| Tuesday 13th | Wednesday 14th | Thursday 15th |
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09:00: Reception 09:30: Gregory Ginot 10:35: Coffee 11:00: Francesca Pratali 12:30: Lunch at Esplanade 14:30: Bertrand Toën 15:30: Coffee 16:00: Xiabing Ruan |
09:45: Pedro Magalhães 10:50: Coffee 11:30: Grigory Solomadin 13:00: Lunch at Esplanade 15:00: Vladimir Dotsenko 20:00: Dinner at Saveurs |
09:00: Marie-Camille Delarue 10:05: Coffee 10:30: Mario Fuentes 12:00: Lunch at Upsidum |
Gregory Ginot: "Centralisers for factorisation algebras "
Résumé : Centers and centralizers (in the derived world) of associative or En structures encode the deformation of the associated structures. The latter are special case (i.e. locally constant case) of factorization algebras which can in turn be used to study the latter structures. This talk aims at explaining general strategy to describe centralizers for specific types of factorization algebras, namely universal enveloping factorization algebras of Lie algebras, which arise in (some approach to) mathematical physics. and if time permits to describe applications to deformation quantization problems.
Francesca Pratali: "The root functor"
Résumé : Given an infinity-operad P and a set W of unary operations, the localization of P at W is 'the' initial infinity-operad in which the operations in W are inverted. Exhibiting an infinity-operad as a localization gives information on the infinity-category of algebras over it; studying the localization of P at W is particularly convenient when P is discrete, i.e. has discrete spaces of operations, avoiding homotopy-coherence issues. A well-known example of this phenomenon is given by the little disks operad E_n: after a result due to Lurie, E_n is equivalent to the localization of the discrete operad Disk(R^n) of disks and disjoint unions at isotopy equivalences. A consequence of this is that the infinity-category of E_n algebras is equivalent to that of locally constant algebras over Disk(R^n).
In this talk, I will show that this phenomenon holds more generally, and that every infinity-operad is equivalent to the localization of a discrete one. By working in the dendroidal formalism, I will provide a model for such discrete resolution and for the associated morphism, which I call the 'root functor'. I will show how this induces a weak operadic equivalence after localization, explaining how this extends an analogous result for infinity-categories due to Joyal. As an application, I will show that the infinity-category of algebras over an infinity-operad is equivalent to that of locally constant algebras over its discrete resolution. I will conclude with some open questions!
Bertrand Toën: "LSym-categories"
Résumé : The notions of symmetric monoidal \infty-categories and \infty-operads are not always well suited for questions coming from algebraic geometry. I'll explain why, and introduce a new notion of LSym-categories that is better behaved. I'll give some examples coming from algebraic geometry, and mention some applications to the study of motives and "exponential homotopy theory". (joint with J. Nuiten)
Xiabing Ruan: " A Computation of the Tamarkin-Tsygan Calculus"
Résumé : In his 2005 lecture notes, Alexander Polishchuk conjectured that a Koszul algebra of finite global dimension d must have at least d generators. However, in 2016, Natalia Iyudu and Stanislav Shkarin constructed a counterexample: a Koszul algebra with three generators x, y, z and relations (x^2+yx, xz, zy), has global dimension 4. In the same year, Vladimir Dotsenko and Soutrik Roy Chowdhury computed a Gröbner basis for this algebra. In this talk, I will present a complete and explicit computation of the Hochschild (co)homology as well as the Tamarkin–Tsygan calculus of this algebra. The computation relies on techniques including Gröbner bases, algebraic Morse theory, and so on.
Grigory Solomadin: "From wonderful compactifications to the Davis-Januszkiewicz spaces"
Résumé : The Deligne--Mumford compactifications of moduli spaces of genus zero curves with marked points form a topological operad: the closed boundary strata are products of smaller Deligne-Mumford spaces, and the inclusions of strata give an operad structure. It is also possible to define the same topological operad using the theory of "wonderful compactifications" of subspace arrangements of De Concini and Procesi. Both of those viewpoints generalize to higher dimensions: Chen, Gibney and Krashen defined spaces parametrizing "pointed trees of d-dimensional projective spaces", which were proved by Gallardo and Routis to be isomorphic to a certain wonderful compactification.
