Séminaires : Séminaire d'Algèbre

A famous result in representation theory is Auslander’s correspondence which connects finite-dimensional algebras of finite representation-type with Auslander algebras. Over the years, many generalisations of Auslander algebras have been proposed: for instance n-Auslander algebras (by Iyama), n-minimal Auslander–Gorenstein algebras (by Iyama and Solberg), among others. All of the concepts above require the existence of a faithful projective-injective module and use classical dominant dimension. Now replace the faithful projective-injective module with a self-orthogonal module and classical dominant dimension with relative dominant dimension with respect to a module and you get a relative Auslander-Gorenstein pair.

In this talk, we introduce relative Auslander-Gorenstein pairs. Further, we will characterise relative Auslander pairs (those whose underlying algebras have finite global dimension) by the existence and uniqueness of tilting-cotilting modules having the highest values of relative dominant and codominant dimension with respect to the self-orthogonal module. At the end, we discuss explicit examples of relative Auslander pairs. (This is joint work with Chrysostomos Psaroudakis.)

This talk will take place in hybrid mode at the Institut Henri Poincaré.

The Gekhtman-Shapiro-Vainshtein conjecture (the GSV conjecture) states that for any any given simple complex algebraic group G and any Poisson bracket from the Belavin-Drinfeld class, there exists a compatible generalized cluster structure. In this talk, I will review the process of constructing compatible generalized cluster structures, as well as the current state-of-the-art on the GSV conjecture. After that, I will describe a construction of generalized cluster structures on SL_n compatible with Poisson brackets induced from the Poisson dual of SL_n endowed with the Poisson structure determined by a BD triple of type A_{n-1}. I will also describe the associated family of birational quasi-isomorphisms. The talk will be based on the preprint arXiv:2312.04859 (joint work with M. Gekhtman). 

This talk will take place on Zoom only.

I will talk about a link between arc spaces of singularities, which are algebro-geometric objects, and identities of integer partitions. This link allows us to discover new partition identities in the spirit of the work of Ramanujan. The talk is accessible to a wide audience. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.

Celebrated work of Bridgeland and Smith shows a correspondence between quadratic differentials on Riemann surfaces and stability conditions on certain 3-Calabi--Yau triangulated categories. Part of this correspondence is that finite-length trajectories of the quadratic differential correspond to stable objects of phase 1. Speaking roughly, these stable objects are then counted by an associated Donaldson--Thomas invariant. Work of Iwaki and Kidwai predicts particular values for these Donaldson--Thomas invariants according to the different types of finite-length trajectories, based on the output of topological recursion. We show that the category recently studied by Christ, Haiden, and Qiu produces Donaldson--Thomas invariants matching these predictions. This is joint work with Omar Kidwai.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

Coulomb branches are certain moduli spaces arising in supersymmetric field theory. They include as special cases many spaces of independent interest such as double affine Hecke algebras, certain open Richardson varieties, multiplicative Nakajima quiver varieties etc. In the four-dimensional case, one expects that their coordinate rings can be categorified by abelian monoidal categories carrying a cluster structure. After reviewing the mathematical construction of these Coulomb branches we will explain how these categories are constructed and why the cluster structure appears. This is joint work with Harold Williams. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.

A complex of stable infinity-categories is a categorification of a chain complex, meaning a sequence of stable infinity-categories together with a differential that squares to the zero functor. Examples of such categorical complexes arise for instance via a categorification of the totalization construction, which produces a categorical complex from a categorical multi-complex, such as a commuting cube of stable infinity-categories. We will then explain how categorified perverse sheaves, also known as perverse schobers, on C^n (with a certain stratification) can be described in terms of categorical cubes and categorical complexes of spherical functors, and what categorical totalization means in this case geometrically. This talk is based on joint work with T. Dyckerhoff and T. Walde. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.

We introduce cluster braid groups, with motivations coming from the study of stability conditions on triangulated categories. In the Coxeter-Dynkin case, they are naturally isomorphic to the corresponding Artin braid groups (1407.5986 and 2310.02871). In the surface case, they are naturally isomorphic to braid twist groups (1407.0806, 1703.10053 and 1805.00030). If time permits, I will mention an application to quadratic differentials.

This talk will take place on Zoom only.

The leitfaden of this talk will be the general problem of determining which invariants of a finite group G are determined by which piece of the representation category of G over a commutative ring R. In the first part of the talk, we will recall the information encoded by the monoidal category of complex representations and its (braided) auto-equivalences. By doing so we will stumble on a question concerning the connection between two types of rigidity associated to G. The first is given by the group of class-preserving outer automorphisms of G and the second is a birational invariant of the quotient variety V/G, where V is a faithful representation of G. The aim of the second part of the talk will be to present some new perspective on them. Thereafter, in the last part, we will explain how the situation changes when taking R to be a number field or its ring of integers. In particular, the role of the theory of arithmetic groups will be emphasized. All along the talk, we will mention some open questions and some recent contributions.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

Exact dg categories allow to enhance extriangulated categories and to perform constructions like functor categories or tensor products for which the extriangulated structure alone does not suffice. In particular, they yield a new approach to and a generalization of higher versions of Auslander correspondences as established by Iyama and by Iyama-Solberg, for example. In this talk, I will give an introduction to exact dg categories and sketch their application to correspondences on the example of 0-Auslander categories. We will see in particular that the framework of exact dg categories allows to enhance the correspondences to equivalences of infinity-groupoids.

This talk will take place on Zoom only.

The theory of stable envelopes, introduced by Maulik and Okounkov, provides a fascinating interplay between the geometry of holomorphic symplectic varieties and integrable systems. We apply this theory to bow varieties which form a rich family of holomorphic symplectic varieties including type A Nakajima quiver varieties. We then discuss a formula for the multiplication of torus equivariant first Chern classes of tautological bundles of bow varieties with respect to the stable envelope basis. This formula naturally generalizes the classical Chevalley-Monk formula and can be expressed in terms of moves on skein-type diagrams that label the stable envelope basis. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.

We utilize freezing operators to establish connections among distinct (quantum) upper cluster algebras. This approach enables us to compare the quantized coordinate rings of different varieties. We prove that these operators send localized (quantum) cluster monomials to localized (quantum) cluster monomials. Furthermore, in many instances, they also preserve bases. Remarkably, the bases constructed via freezing operators coincide with those obtained via localization.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

The Chekanov-Eliashberg dg-algebra yields a powerful isotopy invariant of (possibly singular) Legendrian submanifolds in a class of contact manifolds, and is also intimately related to Fukaya categories of a class of non-compact symplectic manifolds. The goal for this talk is to explain how the relative Ginzburg algebra associated to any ice quiver with trivial potential is quasi-isomorphic to some Chekanov-Eliashberg dg-algebra. The proof is constructive. I will give a gentle introduction to Chekanov-Eliashberg dg-algebras and will discuss how the relation to relative Ginzburg algebras is interesting to contact and symplectic geometers.

This talk will be on Zoom only.

 We will describe a 5-dimensional Lie algebra over an algebraically closed field of characteristic 2 and show that its restricted enveloping algebra is special biserial, hence tame. We obtain an explicit description of all of its families of finite-dimensional indecomposable modules using Crawley-Boevey's description via strings and bands of the indecomposable modules over a special biserial algebra. This is joint work with N. Andruskiewitsch, S. D. Flora and D. Flores.


This talk will take place in hybrid mode at the Institut Henri Poincaré.

