12 to 16 December 2022

Gwyn Bellamy |

Pieter Belmans |

Vladimiro Benedetti |

Pablo Boixeda Alvarez |

Alexander Bravermann |

Gurbir Dhillon |

Tamas Hausel |

David Jordan |

Mikhail Kapranov |

Tasuki Kinjo |

Thang T. Q. Le |

Serguey Mozgovoy |

Andrei Negut |

Simon Riche |

Raphael Rouquier |

Pavel Safronov |

Sarah Scherotzke |

Maxim Smirnov |

Weihong Xu |

Amphi Turing (ground floor). You can find some information to find the building Sophie Germain here

MAP

TRANSPORTATION

Some financial support may be available for young researchers. If you are interested please contact eric.vasserot@imj-prg.fr

MONDAY 12th |

9h30-10h30 : Gwyn Bellamy |

10h30-11h00 : Coffee break |

11h00-12h00 : Pavel Safronov |

14h00-15h00 : Raphael Rouquier |

15h00-16h00 : Coffee break |

16h00-17h00 : Mikhail Kapranov |

TUESDAY 13th |

9h30-10h30 : Alexander Bravermann |

10h30-11h00 : Coffee break |

11h00-12h00 : Vladimiro Benedetti |

14h00-15h00 : Sarah Scherotzke |

15h00-16h00 : Coffee break |

16h00-17h00 : Pablo Boixeda Alvarez |

WEDNESDAY 14th |

9h30-10h30 : David Jordan |

10h30-11h00 : Coffee break |

11h00-12h00 : Maxim Smirnov |

14h00-15h00 : Gurbir Dhillon |

15h00-16h00 : Coffee break |

16h00-17h00 : Pieter Belmans |

THURSDAY 15th |

9h30-10h30 : Serguey Mozgovoy |

10h30-11h00 : Coffee break |

11h00-12h00 : Thang T. Q. Le |

14h00-15h00 : Tasuki Kinjo |

15h00-16h00 : Coffee break |

16h00-17h00 : Simon Riche |

FRIDAY 16th |

9h30-10h30 : Andrei Negut |

10h30-11h00 : Coffee break |

11h00-12h00 : Weihong Xu |

14h00-15h00 : Tamas Hausel |

15h00-16h00 : Coffee break |

## Gwyn Bellamy : Minimal degenerations for quiver varieties. |

Abstract : For any symplectic singularity, one can consider the minimal degenerations between symplectic leaves - these are the relative singularities of a pair of adjacent leaves in the closure relation. I will describe a complete classification of these minimal degenerations for Nakajima quiver varieties. This goes a long way to classifying those leaves in a quiver variety that have normal closure. As an application, I'll explain how one can identify (the normalization of) a leaf closure in a Calogero-Moser space associated to a wreath product group with a Calogero-Moser space for a parabolic subgroup. This confirms a conjecture of Bonnafé-Rouquier for this family of groups. The talk is based on joint work in progress with Ruslan Maksimau and Travis Schedler. |

## Pieter Belmans : Quiver GIT without the GIT |

Abstract : Moduli spaces of quiver representations are usually constructed using GIT, resulting in projective moduli spaces of semistable representations for an acyclic quiver. Because GIT is not always available as a method to construct moduli spaces, it is interesting to find a moduli-theoretic proof of the projectivity in the quiver case, by constructing a natural ample line bundle on the good moduli space of the moduli stack. This then serves as a blueprint for other projectivity proofs. I will illustrate this "Beyond GIT" construction for quivers, and discuss effective bounds on global generation of determinantal line bundles. This is joint work with Chiara Damiolini, Hans Franzen, Vicky Hoskins, Svetlana Makarova and Tuomas Tajakka. |

## Vladimiro Benedetti : Discriminants of theta-representations |

Abstract : Tevelev has given a remarkable explicit formula for the discriminant of a complex simple Lie algebra in terms of the Weyl group. This discriminant can be defined as the equation of the dual hypersurface of the minimal nilpotent orbit inside the projectivized of the Lie algebra itself (also called the adjoint variety). In this talk we will see how to extend this formula to the setting of graded Lie algebras. In this context we will define the equivalent of the minimal nilpotent orbit and express the equation of the corresponding dual hypersurface in terms of the little Weyl group of the graded Lie algebra. As an application, we will explain how this implies for example that the codegree of the Grassmannian G(4, 8) is equal to the number of roots of e7. This is a joint work with Laurent Manivel. |