Algebraically, the homology operad of the operad of genus zero Deligne--Mumford compactifications was studied by Getzler who called it HyperCom. It is known (Drummond-Cole--Vallette, Khoroshkin--Markarian--Shadrin) that this operad represents the homotopy quotient of the operad of Batalin--Vilkovisky algebras by the circle action, which is a differential operator of order 2. In recent work of Dotsenko, Shadrin and Tamaroff, a similar quotient was computed by a differential operator of any given order d. In particular, for order 1, they defined an algebraic operad called 1-HyperCom. On the level of graded vector spaces, components of that later operad can be identified with the so called Davis--Januszkiewicz spaces constructed out of simplicial complexes of nested sets. In this talk, we shall give an introduction to the relevant notions alluded to above and show that, as the dimension d varies, the Chen--Gibney--Krashen spaces interpolate between the Deligne--Mumford compactifications and the Davis--Januszkiewicz spaces. This is a joint work with V. Dotsenko, E. Hoefel, and S. Shadrin.
Pedro Magalhães: "Mixed Hodge vs Pluripotential Homotopy Theory"
Résumé : The rational homotopy type of a complex algebraic variety can be enriched with mixed Hodge structures, capturing additional algebro-geometric information about the variety. On the differential geometric side, the complexified de Rham algebra of a complex manifold naturally carries a bigrading, with its differential decomposing into two components of distinct bidegree. And this bigraded structure reflects the underlying complex geometry. At the intersection of the algebraic and differential realms sit smooth projective varieties and so one can study and compare their respective enrichments homotopically. In this talk, I will survey the homotopical approaches on both sides and explore their interplay in the case of smooth projective varieties.
Vladimir Dotsenko: "Stable rooted trees of projective spaces and homotopy Koszul operads"
Résumé : This is a sequel to the talk of G. Solomadin. In joint work with him, E. Hoefel, and S. Shadrin, we studied the spaces of Chen, Gibney, and Krashen, which generalize the Deligne--Mumford compactifications of moduli spaces of genus zero curves with marked points: genus zero curves are replaced by d-dimensional projective spaces. For the case of genus zero Deligne--Mumford spaces, the operad HyperCom of Getzler is Koszul, with the Koszul dual operad Grav on the shifted homology of the open parts of the moduli spaces (which do not form a topological operad!). Getzler proved this result using purity of mixed Hodge structures; this strategy is no longer available in higher dimensions, since the Hodge structures of the open parts of the spaces of Chen, Gibney, and Krashen are no longer pure. However, combining tools from several different domains, we were able to describe the homology operad for each dimension d completely explicitly. In particular, it turns out that this operad is homotopy Koszul in the sense of Merkulov and Vallette, and the first layer of operations of the Koszul dual homotopy cooperad still comes from the geometry of the open parts (but there are higher layers). I shall explain this result in detail, and also explain a new viewpoint it offers on applications of mixed Hodge structures in homotopy theory.
Marie-Camille Delarue: " Stable homology of variants of the Thompson groups"
Résumé : Thompson's groups are groups of piecewise linear self-homeomorphisms of an interval where the cut points are dyadic rationals. Different variations of these groups exist such as the Higman-Thompson groups or the Brin-Thompson groups. Using Thumann's notion of operad groups, we build a topological model for these groups in the form of categories of 1-"cobordisms" where objects are configurations of points and morphisms are paths of configurations which are allowed to collide in certain ways. We build a scanning map on these topological models to compute the stable homology of Higman- (and Brin-) Thompson groups.
Mario Fuentes: "I need an algebraic model of my space (and I don't know how to construct it)"
Résumé : A central goal in algebraic topology is to construct algebraic objects that faithfully capture the homotopical information of topological spaces. This idea underlies Rational Homotopy Theory, which aims to build differential graded algebras that encode the rational homotopy type of certain spaces. These algebraic models (often associative algebras, Lie algebras, or variations of them) serve as algebraic counterparts to spaces. However, the classical theory initially applied only to a restricted class of spaces, typically requiring some conditions on connectedness (being simply connected / nilpotent / connected) and on finiteness (being finite-type spaces).
