Abstract: We will discuss two families of polytopes with similar flavour. The first ones, very classical, are named associahedra but the second ones, named halohedra, are more recent. One can describe in simple combinatorial terms their graphs of vertices and edges. The construction of polytopes that realize this combinatorics is a long story, with many players and many solutions. The parallel between these two families is not yet complete: there remains to invent a natural orientation of the edges of halohedra, whose existence is suggested by a recent new viewpoint on "Coxeter-Catalan combinatorics".

Abstract: The sheets of a Lie algebra are the irreducible components of its loci of fixed orbit dimension. These were introduced in a program of Borho, Kraft and Jantzen for the study of representation theory via the orbit method. They have since been classified and their orbit spaces have been described via generalisations of Kostant's work for the regular sheet. We consider more detailed geometric properties of sheets and their orbit spaces, and in particular show that they are controlled by a finite group determined by the sheet. In a sense, this group is the obstruction to many nice geometric behaviours which occur in the case of the regular sheet.

Our main application is to describe the Hitchin fibration for certain loci in the moduli space of Higgs bundles which lie entirely over the discriminant locus. The Hitchin fibres for these loci are in a certain sense "non-abelian", but they fibre over spaces of abelian torsors with descriptions similar to Donagi and Gaitsgory's cameral data for the regular locus.  A second potential application would be to relate these finer geometric properties of sheets back to the representation theory of the Lie algebra, in the spirit of the orbit method.

Abstract: For many quantum groups a quantum cluster algebra structure is known, giving a combinatorial framework to study these algebras. On the other hand, such quantum groups can sometimes also be defined in the analytical setting, e.g. as locally compact quantum groups. We present some ongoing work to combine the cluster structure with the analytical structure, leading to applications in the theory of positive representations.

Abstract: Affine W-algebras form a family of vertex algebras generalising affine and Virasoro Lie algebras. They are in 1-to-1 correspondence with some affine Poisson varieties, the Slodowy slices.

In this talk, I will present how one can use Poisson geometry to prove reduction by stages for affine W-algebras. By definition, we say that reduction by stages holds whenever some W-algebra can be obtained as the noncommutative Hamiltonian reduction of another W-algebra.

This work is joint with Naoki Genra.

Abstract: Kostka–Foulkes polynomials $K_{\lambda,\mu}(q)$ are important combinatorial objects which have many incarnations in representation theory. We will characterize them in two different ways: as affine Kazhdan-Lusztig polynomials and as q-analogues of Kotska numbers, which count the weight multiplicities in irreducible representations. Then, we will discuss one of the most long-standing open problems in algebraic combinatorics: finding a closed and positive formula for these polynomials, together with a geometric interpretation. \emph{This includes a brief overview of a recently developed approach which has already been used to solve the problem for type $A$ and $C_2$.}

Given a non-commutative algebra $Q$ and its semiclassical limit $A$, an intriguing question has always been “Do the properties of $Q$ always reflect the (Poisson) properties of $A$?”. Of particular interest is the behaviour of automorphisms. The most famous example is the Belov Kanel-Kontsevich Conjecture, which predicts that the group of automorphisms of the nth-Weyl algebra $A_n$ is isomorphic to the group of Poisson automorphisms of the polynomial algebra $\mathbb{C}[x_1,…,x_{2n}]$. In this talk, I will present my work on a problem similar to the BKK Conjecture. Take a symplectic quotient singularity; the parameter spaces of filtered deformations and the parameter space of filtered quantizations coincide. Do the Poisson isomorphisms between the deformations coincide with the automorphisms between the quantizations? I have a positive answer in the special case of Kleinian singularities of type A and D, but the general problem remains open.

Abstract: Let G be a complex affine algebraic group. If C is a smooth algebraic curve and x is a point in C, the affine Grassmannian is an algebro-geometric object that parametrizes G-bundles on C together with a trivialization outside x. Alternatively, one can define the affine Grassmannian as the quotient G((t))/G[[t]].

