University of North Carolina at Chapel Hill

Mondays/Fridays 4pm, PH-367 or PH-385

The aim of this seminar is to bring speakers from this area and outside to speak on topics related to Representation Theory (specially geometric and topological methods employed in Representation Theory). The speakers are expected to give their talks at a level suitable for graduate students. The seminar is organized by Prakash Belkale, Jiuzu Hong, Shrawan Kumar and Richárd Rimányi.

2026 Spring

Date	Speaker	Affiliation	Title
April 10	Jialiang Zou	MIT	Theta correspondence and Springer correspondence
March 27	Leonardo Mihalcea	Virginia Tech	Quantum K theory of Grassmannians in physics and math
Feb 20	Hyun Kyu Kim	KIAS	Skein algebras of genus zero surfaces and quantized K-theoretic Coulomb branches
Jan 23	Jacob Matherne	NC State	The intersection cohomology of a matroid
Jan 16	Chi Hong Chow	Virginia Tech	Mirror symmetry and Gamma conjectures: flag variety case

Jialiang Zou: Theta correspondence and Springer correspondence
Abstract: Let V and W be an orthogonal and a symplectic space, respectively. The action of G=O(V)\times Sp(W) on V\otimes W provides an example of G-hyperspherical varieties introduced by D. Ben-Zvi, Y. Sakellaridis, and A. Venkatesh (BZSV for short). It is the classical limit of theta correspondence from the perspective of quantization. I will explain a geometric construction motivated by theta correspondence over finite fields, which describes how principal series representations behave under theta correspondence using Springer correspondence. This is joint work with Jiajun Ma, Congling Qiu, and Zhiwei Yun.

BZSV proposed a relative Langlands duality linking certain G-hyperspherical varieties M with their dual G^\vee-hyperspherical varieties M^\vee. A remarkable instance of this duality is that the hyperspherical variety underlying theta correspondence is dual to the hyperspherical variety underlying the branching problem in the Gan-Gross-Prasad conjecture. I will discuss how our results fit into the broader framework of this relative Langlands duality.

Leonardo Mihalcea: Quantum K theory of Grassmannians in physics and math
Abstract: The quantum K theory of Grassmannians is a ring with a
product deforming the usual K theory product. In (mathematical) physics, it is
the coordinate ring of an affine variety given by the logarithmic derivatives of a certain superpotential, giving the Bethe Ansatz equations. These equations may be used to construct idempotents in the quantum K ring, called Bethe vectors, which in turn lead to a quantum version of the equivariant localization theory. I will discuss these constructions, with emphasis on geometric interpretations. Based on work with V. Gorbounov and C. Korff, and with with W. Gu, E. Sharpe, and H. Zou.

Hyun Kyu Kim: Skein algebras of genus zero surfaces and quantized K-theoretic Coulomb branches
Abstract: The Kauffman bracket skein algebra of an oriented surface S is a quantization of the SL2 character variety of S, and is generated by isotopy classes of framed links living in S times an interval, modulo skein relations. The relative skein algebra quantizes the relative character variety, fixing the classes of monodromy along small loops around punctures. We show that the relative skein algebra of a punctured surface of genus zero is isomorphic to the Braverman-Finkelberg-Nakajima quantized K-theoretic Coulomb branch, associated to a certain group G and representation N, built from a specific quiver. This gives a monoidal categorification of the genus zero relative skein algebra, which in particular yields a positive basis through the work of Cautis and Williams, partially answering a question posed by D. Thurston. Based on the joint work with Dylan Allegretti and Peng Shan, arXiv:2505.13332.

Jacob Matherne: The intersection cohomology of a matroid
Abstract: The intersection cohomology IH(M) of a matroid M was recently introduced and used to prove a 1974 conjecture of Dowling and Wilson concerning the shape of a certain poset associated with the matroid, and to prove the nonnegativity of the coefficients of matroid Kazhdan–Lusztig polynomials. Part of this talk will explain the basics of IH(M) and its connection with the Kazhdan–Lusztig theory of matroids. Throughout, I will draw parallels with the classical Kazhdan–Lusztig theory of Coxeter groups and flag varieties.