Recent developments have made it possible to overcome many of these limitations by constructing complete differential graded Lie algebras as models for simplicial sets, even when the spaces are not connected or not of finite type. There are two parallel approaches to this construction. The operadic approach uses the higher structure of the chains on simplices, along with the bar-cobar adjunction, to construct a Lie algebra associated with them. The inductive approach builds the model of a simplex as a quasi-free Lie algebra, imposing the condition that its vertices are Maurer–Cartan elements. A suitable differential is then defined recursively. Remarkably, both methods yield, up to homotopy, the same model.
The purpose of this talk is to present these Lie algebra constructions and share some new results in the area. In particular, we will show how several finite-type restrictions can be removed. We will also discuss ongoing work involving curved Lie algebras, which allows the construction of algebraic models for non-pointed spaces. This includes models of unpointed mapping spaces and a framework to study the composition of morphisms in this broader setting.
If you are interested in attending, please write as soon as possible to Ricardo Campos .
14h-15h : Sergei Merkulov: "On the interrelations between graph complexes"
Résumé : We study Maxim Kontsevich's graph complex as well as its oriented and targeted versions, and show a new proof of the theorems due to Thomas Willwacher and Marko Zivkovic stating isomorphisms of their cohomology groups. Both theorems follow from one and the same surprisingly short argument.
15h30-16h30 : Andy Tonks: "Canonical B∞-algebra structures and a new Milnor-Moore type theorem"
Résumé : The motivating example of a B∞-algebra was given by Baues in his work on iterated loop spaces, when he endowed the bar construction (a free coalgebra) with a compatible differential graded algebra structure. Another famous example is the B∞ structure on the Hochschild cochain complex: the relation of B∞ and G∞ algebras was central in the resolutions of Deligne's Hochschild cohomology conjecture. We present work in progress (joint with M I Gálvez and M O Ronco) investigating the appearance of canonical B∞-algebra structures on any algebra endowed with a not-necessarily compatible dga structure, recovering previous results by Markl on A∞-algebras and by Loday-Ronco on multibrace algebras. Application to a new Milnor-Moore type theorem will then be given, extending the results of Loday-Ronco for non-cocommutative Hopf algebras and their relation to multibrace algebras.
16h30-17h30 : Lander Hermans: "Deforming prestacks: an operadic calculus of rectangles"
Résumé : In his foundational work Gerstenhaber furnishes the guiding example for algebraic deformation theory: for an associative algebra A he defined a dgLie bracket on its Hochschild complex and showed that it controls the deformations of A through the Maurer-Cartan equation. Algebraic geometry motivates the natural question whether a similar story exists for diagrams of associative algebras, e.g. when applied to the structure sheaf of a scheme. In this talk I will explain how the Gerstenhaber-Schack complex fulfils this role, yet also motivates to generalize from diagrams to prestacks (i.e. pseudofunctors) as the most suitable objects to start with. Inspired by the fact that the Lie-structure of the Hochschild complex arises from an underlying operadic calculus, we introduce a new L-infinity structure arising from a rectangular operadic calculus. We show it completes the story: the higher Lie brackets on the GS complex control the deformations of prestacks through the generalized MC equation. Along the way, we introduce a new type of operad which can be seen as an enriched version of Leinster’s fc-multicategories (also called virtual double categories). This is joint work with Hoang Dinh Van and Wendy Lowen.
9h30-10h30 : Clément Dupont: "Operadic posets and their cohomology"
Résumé : We will describe a natural formalism which produces a graded operad from a family of posets equipped with an `operadic structure’. It gives an a priori explanation for the fact that the cohomology of posets of partitions has an operadic structure, in the spirit of work of B. Fresse (generalized by B. Vallette to the case of decorated partitions). Another application of our formalism concerns the family of hypertree posets, whose cohomology has the same underlying S-module as the PreLie operad thanks to a result of B. Delcroix-Oger. We will give a conceptual explanation of this fact, which will shed light on a richer structure and an unexpected character: the PostLie operad introduced by B. Vallette. This is joint work with Bérénice Delcroix-Oger.
11h-12h : Coline Emprin: "Kaledin classes and formality criteria"
Résumé : A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory fully characterizing the formality of associative algebras over a characteristic zero field. In this talk, I will present a generalization of Kaledin classes to any coefficients ring and to other algebraic structures (encoded by operads, possibly colored, or by properads). I will prove new formality criteria based on these classes and give applications.