 In 2014, Leclerc gave a construction of a conjectural cluster structure on open Richardson varieties in types ADE. His construction was categorical in nature, involving preprojective algebra modules. His conjecture inspired work on cluster structures on braid varieties in arbitrary type, which generalize open Richardsons. Two cluster structures on braid varieties were recently constructed. The first one, based on ideas and techniques from symplectic topology, is due to Casals-Gorsky-Gorsky-Le-Shen-Simental. I will discuss the other, which is joint work with Galashin, Lam and Speyer. Our main geometric tool is the Deodhar decomposition. In type A, our quivers are given by "3D plabic graphs", which generalize Postnikov's plabic graphs for the Grassmannian. Time permitting, I will also discuss related work with Serhiyenko, where we show that for type A Richardsons, Leclerc's conjectural categorical construction does in fact give a cluster structure, with quivers again given by 3D plabic graphs.

This talk will be on Zoom only.

We describe the upper seminormal crystal structure for the μμ-supported δδ-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster XX-variety. We show that the crystal structure can be algebraically lifted to the generic basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. We introduce the biperfect bases and the strong biperfect bases in the cluster algebra setting and give a description of all strong biperfect bases.

This talk will be on Zoom only.

A dimer model is a quiver with faces embedded in a surface. Dimer models on the disk and torus are particularly well-studied, though these theories have remained largely separate. Various “consistency conditions” may be imposed on dimer models on the disk or torus with implications relating to 3-Calabi-Yau properties and categorification. We extend many of these definitions and results to the setting of general surfaces with boundary. We show that the completed dimer algebra of a “strongly consistent” dimer model is bimodule internally 3-Calabi-Yau with respect to its boundary idempotent. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of a suitable dimer model categorifies the cluster algebra given by its underlying ice quiver. We give a class of examples of annulus models satisfying the requisite conditions. 

This talk will take place on Zoom only.

I will present new q-deformations of Lie algebras linked to the modular group and the q-rational numbers as defined by Morier-Genoud and Ovsienko. In particular, I will describe deformations of sl2 and the Witt algebra. These deformations are realized as differential operators acting on the hyperbolic plane, giving new insights into q-rationals. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.

In this talk; we discuss the classification problem of spherical “like” objects in various geometric settings including the minimal resolution of an ADE surface singularity and a 3-fold flopping contraction. The classification of spherical objects is related to questions about the autoequivalence groups or Bridgeland stability conditions, but in 3-fold settings, this is not alway a correct problem to ask. In the first half of the talk, we discuss what kind of objects should be classified, and in the second half, a sketch of the proof will be explained. Our new technique can also be applied to the heart of a bounded t-structure, and classifies all t-structures of the associated null category. As a corollary, the connectedness of the space of stability conditions follows. This is all joint work with Michael Wemyss.

This talk will take place on Zoom only.

The minimal monomial lifting is a sort of homogenisation technique, whose goal is to identify a cluster algebra structure on certain "suitable for lifting" schemes, compatibly with a base cluster structure on a distinguished subscheme. This technique allows to recover, by geometric methods, some well known cluster structures. In this talk, we will present this technique and discuss applications to branching problems in representation theory of complex reductive groups.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

We will begin with an introduction to relative Calabi-Yau structures in the sense of Brav-Dyckerhoff, generalizing the notion of a Calabi-Yau triangulated (or dg-) category to functors. Via so-called relative theory, Calabi-Yau functors give rise to extriangulated categories, which are Frobenius 2-Calabi-Yau. We apply this to examples of cluster categories of surfaces, categorifying the surface cluster algebras with coefficients in the boundary arcs. This talk is mostly based on my preprint arXiv:2209.06595. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.

Cluster categories are fundamental objects in representation theory, including such topics as cluster algebras, tilting theory, singularity theory. The theory of Amiot, Guo, and Keller shows that tilting/silting objects in derived categories (of a finite dimensional algebra or of a Calabi-Yau dg algebra) give rise to cluster tilting objects in the cluster category. We study such correspondences between silting objects and cluster tilting objects. We propose a conjecture on the liftability of cluster tilting objects in the cluster category to silting objects, and discuss some evidence for it. This is based on a joint work with Osamu Iyama.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

The quantum Cartan matrix appears ubiquitously as a key combinatorial ingredient in the representation theory of quantum affine algebras. Through the generalized Schur-Weyl duality, it also plays a central role in the one of quiver Hecke algebras and the quantum unipotent coordinate ring of (skew-)symmetric finite type. Even though there are quiver Hecke algebras and quantum unipotent coordinate rings of non (skew-)symmetric finite type, there is no counterpart in representation theory as far as I and my collaborators understand. In this talk, I introduce a non-commutative ring over Q(q1/2Q(q1/2), which is expected to be a quantum Grothendieck ring for a Hernandez-Leclerc category, if such a representation theory exists, by using the t-quantized Cartan matrix. When we consider its heart subalgebra, the algebra is isomorphic to the quantum unipotent coordinate ring of any finite type. This talk is mainly based on joint work with Kashiwara, Jang and Lee.

This talk will take place on Zoom only.

The (quantized) coordinate rings of many interesting varieties from Lie theory are (quantum) cluster algebras. We construct the common triangular bases for these algebras. Such bases provide analogs of the dual canonical bases, whose existence has been long expected in cluster theory. For symmetric Cartan matrices, they are positive and admit monoidal categorification after base change. We employ a unified approach based on cluster algebra operations. Our results apply to algebraic groups, double Bott-Samelson cells, and braid varieties, etc. Additionally, we find applications in representations of quantum affine algebras.

This talk will take place on Zoom only.

Quantum toroidal algebras Uq(g_tor) occur as the Drinfeld quantum affinizations of quantum affine algebras. In particular; they contain (and are generated by) a horizontal and vertical copy of the affine quantum group. In type A, Miki obtained an automorphism of Uq(g_tor) exchanging these subalgebras, which has since played a crucial role in the investigation of its structure and representation theory.

In this talk; we shall construct an action of the extended double affine braid group B on the quantum toroidal algebra in all untwisted types. In the simply laced cases, using this action and certain involutions of B we obtain automorphisms and anti-automorphisms of Uq(g_tor) which exchange the horizontal and vertical subalgebras, thus generalising the results of Miki. We shall then discuss potential extensions of these results, and applications to the representation theory of quantum toroidal algebras.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

 In the 1990s, I classified rigid representations of a quiver by finitely generated free modules over a principal ideal ring. I shall extend the results to representations of a quiver by finitely generated projective modules over an arbitrary commutative ring.

This talk will kindly be shared by the organisation of the conference
Homological algebra and representation theory, cf.
https://sites.google.com/view/samosconferencerep/home

I will discuss several interlocked constructions giving rise to bialgebra structures all of which have parallel algebraic and topological interpretations. The bialgebras considered will be of different flavors depending on the compatibility between the product and coproduct; for instance, we will see examples of Hopf, Frobenius, infinitesimal and Lie bialgebras. These structures appear when analyzing the role of loop spaces in homotopy theory and manifold topology and reveal new results regarding the algebraic nature of geometric space.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

In combinatorics, several famous enumeration results involve a special kind of product formula. The very same kind of product formula gives the Milnor number of an isolated quasi-homogenous singularity. It seems possible that one could relate combinatorics and singularities by means of derived categories: on the one hand, modules over incidence algebras of partially ordered sets (posets) and on the other hand, some kind of Fukaya-like category that should categorify the Milnor fibration. Even if part of this remains very unprecise and vague, this implies many concrete conjectures about derived equivalences between posets.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

In his study of combinatorial features of cluster categories and cluster-tilted algebras, Ringel introduced an analogue of additive functions of stable translation quivers called cluster-additive functions.
In this talk, we will define cluster-additive functions associated to any acyclic mutation matrix, relate them to tropical points of the cluster X-variety, and realise their values as certain compatibility degrees between functions on the cluster A-variety associated to the Langlands dual mutation matrix (in accordance with the philosophy of Fock-Goncharov). This is based on joint work with Peigen Cao and Jiang-Hua Lu.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

I will give a gentle introduction to the amplituhedron, a geometric object that was introduced in the context of scattering amplitudes in N=4 super Yang Mills. I'll then explain some of the connections of the amplituhedron to cluster algebras.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

This work is an attempt to understand the maximal natural generality context for the Koenig-Kuelshammer-Ovsienko construction in the theory of quasi-hereditary algebras by putting it into a category-theoretic context. Given a field k and a k-linear exact category E with a chosen set of nonzero objects F_i such that every object of E is a finitely iterated extension of some F_i, we construct a coalgebra C whose irreducible comodules L_i are indexed by the same indexing set, and an exact functor from C-comod to E taking L_i to F_i such that the spaces Ext^n between L_i in C−comod are the same as between F_i in E (for n > 0). Thus, the abelian category C−comod is obtained from the exact category E by removing all the nontrivial homomorphisms between the chosen objects F_i in E while keeping the Ext spaces unchanged. The removed homomorphisms are then repackaged into a semialgebra S over C such that the exact category E can be recovered as the category of S-semimodules induced from finite-dimensional C-comodules. The construction used Koszul duality twice: once as absolute and once as relative Koszul duality.
 