In this talk we present possible analogs for the affine Grassmannian, in the setting where the curve is replaced by a smooth projective surface, and the trivialization data are specified with respect to flags of closed subschemes. We also obtain parallel descriptions in terms of quotients of the double loop group G((t))((s)). Based on joint works with B. Hennion, A. Maffei and G. Vezzosi.

Abstract: The classical first Brauer-Thrall conjecture states that if there are infinitely many non-isomorphic finite dimensional indecomposable modules over a finite dimensional algebra, then their dimension is unbounded. The first solution of the conjecture was provided by Roiter and a modern proof relies on Auslander-Reiten theory. Both approaches are homological in nature. We extend the first Brauer-Thrall conjecture to constructible subcategories of the module category of a finitely generated algebra over an arbitrary field. A subcategory is constructible if it can be described as all modules fulfilling a certain finite condition.  For example, Hom- and Ext-orthogonals of a finitely presented module are constructible. In general, these subcategories lose the homological properties of the module category, so a new approach is necessary to tackle the problem. We are able to solve it using the Ziegler spectrum of a ring and its connection to the spectrum of a commutative ring. The first step is to find a suitable curve inside the scheme of finite dimensional modules of a fixed dimension. This result is contributed by Andres Fernandez Herrero.

Abstract: Shifted quantum affine algebras are quantum groups parameterized by a coweight of the underlying Lie algebra. Hernandez introduced in 2022 a category O of representations of these algebras and in 2024 Geiss-Hernandez-Leclerc proved that the Grothendieck ring of this category O has a cluster algebra structure. In this talk I will present a new quantization for this cluster algebra that allows us to define the quantum Grothendieck ring of such category O (in the spirit of Nakajima and Varagnolo-Vasserot). If time permits, I will explain how to deduce from this construction the so-called quantum QQ-systems and a quantum cluster algebra structure on the q-oscillator algebra.

Abstract: Hecke algebras are generated by elements whose multiplication is controlled by an underlying Weyl group. In the q = 0 specialisation, it is determined by an operation known as the Demazure product in the Weyl group, which is well understood in the finite and affine cases, but is a priori not well-defined in the double affine case. In this talk, we discuss conjectures on generalising an approach developed by F. Schremmer using the quantum Bruhat graph, and results so far on using this as a definition in the double-affine setting.

Abstract: The (complex) character table of the finite group GL_n(F_q) has been known since the 50s, thanks to the work of Green. Later, Deligne and Lusztig gave a geometric description of it, thanks to Deligne-Lusztig induction and character sheaves.

However, not much is known concerning the computation of multiplicities, i.e. understanding the decomposition of tensor products of irreducible representations. 

This decomposition are well understood just in the so-called generic case, thanks to the work of Hausel, Letellier and R-Villegas, who also related them to character and quiver varieties. 

It is a natural question to ask how to generalize these results to the computation of all multiplicities. In this talk, I would like to present a first result in this direction and suggest possible ways to tackle the problem.

In particular, I will quickly explain how to compute the decomposition of tensor products of the so-called semisimple split representations. This results relates multiplicities to the counting of representations of certain quivers over F_q.

This method does not extend in general. Computations suggest that we should pursue a more geometric approach and study rather certain related quiver moduli spaces. I would like in particular to explain how the general computation could  be related to the understanding of COHAs and BPS algebras for certain quivers.

Abstract: Given a projective variety X, one can construct a geometric object parametrizing the endomorphisms of X, known as the endomorphism scheme. The set of its connected components naturally carries the structure of a monoid. In this talk, I will present some finiteness results concerning such monoids of connected components (for example, the finite subgroups of each maximal subgroup have uniformly bounded orders). I will also discuss basic connections with the geometry of X, and conclude with an open question about whether these monoids can be bounded "globally".