Despite its usefulness, the original construction of IH(M) was via a complicated inductive process. I will give a simpler characterization of IH(M) which works with coefficients in a field of positive characteristic, thus leading to new “p-Kazhdan–Lusztig polynomials” of matroids. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.

Chi Hong Chow: Mirror symmetry and Gamma conjectures: flag variety case
Abstract: Fano mirror symmetry is a duality between Fano manifolds and Landau-Ginzburg models. In this talk, I will focus on flag varieties and discuss (1) an isomorphism between their quantum D-modules and the Gauss-Manin systems of their Rietsch mirrors, (2) the matching of certain integral structures on these D-modules, and (3) how (1) and (2) yield Gamma conjecture I.

2025 Fall

Date	Speaker	Affiliation	Title
Dec 3	Lin Chen	Tsinghua University	An Extension of the Kazhdan-Lusztig Equivalence
Nov 21	Junliang Shen	Yale University	Perverse filtrations for compactified Jacobians
Nov 14	Milen Yakimov	Northeastern	Poisson geometry and representation theory of root of unity quantum cluster algebras
Oct 10	Che Shen	Columbia university	Quasimaps to the Flag Variety and Tilting Modules in Category O
Sep 26	Pierre Godfard	UNC	Hodge structures on conformal blocks
Aug 26	Yujie Xu	Columbia university	Hecke algebras for p-adic groups and applications to the Langlands correspondence

Lin Chen: An Extension of the Kazhdan-Lusztig Equivalence
Abstract: We will explain some of the main ideas in the preprint https://arxiv.org/abs/2111.14606, where the DG category of Iwahori-integrable affine Lie algebra representations and the DG category of representations of the “mixed” quantum group are identified. This is a joint work with Yuchen Fu.

Junliang Shen: Perverse filtrations for compactified Jacobians
Abstract: The perverse filtration plays a crucial role in the study of the topology of the Hitchin system; for example, in the (now proven) P=W conjecture, the perverse filtration for the Hitchin system describes the weight filtration of the character variety of the surface group via the non-abelian Hodge theory. For an algebraic curve with locally planer singularities, the perverse filtration for the compactified Jacobian of this curve is more mysterious. It was conjectured (by Oblomkov, Rasmussen, Shende, Gorsky …) that this “local” perverse filtration behaves similar to that of a Hitchin system, and is related to knot invariants. In this talk I will discuss some recent progress on structures of the “local” perverse filtration associated with a locally plane curve, where a Fourier transform plays a key role. Applications and open questions will be discussed. Based on joint work with Davesh Maulik and Qizheng Yin.

Milen Yakimov: Poisson geometry and representation theory of root of unity quantum cluster algebras
Abstract: We will provide a gentle introduction to Cluster Algebras focusing at their Poisson geometry and representation theory at roots of unity. We will show that all root of unity quantum cluster algebras have canonical structures of Cayley-Hamilton algebras (in the sense of Procesi), which allows the transfer of finite generation between the quantum and classical situations. We will then focus on the geometry of the Gekhtman-Shapiro-Vainshtein Poisson brackets, proving that the spectra of cluster algebra have explicit Zariski open torus orbit of symplectic leaves, which is a far reaching generalization of the Richardson divisor of a Schubert cell in Lie theory. We will finish with a classification of the irreducible representations of quantum cluster algebras at roots of unity of maximal dimension. This is a joint work with Shengnan Huang, Thang Le, Greg Muller, Bach Nguyen and Kurt Trampel.

Che Shen: Quasimaps to the Flag Variety and Tilting Modules in Category O
Abstract: A quasimap from an algebraic curve to a GIT quotient is a map to the stack quotient that is generically stable. The geometry of Laumon spaces (an open subset of quasimaps from P^1 to the flag variety) is closely related to the representation theory of gl_n. In particular, one can construct an action of gl_n on the cohomology of Laumon spaces via geometric correspondences, and this cohomology can be identified with dual Verma modules of gl_n under this action. The full moduli space of quasimaps provides a natural compactification of Laumon spaces. I will explain how to construct an action of gl_n on the equivariant cohomology of these moduli spaces and explore its relation to tilting modules in Category O.