Talk shared by the GAP conference, cf. https://personal.psu.edu/mps16/hirsutes2023/gap2023.html

You can follow the talk using the link (without interaction): https://www.ihp.fr/fr/live-0

We show that among the derived equivalent classes of homologically smooth and proper graded gentle algebras there is only one class whose perfect derived category does not admit silting objects.

As one application  we give a sufficient and necessary condition for any homologically smooth and proper graded gentle algebra under which all pre-silting objects in its perfect derived category may be complete into silting objects.
As another application we confirm a conjecture by Lekili and Polishchuk that the geometric invariants which they construct for homologically smooth and proper graded gentle algebras are a complete derived invariant. Hence, we obtain a complete invariant for partially wrapped Fukaya categories of surfaces with stops.
This is a report on joint work with Sibylle Schroll and Zhengfang Wang.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

I will talk about an, in general conjectural, construction of a X-cluster structure on
certain Hamiltonian reductions of an X-cluster variety. There are two main classes of examples of
such constructions: moduli spaces of framed local systems with special monodromies and phase spaces of
Goncharov-Kenyon integrable systems. The first class includes the phase space of open XXZ chain and
Ruijsenaars integrable systems. The second class includes integrable systems corresponding to the
q-difference Painleve equations.
Based on works in progress and discussions with P. Gavrylenko, A. Marshakov, M. Semenyakin,
A. Shapiro, G. Schrader.

This talk will take place on Zoom only.

The work of Squier and Kobayashi proves that the homology of a monoid can be computed using a so called complete rewriting system, which acts as a convenient presentation of the monoid. Later, Dehornoy and Lafont noted that such a convenient presentation arises in particular when considering monoids satisfying combinatorial assumptions regarding existence of lcms. This gave rise to the so called Dehornoy-Lafont order complex, which was used to compute the homology of complex braid groups by Callegaro and Marin. After giving a quick summary of these works, I will present a generalization of this latter complex to the case of a category which again satisfies convenient combinatorial assumptions. Of course, as my "true" goal is to compute the homology of a group using some associated category, I will also give a link between the homology of a category, that of its enveloping groupoid, and that of a group which is equivalent to the said groupoid. Lastly, I will explain an application to the case of the complex braid group B31B31​, which is studied through its associated Garside category, and which was not directly covered by previous approaches.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

 We introduce a new family of algebras attached to quivers with potentials, using critical K-theory and critical Borel-Moore homology. They generalize the convolution algebras attached to quivers by Nakajima. We give some applications to cohomological and K-theoretical Hall algebras, to shifted quantum loop groups, and to Kirillov-Reshetikhin and prefundamental representations.

This talk will take place in hybrid mode at the IHP.

Let $B/A$ be a pair of commutative rings. We propose an algebraic approach to the cotangent complex $L_{B/A}$. Using commutative semi-free DG ring resolutions of B relative to A, we construct a complex of $B$-modules $LCot_{B/A}$. This construction works more generally for a pair $B/A$ of commutative DG rings. In the talk, we will explain all these concepts. Then we will discuss the important properties of the DG $B$-module $LCot_{B/A}$. It time permits, we'll outline some of the proofs. It is conjectured that for a pair of rings $B/A$, our $LCot_{B/A}$ coincides with the usual cotangent complex $L_{B/A}$, which is constructed by simplicial methods. We shall also relate $LCot_{B/A}$ to modern homotopical versions of the cotangent complex.

 Parametrizations of the  canonical bases, string basis and theta basis, can be obtained by the tropicalization of  the Berenstein-Kazhdan decoration function and the Gross-Hacking-Keel-Kontsevich potential respectively. For  a classical Lie algebra and a reduced decomposition $\mathbf i$,  the decorated graphs are constructed algorithmically, vertices of such graphs are labeled by monomials which constitute the set of monomials of the Berenstein-Kazhdan potential.  Due to this algorithm;  we obtain a characterization of $\mathbf i$-trails introduced by Berenstein and Zelevinsky. Our algorithm uses multiplication and summations only, its complexity is linear in time of writing the monomials of the potential. For SL_n, there is an algorithm due to Gleizer and Postnikov which gets all monomials of the Berenstein-Kazhdan potential using combinatorics of wiring diagrams. For this case, our algorithm uses simpler combinatorics and is faster than the Gleizer-Postnikov algorithm. The cluster algorithm due to Genz, Schumann and me is polynomial in time but it uses divisions of polynomials of several variables.
If time permits, I will report on applications of decorated graphs to analysis of the Newton polytopes of F-polynomials related to the Gross-Hacking-Keel-Kontsevich potentials. The talk is based on joint works with Volker Genz and Bea Schumann and with Yuki Kanakubo and Toshiki Nakashima.

The talk will take place in hybrid mode at the Institut Henri Poincaré

Dimension expanders, introduced by Wigderson and Lubotzky-Zelmanov, are a linear algebra analogue of the notion of expander graphs. We interpret this notion in terms of quiver representations, as a quantitative variant of stability. We use Schofield’s recursive description of general subrepresentations to re-derive existence of dimension expanders and to determine optimal expansion coefficients.

The talk will take place in hybrid mode at the Institut Henri Poincaré
 

The category of dg categories is related by Koszul duality to a certain category of colagebras, so-called pointed curved coalgebras. In this talk we wil review this Quillen equivalence and observe that it is in fact quasi-monoidal. By constructing internal homs of pointed curved coalgebras we can then construct a concrete closed monoidal model for dg categories. In particular this gives natural descriptions of mapping spaces and internal homs between dg categories. This is joint work with A. Lazarev.

Exceptionally, this talk will take place in hybrid mode in room 1013 of the Sophie Germain building (8, place Aurélie Nemours, 75013 Paris).

For any d in {1,…,6}, we prove that the web of conics on a del Pezzo surface of degree d carries a functional identity HLog(7-d) whose components are antisymmetric hyperlogarithms of weight 7-d. Our approach is uniform with respect to d and relies on classical results about the action of the Weyl group on the set of lines on the del Pezzo surface. These hyperlogarithmic functional identities HLog(7-d) are natural generalizations of the classical 3-term and (Abel's) 5-term identities of the logarithm and the dilogarithm, which are the identities HLog(1) and HLog(2) corresponding to the cases d=6 and d=5 respectively.
If time allows, I will give a list of many nice properties enjoyed by the 5-term identity of the dilogarithm and will explain that most of these properties (such as being of cluster type) have natural generalizations which are satisfied by the weight 3 hyperlogarithmic identity HLog(3).
The talk will be mainly based on the preprint arXiv:2301.06775 written with Ana-Maria Castravet.

Exceptionally, this talk will take place in hybrid mode in room 1013 of the Sophie Germain building (8, place Aurélie Nemours, 75013 Paris).

