factorization Algebras (FAs), both of which encapsulate the algebraic structure carried by observables in a QFT. The significance of these frameworks is motivated, among other things, by rigorous programs led by Fredenhagen-Rejzner and Costello-Gwilliam to construct perturbative QFTs using AQFTs and FAs, respectively. Recent contributions from Gwilliam-Rejzner and Benini-Musante-Schenkel establish a relationship between these two programs, vaguely speaking. At a more structural level, Benini-Perin-Schenkel have established an equivalence of 1-categories between specific subcategories of AQFTs and (time-orderable pre-)FAs. The goal of this talk is to present our strategy for establishing an even broader equivalence between ∞-categories of AQFTs and tpFAs. This additional layer of generality is crucial for accommodating gauge theories, which are QFTs with non-trivial homotopical content. This talk is based on ongoing joint work with M.Benini, A.Grant-Stuart and A.Schenkel.

Vadim Lebovici : Théorèmes de décomposabilité des modules de persistance multiparamétriques

La décomposabilité des modules de persistance uniparamétriques en somme directe de modules intervalles — l’existence des fameux “codes-barres” — est la clé de voûte de la théorie de l’homologie persistante et de ses applications en analyse topologique de données et en géométrie symplectique. L’impossibilité de trouver de telles décompositions dans le cas multiparamétrique a suscité le développement de diverses approches. Dans cet exposé, je présenterai l’une d’entre elles, consistant à exhiber des sous-classes de modules de persistance admettant effectivement une décomposition en somme de modules intervalles. En outre, l’appartenance à ses sous-classes peut-être testée localement, i.e., sur des sous-ensembles très simples de l’espace des paramètres. Cet exposé est basé sur des travaux en collaboration avec Magnus B. Botnan, Jan-Paul Lerch et Steve Oudot.

Tasos Moulinos : Lifting the Hilbert additive group to the sphere

The Hilbert additive group scheme arises in algebraic geometry as the “unipotent completion” of the integers over Spec(Z). At the level of functions, it carries a canonical filtration compatible with the group structure. This is intimately related to the HKR filtration on Hochschild homology. I will review the above story and then discuss ongoing work (joint with Alice Hedenlund) on lifting the Hilbert additive group, together with its filtration on functions, to the setting of derived algebraic geometry over the sphere spectrum. This crucially uses the yoga of even filtrations, introduced by Hahn-Raksit-Wilson, which I will briefly review as well.

Victor Roca i Lucio (CNRS / UPCité) : Higher Lie theory in positive characteristic

Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L-infinity-algebras, giving in general infinity-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.

Robin Sroka (Universität Münster) : Scissors automorphism groups and their homology

Two polytopes in Euclidean n-space are called scissors congruent if one can be cut into finitely many polytopic pieces that can be rearranged by Euclidean isometries to form the other. A generalized version of Hilbert’s third problem asks for a classification of Euclidean n-polytopes up to scissors congruence. In this talk, we consider the complementary question and study the scissors automorphism group – it encodes all transformations realizing the scissors congruence relation between distinct polytopes. This leads to a group-theoretic interpretation of Zakharevich’s higher scissors congruence K-theory. By varying the notion of polytope, scissors automorphism groups recover many important examples of groups appearing in dynamics and geometric group theory including Brin–Thompson groups and groups of rectangular exchange transformations. Combined with recently developed computational tools for scissors congruence K-theory, we recover and extend calculations of their homology. This talk is based on joint work with Kupers–Lemann–Malkiewich–Miller.

Juan-Ramón Gómez-García : Turaev’s coproduct and parabolic restriction

Inspired by Jaeger’s composition formula for the HOMFLY polynomial, Turaev defined a coproduct on the HOMFLY skein algebra of a framed surface S, turning it into a bialgebra. Jaeger’s formula can be viewed as a universal version of the restriction of the fundamental representation from GL_{m+n} to GL_m \times \GL_n. The restriction functor is, however, not braided hence it cannot be extended to the skein category of an arbitrary surface. So there was a priori no reason for Turaev’s coproduct to be well-defined.. In this talk, I will explain how to construct a universal version of parabolic restriction on framed surfaces, using skein theory with defects. Precisely, parabolic restriction yields a morphism (bimodule) between the GL_t-skein category and (GL_t \times GL_t)-skein category of S. This construction depends on the choice of the framing, is compatible with gluing surfaces and recovers the Turaev’s coproduct when applied to links, justifying why this is well-defined.

Hyungseop Kim : Some formal gluing diagrams for continuous K-theory

The study of descent properties of K-theory plays an important role in understanding its values and behaviour in many geometric contexts. In this talk, I will explain a construction of certain diagrams arising from formal gluing situations for which continuous K-theory, and more generally all localising invariants, satisfy descent, from the perspective of dualisable categories. I will also discuss how this encompasses both Clausen–Scholze’s gluing result for analytic adic spaces and an adelic descent for dualisable categories.

Maria Yakerson : Fun facts about p-perfection

In this talk we will discuss the structure of $\mathbb E_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen’s $+$-construction, similarly to group completion. This is joint work with Maxime Ramzi.