Pierre Godfard: Hodge structures on conformal blocks
Abstract: Modular functors are collections of vector bundles with flat connections on (twisted) moduli spaces of curves, individually known as conformal blocks, that satisfy strong compatibility conditions with respect to natural maps between these moduli spaces. Such structures arise naturally in the representation theory of affine Lie algebras and quantum groups, where the conformal blocks are known to be semisimple.

Recently, Hodge structures on the genus-0 conformal blocks associated to affine Lie algebras have been studied by Belkale, Fakhruddin, and Mukhopadhyay through a motivic construction. In particular, they computed genus-0 Hodge numbers for $sl_n$.

I will discuss an axiomatic proof of the existence and uniqueness of such Hodge structures and of the semisimplicity of conformal blocks, for any modular functor (i.e. any modular fusion category). If the flat bundles of conformal blocks were rigid and semisimple, a result of Simpson in non-Abelian Hodge theory would imply that they support Hodge structures. However, this is not the case in general. I will explain how a different form of rigidity for modular fusion categories—Ocneanu rigidity—can be used, together with non-Abelian Hodge theory, to tackle these questions. Finally, I will discuss an application to the computation of Hodge numbers for $sl_2$ modular functors of odd level in higher genus and how these numbers are part of (new) cohomological field theories (CohFTs).

Yujie Xu: Hecke algebras for p-adic groups and applications to the Langlands correspondence
Abstract: I will talk about several results on Hecke algebras attached to Bernstein blocks of arbitrary reductive p-adic groups, and their applications to the local Langlands program. One such application is an explicit understanding of the (classical, arithmetic) Local Langlands correspondence with explicit L-packets. If time permits, I will talk about some categorical “upgrades” involving certain coherent Springer sheaves.

2025 Spring

Date	Speaker	Affiliation	Title
April 11	Shrawan Kumar	UNC Chapel Hill	Exposition of Some Works of Kashiwara Related to Representation Theory
March 28	Daping Weng	UNC Chapel Hill	Weighted Cycles on Weaves
March 7	George Lusztig	MIT	An indexing of Weyl group representations

Shrawan Kumar: Exposition of Some Works of Kashiwara Related to Representation Theory
Abstract: I will expose some works of Masaki Kashiwara (the new Abel prize winner) related to representation theory. This will include his work on the proof of Kazhdan-Lusztig conjecture, which is obtained through a study of D-modules on the flag varieties, localization theorem, Riemann-Hilbert correspondence and Hecke algebras. In addition, I will briefly recall his work on crystal basis.

Daping Weng: Weighted Cycles on Weaves
Abstract: Weaves were first introduced by Casals and Zaslow as a graphical tool to describe a family of Legendrian surfaces living inside the 1-jet space of a base surface. Casals, Gorsky, Gorsky, Le, Shen, and Simental later generalized weaves to all Dynkin types such that the original weaves for Legendrian surfaces belong to Dynkin type A, and they use weaves of general Dynkin types to describe the cluster structure on braid varieties. In my previous joint work with Casals, we gave a topological interpretation of the cluster structures associated with weaves of Dynkin type A by associating the quiver with intersections of certain 1-cycles on surfaces and associating cluster variables with merodromies (parallel transports) along dual relative 1-cycles. In this talk, I will generalize this topological interpretation to all general Dynkin types by introducing a new diagrammatic object called “weighted cycles” and constructing an intersection pairing between them. I will define the merodromy along a weighted cycle and explain how to describe cluster variables using merodromies. If time allows, I will also mention a connection to quantum groups and skein algebras.

George Lusztig: An indexing of Weyl group representations
Abstract: Let W be a Weyl group and let Irr(W) be the set of complex irreducible representations of W (up to isomorphism) . It is known that Irr(W) can be partitioned into families (in bijection with the two-sided cells of W) . It turns out that the representations in a given family can be indexed by certain pairs of subgroups of a finite group attached to the family. A related construction gives a new basis of the Grothendieck group of representations of W related to the standard one by a triangular matrix.

Geometric Methods in Representation Theory Seminar