We study self-orthogonal modules, i.e., modules T such that Ext^i(T, T) = 0 for all i > 0. We introduce projectively Wakamatsu-tilting modules (pW-tilting modules) as a generalization of tilting modules. If A is a representation-finite algebra, every self-orthogonal A-module can be completed to a pW-tilting module, and the following classes coincide: pW-tilting modules, Wakamatsu tilting modules, maximal self-orthogonal modules, and self-orthogonal modules T with |T| = |A|. We also prove that every self-orthogonal module over a representation-finite Iwanaga-Gorenstein algebra has finite projective dimension. We finally explain some open conjectures on self-orthogonal modules.

The skein algebra of a marked surface admits the basis of bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta basis from the cluster scattering diagram by Gross-Hacking-Keel-Kontsevich. In a joint work with Travis Mandel, we show that the two bases coincide except for the once-punctured torus. Our results extend to quantum cluster algebras with coefficients arising from the surface even in punctured cases. Long-standing conjectures on strong positivity and atomicity follow as corollaries.

Exceptionally, this talk will take place in hybrid mode in room 1013 of the Sophie Germain building (8, place Aurélie Nemours, 75013 Paris).

The double affine Hecke algebra was introduced by Cherednik around 1995 as a tool in his study of Macdonald polynomials. Its degenerate version, called trigonometric double affine Hecke algebra (TDAHA), has also turned out to be linked to different areas, notably to the representation theory of pp-adic groups. Given a root system, the TDAHA HcHc​ depends on a family of complex parameters cc. Given two families of parameters cc and c′c′ whose difference takes integer values, there exists a triangle equivalence between the bounded derived categories of the corresponding TDAHAs, which we call translation functor. The objective of this talk is to explain the construction of this functor. 

This talk will be on Zoom only.

Abstract: As a part of Langlands duality, certain equations were found in two different areas of mathematics. They are known as Baxter’s TQ systems and the QQ type systems, as they trace back to Baxter’s study on integrable models in the 1970s. During the same decade, similar systems of equations were discovered in the area of ordinary differential equations (ODE) by Sibuya, Voros and others. Today, this remarkable correspondence is realized as a duality between representation theory of nontwisted quantum affine algebras (QAA) and the theory of opers for their Langlands dual Lie algebras.
We are interested in this duality when the roles of the affine Lie algebra and its dual are exchanged. When the nontwisted QAA is of type BCFG, its dual will be a twisted QAA. To exchange their roles amounts to studying representations of twisted QAAs.
In this talk, we will begin by reviewing this story. We will explain the representation theory of twisted QAAs and their Borel algebras. We will explain the expected relationship between twisted and nontwisted types, and we will establish TQ systems and QQ^{~} systems for twisted QAAs.

This talk will take place in hybrid mode at the IHP.

One of the most celebrated theorems in the theory of quiver representations is undoubtedly Gabriel’s theorem, which reveals a deep connection between quiver representations and root systems arising from Lie algebras. In particular, Gabriel’s theorem shows that the finite-type quivers are classified by the ADE Dynkin diagrams and the indecomposable representations are in bijection with the underlying positive roots. Following the works of Dlab—Ringel, the classification can be generalised to include all the other Dynkin diagrams (including BCFG) if one considers the more general notion of valued quivers (K-species) representations instead.

While the theories above relate (valued) quiver representations to root systems arising from Lie algebras, the aim of this talk is to generalise Gabriel’s theorem in a slightly different direction using root systems arising in Coxeter theory. Namely, we shall introduce a new notion of Coxeter quivers and their representations built in (other) fusion categories, where we have a generalised Gabriel’s theorem as follows: a Coxeter quiver has finitely many indecomposable representations if and only if its underlying graph is a Coxeter-Dynkin diagram — including the non-crystallographic types H and I. Using a similar notion of reflection functors as introduced by Bernstein—Gelfand—Ponomarev, we shall also show that the isomorphism classes of indecomposable representations of a Coxeter quiver are in bijection with the positive roots associated to the root system of the underlying Coxeter graph. --

This talk will take place in hybrid mode at the IHP.

We introduce a generalization of $K$-$k$-Schur functions and k-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases promise to be detailed in the future.

We introduce the notion of a lower (upper) bound cluster algebra generated by projective cluster variables. Projective cluster variables are often categorified by projective modules of the corresponding quiver with relations. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case, we use our results to construct a basis for the cluster algebra. We also show that the coincidence between cluster algebra and the lower bound cluster algebra generated by projective cluster variables holds beyond acyclic seeds. Part of this talk is based on joint work with Karin Baur.

2-representations of Kac-Moody algebras arise algebraically and as categories of constructible sheaves. We will discuss two settings involving coherent sheaves: derived cotangent bundles to spaces of quiver representations and spaces of quasi-maps in flag varieties.
This talk will be shared by the conference "Geometric representation theory and quantum topology", cf. http://georep22.imj-prg.fr/

In this talk, we talk about the categorical crystal structure on the Hernandez-Leclerc category $\mathscr{C}_\mathfrak{g}^0$. We define extended crystals for quantum groups and show that there is a braid group action on extended crystals.  We then explain how the set of the isomorphism classes of simple modules in $\mathscr{C}_\mathfrak{g}^0$ has an extended crystal structure, and discuss the braid group action from the viewpoint of the Hernandez-Leclerc category $\mathscr{C}_\mathfrak{g}^0$. This talk is based on a joint work with M. Kashiwara (arXiv: 2111.07255 and 2207.11644).

In these two talks, we will start by introducing a result which establishes the existence and uniqueness of (DG) enhancements for triangulated categories which admit an additive generator whose endomorphism algebra is finite-dimensional (over a perfect field). We will then present a generalisation of this result that allows us to treat a larger class of triangulated categories, which instead admit a generator with a strong regularity property (a so-called dZ-cluster tilting object). We will also explain how our result, combined with crucial theorems of August and Hua-Keller, leads to a positive solution of the Donovan-Wemyss Conjecture for contraction algebras as observed by Keller. We will also comment on some details about the proof.

In these two talks, we will start by introducing a result which establishes the existence and uniqueness of (DG) enhancements for triangulated categories which admit an additive generator whose endomorphism algebra is finite-dimensional (over a perfect field). We will then present a generalisation of this result that allows us to treat a larger class of triangulated categories, which instead admit a generator with a strong regularity property (a so-called dZ-cluster tilting object). We will also explain how our result, combined with crucial theorems of August and Hua-Keller, leads to a positive solution of the Donovan-Wemyss Conjecture for contraction algebras as observed by Keller. We will also comment on some details about the proof.

Topological spaces up to (weak) equivalences are faithfully represented by simplicial combinatorial structures. Through an identification of the $n$-dimensional simplex with the space of probability measures on a finite set of size $n+1$, we investigate what happens when it is replaced by the space of random variables that naturally lies 'above' it. By this procedure, we obtain in particular a simple description of the classifying set of a (discrete) group, and also a new concept of geometric realization. This new one also induces an equivalence of categories up to homotopy with simplicial sets and topological spaces. The 'probability-law' map then defines a natural transformation between the two corresponding Quillen equivalences.

The mutation theory of quivers with potential and their representations, developed around 15 years ago by Derksen-Weyman-Zelevinsky, has had a profound impact both inside and outside the theory of cluster algebras. In this talk I will present results obtained in joint works with Geiss and Schröer, and with de Laporte, about some interesting behaviors of DWZ's mutations of representations. Namely, despite needing several non-canonical choices of linear-algebraic data in order to be performed, they can always be arranged so as to become regular maps on dense open subsets of representation spaces rep(Q,S,d). As a consequence, one obtains the invariance of Geiss-Leclerc-Schröer's 'generic basis' under mutations even in the Jacobi-infinite case, thus generalizing a result of Plamondon. Furthermore, given two distinct vertices k, \ell of a quiver with potential (Q,S), the k-th mutation of representations takes the \ell-th indecomposable projective over (Q,S) to the \ell-th indecomposable projective over \mu_k(Q,S). When a certain 'optimization' condition is satisfied by \ell, this allows to compute certain 'Landau-Ginzburg potentials' as F-polynomials of projective representations.

