Program

Opening

David Hernandez

Title:  Cluster algebras and shifted quantum groups

Abstract: We report on a joint work and an ongoing project with C. Geiss and B. Leclerc. Shifted quantum affine algebras emerged from the study of quantized Coulomb branches. We show that the Grothendieck ring of the category O for the shifted quantum affine algebras has the structure of a cluster algebra. The cluster variables of a class of distinguished initial seeds are certain formal power series defined from a new Weyl group action introduced in a joint work with Frenkel. These cluster variables satisfy a system of functional relations called QQ-system. In a work in progress, we extend the construction to non simply-laced types, and we give a new interpretation of F-polynomials in this context.

Timm Peerenboom

Title: Cohomological Hall Algebras of Torsion Sheaves on Weighted Projective Lines

Abstract: Cohomological Hall algebras for quivers were introduced by Kontsevich--Soibelman. We apply their construction to weighted projective lines and obtain a sequence of infinite dimensional algebras which we compute in terms of generators and relations.

Eric Opdam

Title: Nonsymmetric shift operators for Dunkl-Cherednik operators

Abstract: Trigonometric Dunkl-Cherednik operators provide a representation of Cherednik's trigonometric double affine Hecke algebra. We present a theorem of existence and uniqueness of ''nonsymmetric shift operators'' for these Dunkl-Cherednik operators. These are trigonometric differential-reflection operators which shift the parameters of the Dunkl-Cherednik operators by integers, and which restrict on the space of W-invariant polynomials to the well known hypergeometric shift operators. The latter are instrumental in the theory of symmetric Macdonald polynomials. Joint work in progress with Valerio Toledano Laredo.

Xin Fang

Title: Seshadri stratifications and applications

Abstract: In this talk I will introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure allows us to construct (1). a Newton-Okounkov simplicial complex with an extra integral structure; (2). a flat degeneration of the variety into a reduced union of toric varieties. As an application, I will discuss a combinatorial criterion for the projective normality. Among examples, I will emphasize on the Schubert varieties and toric varieties, where the Lakshmibai-Seshadri paths and the barycentral subdivision of a polytope get algebro-geometric interpretations.

Alexandre Minets

Title: Ersatz parity sheaves and stratifications of algebras

Abstract: KLR algebras can be realized as Ext-algebras of constructible sheaves on a certain space. In characteristic 0, Kato proved that such algebras are (polynomially) quasi-hereditary, under the assumption that the space has finitely many orbits under the action of an algebraic group. This result was extended to characteristic p by McNamara, substituting perverse sheaves techniques for parity sheaves. Unfortunately, this approach does not apply to KLR algebras beyond Dynkin type. I will explain how to extend the theory of parity sheaves to cover the first non-trivial case of Kronecker quiver, and speculate about how to approach other affine types. Based on arXiv:2504.17430, joint with R. Maksimau.

Pierre Baumann

Title: Some properties of the Mirković-Vilonen basis

Abstract: Let G be a complex connected reductive group. Under the Geometric Satake Equivalence, an irreducible representation V of G can be realized as the intersection homology of an affine Schubert variety. In this context, Mirković and Vilonen singled out algebraic cycles, whose classes form a basis of V. This basis shares many properties with the canonical basis of V, but the precise relationship between the two bases is not well understood.

Caroline Lassueur

Title: On the role of p-permutation modules in the modular representation theory of finite groups.

Abstract: Since the last century, the modular representation theory of finite groups has been driven by a series of long-standing and profound conjectures, amongst which we find Broué's Abelian Defect Group Conjucture, Alperin's Weight Conjecture and Puig's Finiteness Conjecture. Over the past decades, it has become increasingly clear that the class  of p-permutation modules plays a major rôle in all of them. In this talk, I will survey these ideas, as well as results I obtained about classifications of p-permutation modules. 

Balázs Szendroi

Title: Enumerative geometry of Quot schemes around the McKay correspondence and representation theory of affine algebras

Abstract: This talk will review some results computing generating series of Euler characteristics of certain singular Nakajima quiver varieties, spaces that can also be identified with Quot schemes of certain modules over cornered preprojective algebras. The proofs involve the study of collapsing fibres in the associated variation of GIT map, as well as (in type A) some combinatorics of coloured partitions, or (in arbitrary type) the study of quantum dimensions of standard modules over loop algebras following Nakajima. We will also point out some intricate representation theoretic questions that arise from the comparison of these two approaches. Based joint works with Lukas Bertsch, Alastair Craw, Søren Gammelgaard and Ádám Gyenge.