Abstract: In the talk, we will introduce the notion of reachability categories. These categories are obtained from path categories of quivers by taking quotients under the "reachability" relation. We will compare reachability categories to path categories, from both a topological and a categorical viewpoint. Then, we will focus on the category algebras of reachability categories, also known as commuting algebras. As application, we will prove that commuting algebras are Morita equivalent to incidence algebras of posetal reflections of reachability categories, a result previously obtained by E. L. Green and S. Schroll. If time allows it, we shall see further connections to magnitude homology, Hochschild cohomology, and persistent homology of graphs. This is joint work with H. Riihimäki.

Abstract: About 10 years ago, Schiffmann proved that the number of absolutely indecomposable vector bundles on a curve over a finite field (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave another proof of this result in a slightly more general formulation, but neither proof generalizes in an obvious way to G-bundles for other reductive groups G.

In joint work with Zhiwei Yun, we generalize the above results to G-bundles. Namely, we express the number of absolutely indecomposable G-bundles on a curve X over a finite field in terms of the cohomology of the moduli stack of stable parabolic G-Higgs bundles on X. For G=GL(n), Schiffmann also proves that this number is given by an explicit polynomial in the Weil numbers of X, thus giving a closed formula for the Poincaré polynomial of the moduli space of stable Higgs bundles. Such explicit calculation remains an open question for general G.

Abstract: Let $A$ be a gentle algebra. For every collection of string and band diagrammes, we consider the constructible subset of the variety of representations containing all modules with this underlying diagramme. For instance, to a band we associate the set of corresponding band modules with all choices of parameters. We study degenerations of such sets and show that some of them are induced by kissing of string and band diagrammes.

This work combines representation theory of finite dimensional algebras with algebraic combinatorics and algebraic geometry. 

Abstract: The study of trace identities, which is closely related to the invariant theory of $n\times n$ matrices, plays a significant role in contemporary mathematics and has crucially contributed to the development of PI-theory. The foundational contributions to this area were made by Procesi and Razmyslov, who independently proved that all trace identities of the full matrix algebra $M_n(F)$ of order $n$ are consequence of a single polynomial derived from the Cayley-Hamilton theorem.

Within PI-theory, a notable class of algebras is formed by upper triangular matrix algebras. In this context, Berele identified a set of generators for their trace identities. However, upper triangular matrix algebras admit a wide range of trace functions. In this talk, we extend Berele’s results, by determining generators of trace identities of $2 \times 2$ upper triangular matrices for all possible

traces and provide general results regarding n × n upper triangular matrices. In particular, a complete generalization of Berele’s results for $n > 2$ is still not known.

Abstract: The geometric model for the derived category of a gentle algebra (Opper-Plamondon-Schroll 2018) uses graded curves on a surface to describe indecomposable objects and morphisms. However, not all curves are gradable. A natural question is to ask if there exists a category with a geometric model using all curves on this surface. In this talk I will first present the geometric model for the derived category of gentle algebras of finite global dimension and how to construct its 1-periodic derived category. This category can be seen as the stable category of maximal Cohen-Macaulay modules over an extension of A and indecomposable objects are parametrized by all curves on the surface.

Abstract: For a smooth curve $C$, the Schur algebras are defined by the Borel-Moore homology

of  Steinberg-like stacks built from coherent sheaves on $C$. In the case $C=\mathbb{P}^1$, we

construct a PBW basis for this algebra. Using the derived equivalence between the category of

coherent sheaves on $\mathbb{P}^1$ and the category $\mathrm{mod}\mathbb{C}Q$ of

quiver representations of the Kronecker quiver Q, we show that its Schur algebra is an inverse

limit of certain quotients of the quiver Schur algebras for Q. This is a joint work with Olivier Schiffmann.

Abstract: One of the best known family of cluster algebras come from the Grassmannians $Gr(k,n)$. These cluster algebras are known to be related through categorification to surface singularities of the form $x^k+y^{n-k}+z^2=0$ . They are finite type only when $(k-2)(n-k-2)<4$ and such cases are exactly when the corresponding singularity is ADE type, and moreover the cluster type matches the singularity type. The combinatorics of the corresponding ADE root system essentially completely controls the properties of the cluster algebra and the singularity.  