This is joint work with Osamu Iyama. Let $A$ be a finite dimensional algebra over an algebraically closed field. Then the numerical torsion pairs of Baumann-Kamnitzer-Tingley give an equivalence relation on the real Grothendieck group of finitely generated projective $A$-modules, which is called TF equivalence. By results of Yurikusa and Bruestle-Smith-Treffinger, we have that the g-vector cone of each 2-term presilting complex is a TF equivalence class. To get more TF equivalence classes, we can use canonical decompositions of elements in the (integral) Grothendieck group of finitely generated projectives introduced by Derksen-Fei. We have showed that the cone defined by the canonical decomposition of each element is contained in some single TF equivalence class. Moreover, we have also obtained that, if $A$ is an E-tame algebra, then this cone is precisely a TF equivalence class. In this talk, I will explain these results and some important steps to prove them.

We define tilting subcategories in arbitrary exact categories to archieve the following. Firstly: Unify existing definitions of tilting subcategories to arbitrary exact categories. Discuss standard results for tilting subcategories: Auslander correspondence, Bazzoni description of the perpendicular category. Secondly: We treat the question of induced derived equivalences separately - given a tilting subcategory T, we ask if a functor on the perpendicular category induces a derived equivalence to a (certain) functor category over T. If this is the case, we call the tilting subcategory ideq tilting. We prove a generalization of Miyashita's theorem (which is itself a generalization of a well-known theorem of Brenner-Butler) and characterize exact categories with enough projectives allowing ideq tilting subcategories. In particular, this is always fulfilled if the exact category is abelian with enough projectives.

We present a rigid cluster model to realize the quantum group U_q(g) for g of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group to a quotient algebra of the Weyl group invariants of a Fock-Goncharov quantum cluster algebra. By applying the quantum duality of cluster algebras, we show that the quantum group admits a cluster canonical basis Theta whose structural coefficients are in Laurent polynomials with non negative integer coefficients in the square root of q. The basis Theta satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution.
 

Frieze patterns are infinite strips of numbers satisfying certain determinantal identities. Originally motivated by Gauss’ “miraculous pentagram” identities, these patterns have since been connected to triangulations, integrable systems, representation theory, and cluster algebras. In this talk, we will review a few characterizations and constructions of frieze patterns, as well as a generalization which allows friezes with a “ragged edge” described by a juggling function. These “juggler’s friezes” correspond to special points in positroid varieties, in direct analogy with how classical friezes correspond to special points in Grassmannians.

In 2020, the pandemic hit, and all around the globe we went into lockdowns of various description. During the first lockdown I carefully read Krause's 2015 article "Deriving Auslander's formula". In this talk, I will outline how the ideas of Krause's paper underpin two articles written in 2020 in collaboration with Canonaco and Stellari. One is about the uniqueness of enhancements of large classes of triangulated categories, while the second offers a counterexample to certain vanishing conjectures in negative K-theory. [This zoom talk is kindly shared by Representation theory and triangulated categories.]

The talk, based on joint work in progress with V. Schechtman, will first
recall our description of perverse sheaves on $Sym^n(\mathbb{C})$, the symmetric
product of the complex line with its natural stratification by multiplicities.
This description proceeds in terms of contingency matrices, which are certain
integer matrices appearing (besides their origin in statistics) in three
different contexts:
-- A natural cell decomposition of $Sym^n(\mathbb{C})$.
-- Compatibility of multiplication and comultiplication in $\mathbb{Z}_+$-graded Hopf algebras.
-- Parabolic Bruhat decomposition for $GL_n$.
Perverse sheaves on $Sym^n(\mathbb{C})$ are described in terms of certain data
of mixed functoriality on contingency matrices which we call Janus sheaves.
I will then explain our approach to categorifying the concept of Janus sheaves,
in which sums are replaced by filtrations with respect to the Bruhat order.
Such data can be called Janus schobers. Examples can be obtained from
$\mathbb{Z}_+$-graded Hopf categories, a concept going back to Crane-Frenkel,
of which we consider two examples related to representations of groups $GL_n$
over finite fields (Joyal-Street) and $p$-adic fields (Bernstein-Zelevinsky).
(Zoom talk shared with https://nc-shapes.info )

Graded gentle algebras are classical objects in representation theory. They are quadratic monomial algebras making them particularly amenable to study and they appear in many different areas of mathematics such as in cluster theory, in N=2 gauge theories and in homological mirror symmetry of surfaces. In this talk, we give a construction of a partial cofibrant dg algebra resolution of a graded quadratic monomial algebra inducing an explicit recollement of their derived categories. We show that for graded gentle algebras, both the left and the right side of such a recollement corresponds to cutting the underlying surface which can be associated to a graded gentle algebra. In the case of homologically smooth and proper graded gentle algebras this recollement can be restricted to the derived categories with finite total cohomology, thus inducing a recollement of the corresponding partially wrapped Fukaya categories. We give some consequences of this construction such as the existence of full exceptional sequences, silting objects and simple minded collections. This is joint work with Wen Chang and Haibo Jin arxiv.org/abs/2206.11196.

Given a plabic graph on R^2, we can choose a conormal lift of its zig-zag strands to the unit cotangent bundle of R^2, obtaining a Legendrian link. If the plabic graph satisfies a “grid” condition, its Legendrian link admits a natural embedding into the standard contact R^3. We study the Kashiwara-Schapira moduli space of microlocal rank 1 sheaves associated with the Legendrian link, and construct a natural (quasi-)cluster structure on this moduli space using Legendrian weaves. In particular, we prove that any braid variety associated with (beta Delta) for a 3-strand braid beta admits cluster structures with an explicit construction of initial seeds. We also construct Donaldson-Thomas transformations for these moduli spaces and prove that the upper cluster algebra equals its cluster algebra. In this talk, I will introduce the theoretical background and describe the basic combinatorics for constructing Legendrian weaves and the (quasi-)cluster structures from a grid plabic graph. This is based on joint work with Roger Casals, cf. arxiv.org/abs/2204.13244.

The notions of a modular tensor category, 2d topological modular functor, and 3d topological quantum field theory are essentially equivalent. Fock and Goncharov conjectured that the quantised higher Teichmüller theory gives rise to an analogue of a modular functor. Their construction in turn yields a family of "positive" representations of quantum groups. I will argue that these representations provide a compelling first step towards constructing an analogue of a modular tensor category. This talk will be based on joint works with Gus Schrader.

 For a finite dimensional algebra A, the 2-term silting complexes of A give a simplicial complex \( \Delta(A)\), which is called the g-simplicial complex.
We study several properties of \( \Delta(A)\) and, in particular, we give tilting theoretic interpretations of the h-vectors and the Dehn-Sommerville equations of  \( \Delta(A)\).Consequently, we can explain a close correspondence between torsion classes and wide subcategories, which can be regarded as a refinement of the Koenig-Yang correspondence. This is joint work with Aoki-Higashitani-Iyama-Kase, cf. arxiv.org/pdf/2203.15213.pdf

The unbounded derived category of (right) modules over a ring is a triangulated category with infinite products and coproducts. As a triangulated category with coproducts it is easy to see that it is generated by the projective modules, and similarly it is generated as a triangulated category with products by the injective modules. I will discuss the question of whether it is generated as a triangulated category with coproducts by the injective modules, or as a triangulated category with products by the projective (or flat) modules. I will describe the relationship with the finitistic dimension conjecture, as well as some more recent results.