14h-15h : Bruno Vallette: "Du calcul opéradique au calcul propéradique"
Résumé : Le calcul opéradique a donné naissance à de nombreux résultats et outils qui se sont montrés utiles dans l’études des algèbres différentielles graduées : théorie de la déformation, algèbre homotopique et formalité. La généralisation au niveau des propérades et de leurs (bi)gèbres différentielles graduées n’est pas automatique. Depuis l’introduction de la notion de propérade (Université de Strasbourg, 2003), très peu de résultats ont été montrés dans cette direction. J’expliquerai néanmoins comment on peut établir le même type de théorèmes à ce niveau : infini-morphismes, description du groupe de jauge, équivalence infini-quasi-isomorphismes/zigzag de quasi-isomorphismes, enrichissement simplicial des (bi)gèbres à homotopie près.
15h30-16h30 : Ricardo Campos: "The embedding of commutative homotopical algebra into non-commutative homotopical algebra"
Résumé : Given a topological space, how much of its homotopy type is captured by its algebra of singular cochains? The experienced rational homotopy theorist will argue that one should consider instead a commutative algebra of forms. This raises the more algebraic question of studying the homotopical properties of the forgetful functor from dg commutative algebras to dg associative algebras. This question turns out to be very subtle. For instance, unlike its classical counterpart, this functor is not full at the homotopical level. In this talk I will show that in characteristic zero this functor deserves to be called an embedding. We will see that this result uses tools very much on the topics of our ANR, namely, it is in its essence a result in derived deformation theory which is naturally addressed with tools from higher (curved) Lie theory. This joint work with Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra; based on arXiv:1904.03585 and 2211.02387.
16h30-17h30 : Joan Bellier-Millès: "On the road to the André—Quillen cohomology of curved algebras"
Résumé : We will begin this talk by presenting two contexts involving curved algebras. The first one concerns symplectic geometry and the second one concerns complex geometry. In these two contexts, we are dealing with homotopy curved algebraic structures and we will present the related operadic calculus. Finally, we will explain the ideas that allow us to define a cohomology in these contexts.
9h30-10h30 : Geoffroy Horel: "Binomial rings and homotopy theory"
Résumé : In a famous paper, Sullivan showed that the rational homotopy theory of finite type nilpotent spaces can be encoded in a fully faithful manner by mapping it to the homotopy category of commutative differential graded algebras over the rational numbers. For integral homotopy theory, a result of Mandell shows that it is faithfully captured by the integral cochain functor equipped with its E-infinity structure. This functor is however not full. I will explain a way of fixing this problem inspired by work of Toën, using cosimplicial binomial rings instead of E-infinity differential graded algebras.
11h-12h : Joana Cirici: "Formality of hypercommutative algebras of Calabi-Yau manifolds"
Résumé : Any Batalin-Vilkovisky algebra with a homotopy trivialization of the BV-operator gives rise to a hypercommutative algebra structure at the cochain level which, in general, contains more homotopical information than the hypercommutative algebra introduced by Barannikov and Kontsevich on cohomology. In this talk, I will explain how to use the purity of mixed Hodge structures to show that the canonical hypercommutative algebra defined on any compact Calabi-Yau manifold is formal. This is joint work with Geoffroy Horel.
14h-15h : Anibal Medina-Mardones: "State sums and higher categories"
Résumé : Topological quantum field theories can be constructed at least in two ways: as higher functors from the bordism category, in line with the cobordism hypothesis, or in terms of fields combined with an action functional, a technique widely used in the physics literature. The goal of this talk is to discuss a bridge between these descriptions using methods from combinatorial topology. This is joint work in progress with Lukas Müller.
9h30-10h30 : Paul Laubie: "Combinatorics of pre-Lie products sharing the Lie bracket"
Résumé :Pre-Lie products sharing the Lie bracket are controlled by the D^nPreLie operad, the coproduct of n copies of PreLie in the category of operads over Lie. From a first inspection on the dimension of D^2PreLie, one may find that it is related to the so called Greg trees. We generalize the Greg trees by labelling their black vertices by a coalgebra, then put an operadic structure on them. Those operads are binary, quadratic, Koszul and have the Nielsen-Schreier property.