## Pablo Boixeda Alvarez : Equivalued affine Springer fibers and the small quantum group |

Abstract : In this talk I will discuss several connections between the small quantum group and a certain affine Springer fiber. I will mainly discuss some relation of the category of graded representations of the small quantum group and microlocal sheaves on the affine Springer fiber as part of ongoing work with R.Bezrukavnikov, M. McBreen and Z. Yun. I will also mention some connection of the center of the small quantum group and the cohomology of the affine Springer fiber, part of joint work with R. Bezrukavnikov, P. Shan and E. Vasserot. |

## Alexander Bravermann : Towards the theory of automorphic functions on curves over local non-archimedian fields - a survey (based on joint work in progress with D.Kazhdan and A.Polishchuk) |

Abstract : The unramified part of the classical theory of automorphic forms for a global field of positive characteristic and a reductive group G is equivalent to the study of the space of (complex-valued) functions on the F_q-points of the moduli space of G-bundles on a smooth projective curve X over F_q and the action of the so called Hecke operators on it. After briefly recalling the usual theory we shall discuss what one can do if F_q is replaced by a local non-archimedian field F (the archimedean case was studied extensively in the recent works on Etingof, Frenkel and Kazhdan). We shall address the following questions: 1) How do define some good space of functions on the F-points of the moduli space of bundles on X and how to define the corresponding Hecke operators? 2) What can one say about the eigen-values and of the Hecke operators? Some interesting connection to the geometry of the Hitchin system and the notion of very stable bundles will also be discussed. |

## Gurbir Dhillon : On the log Kazhdan--Lusztig correspondence |

Abstract : An influential conjecture of Feigin--Gainutdinov--Semikhatov--Tipunin from the mid 2000s relates representations of small quantum groups and triplet vertex algebras at positive integer values of the Kac--Moody level. We will formulate an extension of the conjecture to all Kac--Moody levels and will describe a way to prove it conditional on some (in-reach) conjectures in the quantum geometric Langlands program. We will also describe some interesting modular properties of characters of triplet vertex algebras at negative and positive levels, and their relations to Zagier duality and semi-infinite homological duality. |

## Tamas Hausel : Mirror symmetry and big algebras |

Abstract : First we recall the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we discuss a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety. |

## David Jordan : Quantum Hotta--Kashiwara modules, and skeins on tori. |

Abstract : The 1984 Inventiones paper of Hotta and Kashiwara introduces a system of differential equations on a simple Lie algebra which computes the fibres of the Grothendieck--Springer resolution. In this talk (mostly focused on SL_N and GL_N) I will introduce a natural q-deformation which interpolates between the universal Hotta--Kashiwara's D-module and the algebra of functions on the commuting variety, and which generates a subcategory Morita equivalent to modules over the SL_N- and GL_N-skein algebra of the two-torus T^2. Using representation theory of affine and double affine Hecke algebras, I will then conclude formulas for the dimensions of the SL_N- and GL_N-skein modules of the three-torus T^3, generalising a well-known formula (the number 9) for SL_2. This is joint work with Sam Gunningham, Monica Vazirani and Haiping Yang. |

## Mikhail Kapranov : The PROB of graded bialgebras, perverse sheaves on configuration spaces and Hecke algebroids. |

Abstract : I will first recall the earlier work with V. Schechtman relating perverse sheaves on the adjoint quotient h/W of a reductive group with the algebra of parabolic induction and restriction. For the A-series this can be encoded by the braided category (PROB) generated by the universal graded bialgebra. I will then explain the relation of our approach with another point of view on the Ind/Res formalism, that of Hecke/Schur algebroids. |

## Tasuki Kinjo : Cohomological Donaldson-Thomas theory for 2-Calabi-Yau categories |

Abstract : Cohomological Donaldson-Thomas (CoDT) invariants were introduced by Kontsevich-Soibelman and Brav-Bussi-Dupont-Joyce-Szendroi as categorifications of the Donaldson-Thomas invariants counting objects in 3-Calabi-Yau categories. In this talk, I will explain applications of the CoDT theory to the cohomological study of the moduli of objects in 2-Calabi-Yau categories. Among other things, I will construct a coproduct on the Borel-Moore homology of the moduli stack of objects in these categories and establish a PBW-type statement for the Kapranov-Vasserot cohomological Hall algebras. This talk is based on a joint work in progress with Ben Davison. |

## Thang T. Q. Le : Quantum trace homomorphisms for SL_n skein algebras of surfaces |

Abstract : For a punctured surface there are two quantizations of the SL_n character variety: The firs is the SL_n skein algebra (Sikora, MOY graph), and the second is the Fock-Goncharov quantization of the X-variety. When n=2 Bonahon and Wong showed that there is an algebra homomorphism, known as the quantum trace, from the first quantized algebra to the second one. We show that for general n a similar quantum trace map exists, and for many surfaces an A-version of the quantum trace exists. Our quantum trace homomomorphisms can be defined over any ground ring. This is a joint work with T. Yu. |

