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.

Fall 2026

Date	Speaker	Affiliation	Title
Aug 28	Nikolay Grantcharov	UGA	Symmetries of cotangent bundles of generalized base affine spaces
Aug 21	Aaron Slipper	Duke	Geometrization of the Minimal Representation of the Even Orthogonal Group

Nikolay Grantcharov: Symmetries of cotangent bundles of generalized base affine spaces
Abstract: Given an affine algebraic variety X over C, a natural question to ask is what symmetries its cotangent bundle admits. One fundamental source of such symmetries is the Fourier transform. We will explain how these Fourier transforms can be adapted to produce symmetries of the affinization of T^*X in the case of the base affine space X=SL_n/U and more generally the case of X=SL_n/[P,P] for P a standard parabolic subgroup. As a byproduct, we construct many examples of affine algebraic varieties that are not isomorphic, but whose cotangent bundles are isomorphic. We will also discuss an application of the Fourier transforms in producing a generating set of the algebra of functions on T^*X.

Aaron Slipper: Geometrization of the Minimal Representation of the Even Orthogonal Group
Abstract: On a smooth affine variety in characteristic zero, Grothendieck differential operators are generated by functions and derivations. On singular varieties, their algebra can be pathological: for the cubic cone (x^3+y^3+z^3=0), it is neither Noetherian nor finitely generated. We study an intermediate case, the quadric cone (C\subset V\cong\mathbb A^{2k}). Although singular, (C) has a finitely generated algebra (D(C)) of differential operators carrying a natural action of (G=O(2k+2)), even though (G) does not act on (C) itself.
This “extra orthogonal symmetry” also appears in (\mathcal O(T^*C^{\mathrm{sm}})) and in (L^2(C)), the Schrödinger model of the minimal representation of (G). In each setting, a Weyl lift (w_0) acts by a mysterious “quadric Fourier transform.”
We will explain how (D(C)\text{-mod}) categorifies the minimal representation and identify it with two geometric models: a Kazhdan–Laumon-type category glued using the quadric Fourier transform, and a category of harmonic twisted (D)-modules on the conformal compactification of (V). Twisted (D)-modules and the algebraic theory of solutions to the Laplace equation explain the hidden (G)-action and yield a new proof that (D(C)) is finitely generated. If time permits, we will discuss work in progress on the (\ell)-adic counterpart of this story.

Spring 2026

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.

Geometric Methods in Representation Theory Seminar