11h-12h : Khalef Yaddaden: "Torseur de double mélange de multizêtas cyclotomiques et stabilisateurs de coproduits de Rham et Betti"
Résumé : Racinet décrit les relations de double mélange et régularisation entre valeurs polylogarithmes multiples aux racines de l'unité via un Q-schéma DMRι où ι: G → C× est un plongement de groupe d'un groupe cyclique fini G dans C×. Ensuite, Enriquez et Furusho montrent, dans le cas G={1}, qu'un sous-schéma DMRι× est un torseur d'isomorphismes entre objets Betti et de Rham. Dans cet exposé, on établit une généralisation de ce résultat en version cyclotomique. On commencera par expliciter la structure de torseur de DMRι× puis on introduira dans ce contexte les objets de Rham et Betti adéquats : les premiers sont issus d'une algèbre produit croisé et permettent une reformulation du coproduit harmonique de Racinet plus proche du formalisme introduit par Enriquez et Furusho; quant aux les seconds, ils sont issus d'une algèbre de groupe du groupe fondamental orbifold C×\ μ|G| / μ|G| où μ|G| désigne le groupe des racines |G|-ièmes de l'unité. Enfin, on démontrera, au sein du formalisme Betti, l'existence de deux coproduits de coalgèbre et d'algèbre de Hopf tels que DMRι× est un torseur des isomorphismes reliant ces coproduits Betti aux coproduits de Rham.
14h-15h : Benjamin Enriquez: "Sur les liens entre homologie singulière relative et quotients nilpotents du groupe fondamental des espaces topologiques"
Résumé : Pour X une variété, a,b deux points et n≥0 un entier, on sait, en utilisant des techniques de cohomologie de faisceaux, établir un isomorphisme entre (a) le dual du n+1-ième quotient correspondant à la suite centrale descendante du Q-espace vectoriel construit sur le torseur π1(X;a,b), et (b) le n-ième groupe de cohomologie relative du couple formé par Xn et une sous-variété Y(n)a,b construite à partir de a,b et des diagonales consécutives (Beilinson, rédigé par Deligne et Goncharov, puis avec tous les détails par Burgos Gil et Fresan). Nous montrons comment construire, dans le cas où X est un espace topologique, un morphisme de groupes abéliens entre (a) le n+1-ième quotient du Z-module libre construit sur π1(X;a,b), et (b) le n-ième groupe d'homologie relative du couple (Xn,Y(n)a,b) , morphisme dont le tensorisé par Q est celui décrit ci-dessus. La construction repose sur l'étude combinatoire d'opérations de "division" dans les complexes de chaînes. (Travail commun avec Florence Lecomte.)
14h-15h30 : Christine VESPA "Wheeled PROP structure on stable cohomology"
Abstract : Wheeled PROPs, considered by Markl, Merkulov and Shadrin are PROPs equipped with extra structures which can treat traces. In this talk, after explaining the notion of wheeled PROPs, I will describe a wheeled PROP structure on stable cohomology of automorphism groups of free groups with some particular coefficients. I will explain how cohomology classes constructed previously by Kawazumi can be interpreted using this wheeled PROP structure and I will construct a morphism of wheeled PROPs from a PROP given in terms of functor homology and the wheeled PROP evoked previously. This is joint work with Nariya Kawazumi.
16h-17h30 : Eric HOFFBECK "Homologie d'infini-opérades"
Résumé : Après des rappels sur la catégorie Omega et les infini-opérades dans le contexte dendroidal, je définirai une notion d'homologie pour les infini-opérades et les ensembles dendroidaux. L'homologie ainsi définie généralise celle du cas usuel des opérades algébriques, définie par l'homologie de la construction bar. Notre homologie provient également d'une construction bar, que je détaillerai, ainsi qu'une construction cobar. Celles-ci vérifient un théorème d'adjonction. Ceci est un travail en commun avec Ieke Moerdijk.