Algebras and their representations can often be constructed geometrically in terms of convolution of cycles. For example, the Springer correspondence describes how irreducible representations of a Weyl group can be realised in terms of a convolution action on the vector spaces of irreducible components of Springer fibers. Similar situations yield the affine Hecke algebra, quiver Hecke algebra (KLR algebra), quiver Schur algebra or Soergel bimodules. In this spirit, we show that these algebras and their representations can be realised in terms of certain equivariant motivic sheaves called Springer motives. On our way, we will discuss weight structures and their applications to motives. This is joint work with Catharina Stroppel.

We prove that the notion of a curved pre-Calabi–Yau algebra is equivalent to the notion of a curved homotopy double Poisson gebra, thereby settling the equivalence between the two ways to define derived noncommutative Poisson structures. We actually prove that the respective differential graded Lie algebras controlling both deformation theories are isomorphic. This allows us to apply the recent developments of the properadic calculus in order to establish the homotopical properties of curved pre-Calabi–Yau algebras: infini-morphisms, homotopy transfer theorem, formality, Koszul hierarchy, and twisting procedure. (Joint work with Johan Leray available at arxiv.org/abs/2203.05062).

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at arxiv.org/abs/2201.04606.

A Calabi-Yau structure on a smooth algebra allows one to identify Hochschild homology with Hochschild cohomology. With this identification Hochschild homology acquires an additional Gerstenhaber algebra structure. One way to formulate the amount of structure one has on Hochschild homology is to encode it into a 2d TFT. This explains some of the string topology operations on the free loop space of a manifold, but not the string coproduct. If the algebra has additional structure (trivialization of its Hattori-Stalling Euler characteristic) one obtains an extra secondary operation on Hochschild homology, which recovers the string coproduct. Finally, in the free loop space setting, this additional structure can either be recovered from intersection theory of the manifold or from its underlying simple homotopy type, thus relating the two. Using this last relation one can express the difference between the string coproduct of two homotopic but not necessarily homeomorphic manifolds in terms of Whitehead torsion.
This is joint work with Pavel Safronov

In this talk, we investigate an application of the theory of deformed Calabi--Yau completion to enumerative geometry. The notion of Calabi--Yau completion was first introduced by Keller as a non-commutative analogue of the canonical bundle. In the same paper, he also introduced a deformed version of the Calabi--Yau completion. We will explain that the deformed Calabi--Yau completion is a non-commutative analogue of an affine bundle modeled on the canonical bundle. Combining this observation with a recent work of Bozec--Calaque--Scherotzke, we prove that the moduli space of coherent sheaves on a certain non-compact Calabi--Yau threefold is described as the critical locus inside a smooth moduli space. This description has several applications in Donaldson--Thomas theory including Toda's \chi-independence conjecture of Gopakumar--Vafa invariants for arbitrary local curves. By dimensional reduction, it implies (and extends) Hausel--Thaddeus's cohomological \chi-independence conjecture for Higgs bundles.This talk is based on a joint work with Naruki Masuda and another joint work with Naoki Koseki.

Schur-Weyl duality is often used to relate type A Lie groups (or quantum groups) to symmetric groups (or Hecke algebras). In this talk, I will use ribbon calculus and skein modules to describe an instance of this Schur-Weyl duality between representations of the type A quantum group at roots of unity and representations of the Double Affine Hecke Algebra. This is based on joint work with A. Chandler, A. Mellit and C. Novarini.

In this talk, we will first recall the constructions of triangular extension and of source-sink extensio for skew-symmetrizable matrices and some invariants under these constructions. Secondly, we will recall the string diagrams introduced by Shen-Weng, which are very useful to describe many interesting skew-symmetrizable matrices closely related with Lie theory. Thirdly, we will sketch the proof of our main result: the skew-symmetrizable matrices from string diagrams are in the smallest class of skew-symmetrizable matrices containing the (1 times 1) zero matrix and closed under mutations and source-sink extensions. This result applies to the exchange matrices of cluster algebras from double Bruhat cells, unipotent cells, double Bott-Samelson cells among others. Finally, some immediate applications regarding reddening sequences and non-degenerate potentials for many quivers from Lie theory are given.

Higher homological algebra was introduced by Iyama in the late 2000's. His point-of-view was that some classical results by Auslander and Auslander--Reiten were somehow 2-dimensional and should have n-dimensional equivalents. This new theory quickly attracted a lot of attention, with many authors generalising classical notions to the setting of higher homological algebra. Examples of such generalisations are the introduction of n-abelian categories by Jasso, n-angulated categories by Geiss--Keller--Oppermann, and n-torsion classes by Jørgensen.Recently, it was shown by Kvamme and, independently, by Ebrahimi and Nasr-Isfahani, that every small n-abelian category is the n-cluster-tilting subcategory of an abelian category. In this talk we will focus on the relation between n-torsion classes in an n-abelian category MM and (classical) torsion classes of the abelian category AA in which MM is embedded. By considering functorially finite torsion classes, this will allow us to relate n-torsion classes with maximal tau_n-rigid objects in MM.Some of the results presented in this talk are part of a joint work by J. Asadollahi, P. Jørgensen, S. Schroll, H. Treffinger. The rest corresponds to an ongoing project by J. August, J. Haugland, K. Jacobsen, S. Kvamme,Y. Palu and H. Treffinger.

Higher homological algebra was introduced by Iyama in the late 2000's. His point-of-view was that some classical results by Auslander and Auslander--Reiten were somehow 2-dimensional and should have n-dimensional equivalents. This new theory quickly attracted a lot of attention, with many authors generalising classical notions to the setting of higher homological algebra. Examples of such generalisations are the introduction of n-abelian categories by Jasso, n-angulated categories by Geiss--Keller--Oppermann, and n-torsion classes by Jørgensen.Recently, it was shown by Kvamme and, independently, by Ebrahimi and Nasr-Isfahani, that every small n-abelian category is the n-cluster-tilting subcategory of an abelian category. In this talk we will focus on the relation between n-torsion classes in an n-abelian category MM and (classical) torsion classes of the abelian category AA in which MM is embedded. By considering functorially finite torsion classes, this will allow us to relate n-torsion classes with maximal tau_n-rigid objects in MM.Some of the results presented in this talk are part of a joint work by J. Asadollahi, P. Jørgensen, S. Schroll, H. Treffinger. The rest corresponds to an ongoing project by J. August, J. Haugland, K. Jacobsen, S. Kvamme,Y. Palu and H. Treffinger.

A preprojective algebra of non-Dynkin type has a family of tilting modules associated with the elements in the corresponding Coxeter group W. This family is useful to study the representation theory of the preprojective algebra and also to categorify cluster algebras.
In this talk, I will discuss tilting theory of a contracted preprojective algebra, which is a subalgebra eAe of a preprojective algebra A given by an idempotent e of A. It has a family of tilting modules associated with the chambers in the contracted Tits cone. They correspond bijectively with certain double cosets in W modulo parabolic subgroups. 
I will apply these results to classify a certain family of reflexive modules over a cDV singularities R, called maximal modifying (=MM) modules. We construct an injective map from MM R-modules to tilting modules over a contracted preprojective algebra of extended Dynkin type. This is bijective if R has at worst an isolated singularity. We can recover previous results (Burban-I-Keller-Reiten, I-Wemyss) as a very special case.
This is joint work with Michael Wemyss.

In this talk, we will explain relations between exact structures on an additively finite additive category and degenerations of the associated Hall algebras. The first part of the talk will be devoted to the main motivation provided by concrete examples of degenerations of negative parts of quantum groups arising as Hall algebras of quiver representations. We will then turn to Lie theory in order to establish a link from these examples to tropical flag varieties and certain quiver Grassmannians. In the second part of the talk we will present results in the general case and sketch their proofs based on Auslander-Reiten theory. If time permits, we will briefly discuss further conjectural examples and generalizations.

