# Séminaire Algèbre et topologie

### organisé par l'équipe Algèbre, représentations, topologie

• ##### Anne Moreau

Orbites nilpotentes provenant d'algèbres vertex affines admissibles.

18 janvier 2022 - 14:00Salle de séminaires IRMA

Résumé : dans cet exposé, je donnerai une description simple, en terme d’idéaux primitifs, des adhérences d’orbites nilpotentes qui apparaissent comme variétés associées aux algèbres vertex affines de niveaux admissibles. Ces variétés sont également liées à la cohomologie du petit groupe quantique associée à une racine q-ième de l’unité. Il s’agit d’un travail en commun avec Tomoyuki Arakawa et Jethro van Ekeren.
• ##### Dimitri Zvonkine

Invariants de Gromov-Witten des intersections complètes

25 janvier 2022 - 14:00Salle de séminaires IRMA

Résumé : Nous présentons un algorithme qui permet de calculer tous les invariants de Gromov-Witten (GW) de toutes les intersections complètes. L'outil principal est la formule de dégénérescence de Jun Li, qui exprime les invariants de GW d'une intersection complète à partir de ceux de plusieurs intersections complètes plus simples. La difficulté principale est que cette formule ne s'applique pas aux classes de cohomologie primitives. Nous arrivons à contourner ce problème en introduisant les invariants de GW nodaux. Nous montrons que les invariants nodaux sans insertions primitives peuvent toujours être calculés par la formule de dégénérescence. D'autre part, ces invariants contiennent assez d'informations pour retrouver tous les invariants de GW. Travail commun avec H. Argüz, P. Bousseau et R. Pandharipande.
• ##### Sylvain Douteau

Entrelacs homotopiques et théories de l'homotopie stratifiée

1 février 2022 - 14:00Salle de séminaires IRMA

L'étude des espaces stratifiés, et de leurs invariants, débute avec le théorème de Whitney, qui garantit que toute variété algébrique, réelle ou complexe, peut être décomposée en variétés lisses, satisfaisant des conditions de recollement. S'en sont suivies de nombreuses généralisations d'invariants classiques au cas stratifié : la signature, la cohomologie d'intersection et la catégorie des chemins sortants par exemple. Ces nouveaux invariants n'étant compatibles qu'avec les homotopies préservant la stratification, il s'impose alors de définir une théorie homotopique adaptée aux espaces stratifiés. Dans cet exposé, je présenterai deux approches, produisant deux théories de l'homotopie stratifiée a priori distinctes. A travers l'étude des entrelacs - des objets encodant les instructions de recollement entre strates - j'expliquerai pourquoi ces deux théories coïncident. L'exposé sera en parti basé sur des travaux en commun avec Lukas Waas (Université d'Heidelberg).
• ##### Bernhard Keller

Catégorification du tressage grassmannien

8 février 2022 - 14:00Salle de conférences IRMA

Résumé : Chris Fraser a découvert une action du groupe de tresses affine
étendu à d brins sur la Grassmannienne des sous-espaces de dimension k dans un espace de
dimension n. Ici, l'entier d est le pgcd de k et n. Nous relevons cette action en une
action sur la catégorie amassée correspondante construite d'abord par Geiss-Leclerc-Schroeer en
2008. Pour cela, nous nous servons de la description de cette catégorie comme catégorie
de singularités au sens de Buchweitz/Orlov donnée par Jensen-King-Su en 2016. Nous
conjecturons une action du même groupe de tresses sur l'algèbre amassée associée à un couple
arbitraire de diagrammes de Dynkin dont les nombres de Coxeter sont k et n-k. Travail
en cours avec Chris Fraser.
• ##### Ioannis Lavdas

Holomorphic boundary conditions for topological field theories via branes in twisted supergravity

17 mai 2022 - 14:00Salle de séminaires IRMA

Three-dimensional N =4 supersymmetric field theories admit a natural class of chiral half-BPS boundary conditions that preserve N = (0, 4) supersymmetry. While such boundary conditions are not compatible with topological twists, deformations that define boundary conditions for the topological theories were recently introduced by Costello and Gaiotto. Not all N = (0, 4) boundary conditions admit such deformations. We revisit this construction, by engineering the holomorphic theory on the worldvolume of a D-brane. Our brane engineering approach combines the intersecting brane configurations of Hanany–Witten with recent work of Costello and Li on twisted supergravity. The latter approach allows to realize holomorphically and topologically twisted field theories directly as worldvolume theories in deformed supergravity backgrounds, and we make extensive use of this.
• ##### Bora Yalkinoglu