9h-10h30 : Guillaume LAPLANTE-ANFOSSI "La diagonale du multiplaèdre et le produit tensoriel de catégories A-infini"
Résumé : La structure d’algèbre associative à homotopie près, ou algèbre A-infini, est encodée par les associaèdres. Les morphismes entre algèbres A-infini sont encodés par une autre famille de polytopes, d’abord introduite par Stasheff : les multiplaèdres. Dans un travail en commun avec Thibaut Mazuir et Naruki Masuda, nous définissons une approximation cellulaire de la diagonale des multiplaèdres, et nous décrivons son image de manière combinatoire. Cela nous permet de définir un produit tensoriel de morphismes A-infini, compatible avec celui des algèbres A-infini, par des formules explicites. Ce résultat ouvre les portes à des calculs explicites en topologie symplectique, notamment l’étude de la catégorie de Fukaya formée par les produits de variétés symplectiques.
11h-12h30 : Bérénice DELCROIX-OGER "Structures tridendriformes sur les faces des associaèdres d'hypergraphes"
Résumé :En 2004, Jean-Louis Loday et Maria Ronco ont introduit la notion d'algèbre tridendriforme et muni l'espace vectoriel des arbres plans d'une structure d'algèbre tridendriforme. Emily Burgunder et Maria Ronco ont ensuite étendu leur définition aux surjections en 2010. Les arbres plans et les surjections forment les faces de deux polytopes bien connus : l'associaèdre et le permutoèdre. Dans un travail récent avec Pierre-Louis Curien et Jovana Obradovic, nous avons étendu ces constructions aux faces de certains associaèdres d'hypergraphes appelés clans associatifs. Après avoir présenté le contexte et les définitions nécessaires, je présenterai cette construction et des exemples de celle-ci.
14h30-16h : Vladimir DOTSENKO "Une trinité des quotients homotopiques"
Résumé : Je parlerai de mon travail en cours avec Sergey Shadrin et Pedro Tamaroff dans lequel on étudie certains algèbres muni d'un opérateur impair d'ordre fini dont le carré est l'endomorphisme nul. La catégorie des algèbres où cet opérateur est trivial à l'homotopie près nous mène aux généralisations de la notion d'une théorie cohomologique des champs.
9h-10h30 : Najib IDRISSI "Curved Koszul duality and factorization homology"
Résumé : Koszul duality is a powerful theory that can be used – among other things – to produce resolutions of algebras. Usual Koszul duality applies to quadratic algebras, i.e., algebras equipped with a presentation where relations are all quadratic. As soon as relations involve linear and especially constant terms, the theory becomes more involved: the Koszul dual of an algebra is not a mere coalgebra anymore, but a curved coalgebra. In this talk, I will explain how curved Koszul duality can be generalized to algebras over unital binary quadratic operads, based on ideas developed by Millès and Hirsh–Millès. I will then apply this theory of n-Poisson algebras in order to compute factorization homology of universal enveloping n-algebras of Lie algebras.
11h-12h30 : Joan BELLIER-MILLES "Contexte homotopique pour les algèbres courbées"
Résumé : Les algèbres courbées ne vérifient pas l’identité fondamentale : d^2 = 0. Parler d’homologie pour ces algèbres n’a donc pas de sens a priori. Elles apparaissent cependant dans des contextes (en géométrie par exemple) où un analogue d’une théorie de l’homotopie est requis. Nous proposerons un contexte homotopique dans lequel il est possible d’étendre la dualité de Koszul aux opérades courbées. Nous pourrons ainsi décrire une théorie de l’homotopie pour des algèbres courbées.
14h30-16h : Victor ROCA i LUCIO "A groupoid-colored approach to curved operadic calculus"
Résumé : Curved algebras appear in many different areas of mathematics. If one wants to encode them using operads, the one needs to introduce a new type of operads, called curved operads. In this talk, I will present how to generalize different standard results concerning operads to curved operads. Then, I will explain how to encode this objects using (curved) groupoid-colored operads. Using the Koszul duality of Hirsh--Millès at the groupoid-colored level, we construct a new Bar-Cobar adjunction involving curved operads and counital cooperads. Finally, we conclude by studying different homotopical properties and establish duality results that relate this adjunction to the adjunction introduced by Le Grignou between operads and curved conilpotent cooperads. Meanwhile, we construct the cofree not necessarily conilpotent cooperad.