In this talk, we will explain relations between exact structures on an additively finite additive category and degenerations of the associated Hall algebras. The first part of the talk will be devoted to the main motivation provided by concrete examples of degenerations of negative parts of quantum groups arising as Hall algebras of quiver representations. We will then turn to Lie theory in order to establish a link from these examples to tropical flag varieties and certain quiver Grassmannians. In the second part of the talk we will present results in the general case and sketch their proofs based on Auslander-Reiten theory. If time permits, we will briefly discuss further conjectural examples and generalizations.

Let $G$ be a reductive algebraic group over a field $k$. When $k=\mathbb{C}$, R. K. Brylinski constructed a filtration of weight spaces of a $G$-module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's $q$-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. Recently, we managed to generalise this result to the case where $k$ is a field of good positive characteristics. I will give a brief introduction to both historical results and our new result in the talk.

Let $g_0$ be a simple Lie algebra of type $ADE$ and let $U′_q(g)$ be the corresponding untwisted quantum affine algebra. We found an action of the braid group $B(g_0)$ on the quantum Grothendieck ring $K_t(g)$ of Hernandez-Leclerc's category $C^0_g$. In the case of $g_0=A_{N−1}$, we construct a monoidal autofunctor $S_i$ for each integer $i$ on a category $T_N$ arising from the  quiver Hecke algebra of type $A_\infty$. 
Since there is an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t(A^(1)_{N−1})$, the functors $S_i$, $(i=1, ..., N-1)$, recover the action of the braid group $B(A_{N−1})$. 
This is a joint work with Masaki Kashiwara, Euiyong Park and Se-jin Oh.

This is a report on joint work with D. Labardini-Fragoso and J. Schröer. We show that for most marked surfaces with non-empty boundary, possibly with punctures, the generic Caldero-Chapoton functions form a basis of the corresponding cluster algebras for any choice of geometric coefficients. For surfaces without punctures the $\tau$-reduced components of the corresponding gentle Jacobian algebra are naturally parametrized by X-laminations of the surface, and it is easy to see that for principal coefficients, the generic basis coincides with the bangle basis introduced by Musiker-Schiffler-Williams.

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott as a tool to study highest weight theories which arise in the representation theories of semi-simple complex Lie algebras and reductive groups. Surprisingly, there are now many examples of such algebras, such as Schur algebras, algebras of global dimension at most two, incidence algebras and many more. 

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In the early examples the partial order predated (and motivated) the theory, so the choice was clear. However, there are instances of quasi-hereditary algebras where there is no natural choice for the partial ordering and even if there is such a natural choice, one may wonder about all the possible orderings.

In this talk we will explain that all these choices for an algebra $A$ can be organized in a finite partial order which is in relation with the tilting theory of $A$. In a second part of the talk we will focus on the case where $A$ is the path algebra of a Dynkin quiver.

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott as a tool to study highest weight theories which arise in the representation theories of semi-simple complex Lie algebras and reductive groups. Surprisingly, there are now many examples of such algebras, such as Schur algebras, algebras of global dimension at most two, incidence algebras and many more. 

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In the early examples the partial order predated (and motivated) the theory, so the choice was clear. However, there are instances of quasi-hereditary algebras where there is no natural choice for the partial ordering and even if there is such a natural choice, one may wonder about all the possible orderings.

In this talk we will explain that all these choices for an algebra $A$ can be organized in a finite partial order which is in relation with the tilting theory of $A$. In a second part of the talk we will focus on the case where $A$ is the path algebra of a Dynkin quiver.

It is well known that the partially wrapped Fukaya category of a marked
disk is equivalent to the perfect derived category of a Dynkin quiver of
type A. In this talk I will present a higher-dimensional generalisation
of this equivalence which reveals a connection between three a
priori unrelated subjects:

* Floer theory of symmetric products of marked surfaces
* Higher Auslander-Reiten theory in the sense of Iyama
* Waldhausen K-theory of differential graded categories

If time permits, as a first application of the above relationship, I
will outline a symplecto-geometric proof of a recent result of Beckert
concerning the derived equivalence between higher Auslander algebras of
different dimensions. This is a report on joint work with Tobias
Dyckerhoff and Yankı Lekili.

The theory of Newton-Okounkov bodies is a generalization of 
that of Newton polytopes for toric varieties. One of the ingredients for 
the definition of a Newton-Okounkov body is a valuation on the function 
field of a given projective variety. In this talk, we consider Newton-
Okounkov bodies of Schubert varieties defined from specific valuations 
which generalize extended g-vectors in cluster theory. We show that they 
provide polytopes unimodularly equivalent to string polytopes and 
Nakashima-Zelevinsky polytopes, both of which are well-known polytopes 
in representation theory. Indeed, this framework allows us to connect 
string polytopes with Nakashima-Zelevinsky polytopes by tropicalized 
cluster mutations. 
This talk is based on a joint work with Naoki Fujita.

Fomin and Zelevinsky invented cluster algebras, which are algebras with distinguished generators called cluster variables. For any symmetrizable Kac-Moody algebra and Weyl group element, the corresponding quantum unipotent subgroup possesses the dual canonical basis, and it can be viewed as a (quantum) cluster algebra. As a main motivation by Fomin and Zelevinsky, it has been long conjectured that the quantum cluster monomials (certain monomials of cluster variables) belong to the dual canonical basis up to scalar multiples. We sketch a proof of this conjecture in full generality.

Dualizability and invertibility are two natural properties one can ask for objects in (possibly
higher) symmetric monoidal categories. On the one hand, it recovers as special cases various
important notions in geometry and representation theory. On the other hand, it connects those
notions to topology via the cobordism hypothesis. I will explain various examples of this philosophy, with an emphasis on applications to finite braided tensor categories. This is based on joint work with D. Jordan, P. Safronov and N. Snyder. 

Friezes with coefficients are maps assigning numbers to the edges and diagonals of a regular polygon such that all Ptolemy relations for crossing diagonals are satisfied. These are relevant for example for the study of cluster algebras, in a special case they may also be viewed as root systems of certain quantum groups. 
In this talk I will report on recent results on subpolygons of friezes. Depending on the domain of the entries of the friezes, these subpolygons satisfy interesting arithmetic obstructions.

The associahedron is a polytope that encodes Catalan families. One of its many avatars is as a polytopal realization of the g-vector fan of a cluster algebra of type A.  Given a cluster algebra with fixed initial seed, the space of all polytopal realizations of its g-vector fan has been of interest to physicists, appearing for instance in the work of Arkani-Hamed, Bai, He and Yan. 

In this talk, we will see how a description of the set of all polytopal realizations of the g-vector fan of any cluster algebra of finite type with any initial seed can be described by looking for a minimal set of relations between its g-vectors.  To find such a set, we will see how a sub-extriangulated structure of the triangulated structure of the cluster category allows for a categorification of g-vectors, and find all relations in its Grothendieck group. 

This is a report on a joint work with Arnau Padrol, Yann Palu and Vincent Pilaud.

A dg algebra $A$ admits a geometric realization if the category of perfect dg $A$-modules can be embedded into the bounded derived category of a smooth projective variety. In this talk, I will first give an overview of Orlov's results on geometric realizations of dg algebras, and then explain how all dg algebras with finite dimensional cohomology, which are moreover concentrated in non-positive degrees, admit such realizations. The proof is based on a generalization of the Auslander algebra of a finite dimensional algebra to the setting of finite-dimensional A-infinity algebras. If time allows, I will discuss several corollaries related to finite-dimensional models, noncommutative motives, and non-Fourier-Mukai functors. This is based on joint work with Alice Rizzardo, Greg Stevenson, and Michel Van den Bergh.