Un pont entre nombres premiers et nœuds

24 mai 2022 - 14:00Salle de séminaires IRMA

On va expliquer comment le célèbre flot de Toda (qui apparaissait déjà dans les travaux de Frobenius) donne un lien (conjectural) entre les nombres premiers et des nœuds. Si le temps le permet on va aussi expliquer comment le flot de Toda est lié à d’autres aspects arithmétiques.
• ##### Andrew Baker

P-algebras and their modules with some topological applications

31 mai 2022 - 14:00Salle de séminaires IRMA

P-algebras were introduced by Margolis, building on earlier work of Moore and Peterson on nearly Frobenius algebras. The motivating examples are certain large subHopf algebras of the Steenrod algebra at a prime. A (graded, connective) Hopf algebra over a field is a P-algebra if it is a union of finite dimensional subHopf algebras; since finite dimensional Hopf algebras are Poincar\'e duality algebras (the graded version of Frobenius algebras), and satisfy a flatness condition, such a P-algebra is coherent but not Noetherian. Nevertheless, bounded below modules have reasonably tractable properties, generalising well-known results for Poincar\'e duality algebras and I will discuss particularly finite dimensional modules and coherent modules with an emphasis on calculation of Ext groups. Armed with this technology it is now easy to prove vanishing results for homotopy mappings sets $[X,Y]^*$ for various pairs of $p$-complete spectra, some of which seem not to have been noted previously.
• ##### Nathalie Wahl

String topology operations: examples

7 juin 2022 - 14:00Salle de séminaires IRMA

String topology is the study of certain types of operations on the homology of the free loop space of manifolds. By now we know many non-trivial string operations. I'll give a glimpse of the current state of the subject, illustrated by examples.
• ##### Leonid Rybnikov

Bethe subalgebras in Yangians and wonderful compactifications

14 juin 2022 - 14:00Salle de séminaires IRMA

Résumé : Bethe subalgebras form a family of maximal commutative subalgebras in the Yangian Y(g) of any simple Lie algebra g, parametrized by regular elements of the corresponding Lie group G. The generators of these subalgebras can be regarded as integrals of the (generalized) XXX Heisenberg magnet chain. We extend the parameter space for these subalgebras by considering certain limits of the subalgebras from this family. In particular, we get a family of maximal commutative subalgebras in Y(g) parametrized by the toric version of the De Concini - Procesi wonderful closure of the complement to a root hyperplane arrangement, recently introduced by De Concini and Gaiffi. This (conjecturally) gives a Kirillov-Reshetikhin crystal structure on the solutions of Bethe ansatz for the XXX chain and explains the action of the fundamental group of the real form of the De Concini - Gaiffi compactification on KR crystals. This is joint project with Aleksei Ilin.
• ##### Lukas Brantner

Purely inseparable Galois theory

6 juillet 2022 - 11:00Salle de séminaires IRMA

Attention: Horaire inhabituel Résumé: An algebraic extension of fields F/K of characteristic p is purely inseparable if for each x in F, some power x^{p^n} belongs to K. Using homotopical methods, we construct a Galois correspondence for finite purely inseparable field extensions F/K, generalising a classical result of Jacobson for extensions of exponent one (where x^p belongs to K for all x in F). This is joint work with Waldron.
• ##### Guchuan Li

Higher real K-theories from chromatic homotopy theory at prime 2

13 septembre 2022 - 14:00Salle de séminaires IRMA

Résumé : Chromatic homotopy theory is a powerful tool to study periodic phenomena in the stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the homotopy fixed points of Lubin--Tate theories E_h. These fixed points are generalizations of real K-theories with periodic homotopy groups that can be computed via homotopy fixed points spectral sequences. In this talk, we prove that at the prime 2, for all heights h and all finite subgroups G of the Morava stabilizer group, the G-homotopy fixed point spectral sequence of E_h collapses after the N(h,G)-page and admits a horizontal vanishing line of filtration N(h,G) where N(h,G) are specific numbers. Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem. If time allows, I will present a computation of topological modular forms based on this vanishing result. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.
• ##### Xabier García Martínez

Characterising the variety of Lie algebras

27 septembre 2022 - 14:00Salle de séminaires IRMA

Résumé : The variety of Lie algebras is the central example of a variety of non-associative algebras. Many properties, studied from a universal algebra point of view, were firstly introduced to Lie algebras and then generalised to other different structures. One of the many aims of categorical algebra is to give general definitions that can help to understand these notions and their generalisations. The aim of this talk is to discuss two categorical algebraic ideas, representability of actions and the existence of algebraic exponents. They are both interesting in the following way: we can characterise the variety of Lie algebras amongst all varieties of non-associative algebras over an infinite field. Moreover, these characterisations are categorical; in the sense that no elements are used, only morphisms and universal properties. We will discuss the algebraic meanings of these properties, the computational methods used to obtain the characterisations, and what can be done beyond them. Joint work with Matsvei Tsishyn, Corentin Vienne and Tim Van der Linden.
• ##### Anthony Giaquinto