## Serguey Mozgovoy : DT invariants and vertex algebras |

Abstract: Cohomological Hall algebras (CoHAs) can be understood as a mathematical incarnation of algebras of BPS states in string theory. Their Poincare series can be used to determine DT invariants of the corresponding categories. For a symmetric quiver Q, the corresponding CoHA is commutative and I will explain how its dual can be naturally equipped with a structure of a vertex bialgebra. It can also be identified with 1) the universal enveloping algebra of some Lie algebra, 2) the universal enveloping vertex algebra of some vertex Lie algebra, 3) the principal free vertex algebra embedded into some lattice vertex algebra. This identification leads to a new proof of the positivity of DT invariants. |

## Andrei Negut : Quantum loop groups for quivers with general parameters |

Abstract: Since usual quantum groups are given by explicit generators modulo explicit q-Serre relations, one might seek to give similar presentations for quantum loops groups. The generalization turns out to require a lot of new ideas, and we will present a framework that interprets the sought-for generators and relations as being dual to a shuffle algebra construction. We will make this explicit in certain situations of interest (Hall algebras of curves over finite fields, preprojective K-theoretic Hall algebras, quantum loop groups associated to symmetric Cartan matrices) and shed some light on the general case as well. |

## Simon Riche : Modular and integral ramified geometric Satake equivalences |

Abstract : Zhu (under a technical assumption) and Richarz (in full generality) have studied a variant of the geometric Satake equivalence which describes a category of equivariant Q_l-perverse sheaves on the affine Grassmannian of a (non necessarily split) reductive group over k((z)) and a choice of special parahoric subgroup. I will report on a joint work with P. Achar, J. Lourenço and T. Richarz aiming at obtaining a version of this equivalence for integral and modular coefficients. The novelty compared with the usual Satake equivalence is that the Tannakian group describing the category is not necessarily reductive. |

## Raphael Rouquier : Coherent realizations of 2-representations |

## Pavel Safronov : Critical cohomology and deformation quantization |

Abstract : For any stack equipped with a d-critical structure Joyce and collaborators have defined a mixed Hodge module which allows one to define the critical cohomology of the stack. It features prominently in the theory of cohomological DT invariants. This critical cohomology is notoriously difficult to compute. In this talk I will explain a relationship between the critical cohomology and deformation quantization. For instance, in the case of a Lagrangian intersection it allows one to compute critical cohomology in terms of DQ modules. As an example of these ideas, I will explain a relationship between the critical cohomology of the character stack of a closed oriented 3-manifold, complexified Floer homology of Abouzaid--Manolescu and skein modules of Przytycki--Turaev. This is a report on work in progress, joint with Sam Gunningham. |

## Sarah Scherotzke : Cotangent complexes of moduli spaces |

Abstract : We explain how shifted symplectic structures on derived stacks are connected to Calabi-Yau structures on differential graded categories. More concretely, we will show that the cotangent complex to the moduli stack of a differential graded category A is isomorphic to the moduli stack of the *Calabi-Yau completion* of A, answering a conjecture of Keller-Yeung. |

## Maxim Smirnov : Quantum cohomology and derived categories of coadjoint varieties |

Abstract : We will discuss properties of quantum cohomology, both small and big, of coadjoint varieties of simple algebraic groups and how they relate to the structure of Lefschetz collections in the derived categories of these varieties.Some general conjectures pertaining to this will be formulated. The talk is based on the joint works with Alexander Kuznetsov and Nicolas Perrin. |

## Weihong Xu : Quantum K-theory of Incidence Varieties |

Abstract : Buch, Mihalcea, Chaput, and Perrin proved that for cominuscule flag varieties, (T-equivariant) K-theoretic (3-pointed, genus 0) Gromov- Witten invariants can be computed in the (equivariant) ordinary K-theory ring. Buch and Mihalcea have a related conjecture for all type A flag varieties. In this talk, I will discuss work that proves this conjecture in the first non-cominuscule case--the incidence variety X=Fl(1,n-1;n). The proof is based on showing that Gromov-Witten varieties of stable maps to X with markings sent to a Schubert variety, a Schubert divisor, and a point are rationally connected. As applications, I will discuss positive formulas (an equivariant Chevalley formula and a non-equivariant Littlewood-Richardson rule) which determine the multiplicative structure of the (equivariant) small quantum K-theory ring of X. If time permits, I will also discuss current joint work with Gu, Mihalcea, Sharpe, and Zhang on giving a presentation for this ring with inspiration from supersymmetric gauge theory. |