In this talk I will present some recent developments building on the topic of the paper https://arxiv.org/abs/1605.02721. I will introduce the concept of absolute and relative Calabi-Yau structures on dg categories/algebras defined by Brav and Dyckerhoff, and then show how these structures come from a more general definition of an orientation on a (co)sheaf of categories. This will be a noncommutative version of the usual notion of an orientation on a manifold, and gives the desired Calabi-Yau structures upon integration.

Avec Rachel Taillefer (arXiv :1905.07906), nous avons mis en évidence une dualité de 
Poincaré-Van den Bergh des algèbres préprojectives relativement à l'homologie et cohomologie 
de Koszul. Cette dualité est nouvelle quand le graphe est de type Dynkin ADE, ce qui correspond à une

algèbre non Koszul, et elle peut être exploitée en toute généralité en termes de catégorie dérivée.

With Rachel Taillefer  (arXiv :1905.07906), we have exhibited a Poincaré-Van den Bergh duality for

Koszul homology and cohomology of the preprojective algebras. This duality is new when the underlying graph

is of Dynkin type ADE, hence the preprojective algebra is not Koszul, and it can be exploited in full generality

in terms of derived categories.

Motivated by (mutation) deficiencies in classical tilting theory, Adachi, Iyama and Reiten introduced the theory of support tau-tilting modules. In this talk we will be concerned with the problem of determining all support tau-tilting modules (or equivalently all basic two-term silting complexes) for various finite dimensional algebras A over an algebraically closed field. To this end, I will explain a tool that allows to reduce the problem for a given algebra to a more tractable one by repeatedly taking quotients by central elements (in the radical). In practice, it turns out that this process often spits out a string algebra for which I will present an explicit combinatorial description of the support tau-tilting modules. Also various applications, such as tau-tilting finiteness and iterated tilting mutation, to blocks of group algebras and special biserial algebras will be presented.  This is based on joint work with Florian Eisele and Theo Raedschelders.

Un diagramme de Postnikov est une collection de brins dans un disque, soumise à certaines règles concernant les croisements des brins. Le diagramme détermine beaucoup d'autres objets et notamment une algèbre amassée. Un théorème récent de Galashin et Lam affirme que cette algèbre est isomorphe à l'anneau des fonctions régulières sur une sous-variété de la grassmannienne dite variété de positroïde. Dans cet exposé, je vais expliquer la construction d'une catégorification de cette algèbre amassée basée sur une propriété de Calabi-Yau d'une autre algèbre (non commutative) associée au diagramme de Postnikov.

We will talk about the enough g-pairs property of  cluster algebras and its explanations in terms of categories and surfaces.  We will give its applications to d-vectors and characterization for clusters.

Je vais présenter une équivalence de categories abéliennes graduées entre une catégorie de modules pour l’algèbre de Hecke Carquois de type A_1^(1) et la catégorie des faisceaux pervers cohérents sur le conenilpotent de type A. Je ferai ensuite le lien avec les catégorifications de Kashiwara-Kim-Oh-Park et de Cautis-Williams de la cellule ouverte unipotente quantique de type A_1^(1).

The enough g-pairs property of cluster algebras and applications. We will talk about the enough g-pairs property of  cluster algebras and its explanations in terms of categories and surfaces.  We will give its applications to d-vectors and characterization for clusters.

A triangulation of marked surfaces induces a Jacobian algebra. We study g-vector cones associated with its tau-tilting modules. We determine the closure of the union of g-vector cones associated with all tau-tilting modules. It is equal to the whole space. Our main ingredients are laminations, their shear coordinates and their asymptotic behavior under Dehn twists. The same result holds for gentle algebras by using their geometric realization.

Des déformations d'algèbres amassées avec plusieurs paramètres quantiques (algèbres amassées toroidales) apparaissent naturellement dans l'étude des représentations d’algèbres affines quantiques. Par ailleurs, la construction algébrique de l’anneau de Grothendieck quantique par Hernandez suggère que nous pouvons avoir aussi assez naturellement des déformations avec plusieurs paramètres de ces anneaux (anneaux de Grothendieck toroidaux). Ces objets sont fortement liés; en particulier lorsque les anneaux de Grothendieck fournissent des exemples de catégorification monoïdale d’algèbres amassées. Nous allons construire l’anneau de Grothendieck toroidal pour une algèbre affine quantique simplement lacée et nous allons prouver  que, de manière remarquable, pour une certaine catégorie monoidale C_1, il y a une structure naturelle d’algèbre amassée toroidale. Ce travail est en collaboration avec D. Hernandez.

Quiver Schur algebras are a generalization of Khovanov-Lauda-Rouquier algebras, well known for their role in the categorification of quantum groups. In this talk I will discuss their basic structural properties, as well as their connection to the cohomological Hall algebras defined by Kontsevich and Soibelman.

A pentagram map is a discrete integrable transformation on the space of projective classes of (twisted) n-gons in projective plane. We will discuss a classical and non-commutative version of pentagram map and its integrability properties. 

In the first part of the talk, we prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$-categories with respect to the corresponding classes of quasi-equivalences are all equivalent. As an application, in the second part, we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital $A_\infty$-functors between them. This is a joint work with Prof. A. Canonaco and Prof. P. Stellari arXiv:1811.07830.

In this talk, we discuss our approach to ``higher monoidal Deligne conjecture''.
An interest to this statement raised after the proof of Kontsevich formality theorem given by D.Tamarkin. In this case, it says that the cohomological Hochschild complex of any associative algebra A has a structure of a homotopical 2-algebra. The corresponding monoidal abelian category is the category of A-bimodules.
 We found a proof of a more general statement, which works for a general monoidal abelian category. We adapt the approach of Kock and Toen in their ``simplicial Deligne conjecture''. In the linear situation, we replace Segal monoids used by Kock and Toen by rather exotic objects called Leinster monoids. Rectification of Leinster monoids is one of our technical tools.
Another technical tool is the Drinfeld dg quotient and its refined version, which enjoys a nicer monoidal behaviour. If time permits, we discuss some conceptual problems towards a higher-monoidal generalisation of our results.

Les modules fidèles et équilibrés, qui sont parfois appelés ``modules ayant la propriété de double centralisateur'' apparaissent à de nombreux endroits dans la littérature sur la théorie des anneaux comme par exemple dans la dualité de Schur-Weyl et la notion d'algèbre QF1 de Thrall.
 Dans cet exposé, nous allons étudier ces modules pour les algèbres des matrices triangulaires supérieures et plus généralement pour les algèbres de Nakayama. Nous verrons que le nombre de modules fidèles et équilibrés pour l'algèbre des matrices triangulaires supérieures de taille $n$ est le q-analogue de $n!$ en $q=2$. Parmi ces modules, il y a exactement $n!$ modules ayant exactement $n$ facteurs directs indécomposables. Ces derniers semblent particulièrement intéressants car on peut les munir d'une structure de treillis qui étend naturellement le treillis de Tamari.
 Il s'agit d'un travail en commun avec William Crawley-Boevey, Biao Ma et Julia Sauter.

Drinfeld a montré que la R-matrice universelle du Yangien Yg d’une algebre de Lie semisimple g donne lieu a des solutions rationnelles des equations de Yang-Baxter sur chaque representation irreductible V de dimension finie de Yg. Etonnament, ceci n’est plus le cas si V est réductible, du moins si l’on demande que ces solutions soient naturelles et compatibles avec le produit tensoriel. J’expliquerai que l’on peut par contre obtenir des solutions méromorphes pour tout V. Cette construction se base sur la resommation de la partie abélienne de R, obtenue avec Sachin Gautam, et la réalisation de sa partie triangulaire inférieure comme twist qui conjuge le coproduit standard de Yg en le coproduit de Drinfeld deformé. Travail en commun avec Sachin Gautam et Curtis Wendlandt, basé sur arXiv:1907.03525