Following the program of Arnold, surface singularities beyond ADE type are classified into various families based on “modality” and other invariants. Moreover following work of Ebeling and others, Dynkin diagrams for some of these singularities have been constructed. The question I wish to propose is essentially “what is the “type” of the Grassmannian cluster algebras beyond the finite cases?”. I propose that the answer should be found by further studying the connection between Grassmannian cluster algebras and surface singularities. The first step towards understanding this should be constructing special seeds of the cluster algebra which correspond to these Dynkin diagrams. With this we can hope to have a much better understanding of the combinatorics and properties of the infinite type Grassmannian cluster algebras. 

Abstract: Two quantum affine algebras are said to be Langlands dual if their

associated generalized Cartan matrices are transposes of each other. We

focus on the Langlands dual quantum affine algebras of types $B_n^{(1)}$

and $A_{2n-1}^{(2)}$. Although these two algebras are not known to be

related at the algebraic level, their representation theories exhibit

intriguing connections.

More precisely, a conjecture by Edward Frenkel and David Hernandez

states that for any irreducible finite-dimensional representation $V$ of

the quantum affine algebra of type $B_n^{(1)}$, its character $\chi V$

coincides with the character of some representation of the quantum

affine algebra of type $A_{2n-1}^{(2)}$, under an identification of

corresponding weight lattices. In joint work with Jingmin Guo and

Jianrong Li, we proved this conjecture in the case where $V$ is a snake

module. Moreover, we provide an explicit formula that decomposes $\chi

V$ as a sum of characters of irreducible representations of the quantum

affine algebra of type $A_{2n-1}^{(2)}$. Therefore, we refer to this

result as the Langlands branching rule.

In this talk, I will introduce the background of the problem, emphasize

this unexpected connection between Langlands dual algebras, and present

the tools we use to address it. I will conclude with the branching rule formula, illustrated by a few explicit examples.

Abstract: The Gieseker space is a generalization of the Hilbert point scheme. We will present combinatorial correspondences between the irreducible components of the locus of fixed points of the Gieseker space and the block theory of the Ariki-Koike algebra. First, we will describe the locus of fixed points in terms of Nakajima quiver varieties over the McKay quiver of type $A$. Then, we will present how to recover the combinatorics of cores of charged multipartitions, as defined by Fayers and developed by Jacon and Lecouvey, on the Gieseker side. In addition, we will present a new way to compute the multicharge associated with the core of a charged multipartition and how the notion of core blocks, discovered by Fayers, can be interpreted geometrically. Finally, if time permits, we will present the work in progress concerning the generalization to cyclotomic KLR algebras of other types and the potential deepening of these connections.

Abstract: Ever since being defined in the 1970s, the Kazhdan–Lusztig polynomials have been a key source of research problems for representation theorists. Recently, 'categorifying' the polynomials has resulted in proof of the famous Kazhdan–Lusztig positivity conjecture and helped produce a counter-example to Lusztig’s equally famous conjecture.

In this talk, we'll discuss what we might mean by categorification and try to motivate using categorical structure to gain insights into algebraic questions.

Then we'll discuss this in the context of Kazhdan–Lusztig polynomials, presenting a 'nice' combinatorial answer to our question using (the top half of) oriented Temperley-Lieb diagrams. This will also allow us to create an isomorphism between the category we exploit for this, the Hecke category, and the Khovanov arc algebra (of knot theory fame) in type D, which this talk will focus on, although similar results for types A and B do also exist.

Abstract: Hecke modifications of vector bundles on curves have been studied for decades and have connections with automorphic forms, Hall algebras and parabolic bundles. In rank 2 there exists a correspondence between Hecke algebras and quasi-parabolic bundles. In this talk, I will discuss this correspondence, and its generalisation for higher rank, applications to other topics and some open questions. A part of this is joint work with Roberto Alvarenga and Leonardo Moco.

Abstract: In representation theory, it is fundamental to understand the simple objects. In a highest weight category, some information about simples can be read off from their extension groups to “costandard objects”. This (graded) vector space is usually encoded by singularities of some algebraic geometric space.