Schur-Weyl Duality for Braid and Twin Groups

4 octobre 2022 - 14:00Salle de séminaires IRMA

The symmetric group admits two natural covering groups: the braid group and the twin group. These are obtained, respectively, by removing the involution and cubic relations in the Coxeter presentation of the symmetric group. There is a natural Burau representation for each group, which for the braid group is a q-analog of the permutation representation of the symmetric group and for the twin group is a related orthogonal representation generated by complex reflections. New instances of Schur-Weyl duality are found by examining the diagonal action of these groups on tensor powers of the Burau representation. The centralizer algebra of the action of each group is described diagramatically in terms of partial permutation and partial Brauer algebras. As a result, we obtain many representations of the braid and twin groups that can be combinatorially constructed. This is joint work with Stephen Doty.
• ##### Geoffrey Janssens

A polynomial identity perspective on the classification of finite dimensional algebras

11 octobre 2022 - 14:00Salle de séminaires IRMA

Résumé : Given a set S of algebras, a natural problem is to discover which algebras from S are (not) isomorphic. A classical way to attack such `distinguishing problems' is by means of invariants. In this talk we will associate to any finite-dimensional algebra A two invariants originating from the asymptotics of the sequence of codimensions c_n(A). This sequence structures algebraic information contained in the ideal of polynomial identities satisfied by A. It was proven by Berele and Regev that the sequence c_n(A) grows asymptotically as c n^t d^n for some constants c, t, and d depending on A. Surprisingly the invariant t is an half-integer and the invariant d even an integer. In the first half of the talk, we will give an introduction to these concepts and the role of S_n-representation theory hereby. The second part will be based on joint work with Aljadeff and Karasik. Concretely, we will explain the representation and ring theoretical information contained in the numbers t and d. Hereby, the aim will especially be to recall and explain the role of Kemer's theory (developed for his deep solution to the Specht problem) in our work and comment on further use of it.
• ##### Pierre Guillot

Les actions binaires des groupes simples

18 octobre 2022 - 14:00Salle de séminaires IRMA

Résumé : une action de groupe est binaire lorsqu'on peut trouver une structure de graphe homogène sur l'ensemble portant l'action, de sorte que le groupe soit précisément formé des automorphismes de la structure. Plus généralement, on peut définir la complexité d'une action de groupe, le cas binaire correspondant à la complexité 2. On sait par des principes généraux qu'il existe des classifications élégantes des groupes de permutations de complexité donnée, mais aucun théorème explicite n'a été prouvé. Dans un travail en commun avec N. Gill, nous étudions les actions binaires des groupes simples et prouvons qu'elles sont rares. En particulier, les groupes alternés n'ont aucune action binaire non triviale.
• ##### Aleksandar Milivojević

Formality, Massey products, and non-zero degree maps

8 novembre 2022 - 14:00Salle de séminaires IRMA

Résumé : I will discuss results related to the following basic question in rational homotopy theory: given a non-zero degree map of rational Poincaré duality spaces, does formality of the domain imply formality of the target? As a helpful tool, which might be of independent utility, I will describe a construction that takes in a cohomologically finite dimensional and connected commutative differential graded algebra and produces one that satisfies Poincaré duality. This is joint work with Jonas Stelzig and Leopold Zoller.
• ##### Ilia Itenberg

Real enumerative invariants and their refinement

15 novembre 2022 - 14:00Salle de séminaires IRMA

Résumé : The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of toric surfaces) that arise as results of an appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss the combinatorics of tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them. This is a joint work with Eugenii Shustin.
• ##### Medina Mardones Anibal

Effective algebro-homotopical constructions and applications

22 novembre 2022 - 15:15Salle de séminaires IRMA

Résumé : In order to incorporate ideas from algebraic topology in concrete contexts such as topological data analysis and topological lattice field theories, it is necessary to have effective constructions of concepts defined only indirectly or transcendentally. This talk will discuss how to effectively construct homotopy invariants using the broken symmetry of the diagonal map of a cellular space. More specifically, from an algebra structure on its cochains that is both commutative and associative up to coherent homotopies. This algebraic structure encodes the entire homotopy type of the space by a result of Mandell, and during the talk it will be described in terms of a finite number of generating multi-operations. Time permitting, this talk will also outline some applications of such constructions.
• ##### Vyacheslav Futorny

Tame and strongly tame representations of simple affine vertex algebras of type A

6 décembre 2022 - 14:00Salle de séminaires IRMA

Résumé : We will apply the localization technique to construct explicit Gelfand-Tsetlin realizations of Zhu’s algebras for admissible representations of simple affine vertex algebras of type A.