GEOMETRIC REPRESENTATION THEORY AND QUANTUM TOPOLOGY

12 to 16 December 2022

SPEAKERS

Gwin 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

ORGANIZERS

The workshop is organized by the ANR research project CATORE. You can find some information HERE.

PRACTICAL INFORMATIONS

The workshop is hosted by the University Paris Cite, in the center of Paris, building Sophie Germain (campus PRG),
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

SCHEDULE

MONDAY 12th
9h30-10h30 : Gwin 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 : Pablo Boixeda Alvarez
15h00-16h00 : Coffee break
16h00-17h00 : Sarah Scherotzke
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

PROGRAM

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Gwin 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

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

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

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

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

Pavel Safronov

Sarah Scherotzke

Maxim Smirnov

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.