Geometric Methods in Representation Theory Seminar
University of North Carolina at Chapel Hill
Mondays/Fridays 4pm, PH367 or PH385
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.
2024 Fall
Date  Speaker  Affiliation  Title 

Nov 1  Syu Kato  Kyoto  Geometry of Dyck paths 
Oct 25  Mikhail Kapranov  Kavli IPMU  Operadic analysis of parabolic induction 
Oct 4  Phil Tosteson  UNC  Homology of spaces of curves on blowups 
Sept 27  Nicola Tarasca  Virginia Commonwealth  Higher Rank Series and Root Puzzles for Plumbed 3Manifolds 
Sept 13  Jakub Koncki  IMPAN Warsaw & UNCCH  Multiplicative structure of the Ktheoretic McKay correspondence for Hilbert schemes of points 
Aug 30  Tom Gannon  UCLA  Quantization of the Ngô morphism 
Syu Kato: Geometry of Dyck paths Abstract: There are two major research trends in the theory of symmetric functions arising from Dyck paths. One is the theory of Catalan symmetric functions and its geometric realization conceived by ChenHaiman, following the works of Broer and ShimozonoWeyman. The symmetric function part of this story was established by a series of works of BlasiakMorsePunSummers. Another is the study of chromatic symmetric functions of graphs, that utilize the fact that Dyck paths roughly correspond to unit interval graphs. We first exhibit a family of smooth algebraic varieties that realize Catalan symmetric functions, and explain how it imply the geometric predictions of Broer, ShimozonoWeyman, ChenHaiman, and BlasiakMorsePun. We then exhibit that our variety also naturally realize the chromatic symmetric functions of unit interval graphs. Thus, in a sense, our varieties might be understood as a geometric realization of the world of Dyck paths (which are related but not the same as the Hessenberg varieties). This talk is based on arXiv:2301.00862 and arXiv:2410.12231. 
Mikhail Kapranov: Operadic analysis of parabolic induction Abstract: The construction of various types of Hall algebas can be seen as an instance of the parabolic induction formalism. More precisely, this formalism is used to arrange various linear data associated to groups GL_n for all n (e.g., cohomology of the moduli spaces of vector bundles) into an associative algebra. But parabolic induction makes sense for arbitrary reductive groups. The talk, based on joint work in progress with V. Schechtman, O. Schiffman and J. Yuan, proposes an operadstyle approach to formalize the algebralike structures provided by all reductive groups taken together. The corresponding “double” structures (reducing to graded bi/Hopf algebras for the GL_n series) are important for the problem of classifying perverse sheaves on the adjoint quotients h/W. 
Phil Tosteson: Homology of spaces of curves on blowups Abstract: Let C be a smooth projective curve and X be a smooth projective variety. We will consider the space degree d algebraic (or holomorphic) maps from C to X. When X is a projective space, Segal discovered an interesting phenomenon: as the degree increases, the homology of the space of algebraic maps approximates that of the space of continuous maps. Recently, EllenbergVenkatesh observed that this phenomenon is related to Manin’s conjectures about rational points on Fano varieties, suggesting it holds more generally. I will talk about joint work with Ronno Das considering the case where X is a blowup of a projective space at finitely many points (in particular the case of del Pezzo surfaces). 
Nicola Tarasca: Higher Rank Series and Root Puzzles for Plumbed 3Manifolds Abstract: The WittenReshetikhinTuraev (WRT) invariants provide a powerful framework for constructing a family of invariants for framed links and 3manifolds. An ongoing pursuit in quantum topology revolves around the categorification of these invariants. Recent progress has been made in this direction, particularly through a physical definition of a new series invariant for negative definite plumbed 3manifolds. These invariants exhibit a convergence towards the WRT invariants in their limits. In this talk, I will present a refinement of such series invariants and show how one can obtain infinitely many new series invariants starting from the data of a root lattice of rank at least 2 and a solution to a combinatorial puzzle defined on that lattice. This is joint work with Allison Moore. 
Jakub Koncki: Multiplicative structure of the Ktheoretic McKay correspondence for Hilbert schemes of points Abstract: The Hilbert scheme of points in the complex plane is a classical object of study in algebraic geometry. McKay correspondence provides an isomorphism between its Ktheory (or cohomology) and the space of symmetric functions, creating a bridge between geometry and combinatorics. Multiplication by a class in the Ktheory induces an endomorphism of the space of symmetric functions. In the cohomological case, compact formulas for such maps were found by Lehn and Sorger. The Ktheoretical case was studied by Boissiere using torus equivariant techniques. He proved a formula for multiplication by the class of the tautological bundle and stated a conjecture for the remaining generators of Ktheory. In the talk, I will show how torus action simplifies this problem and prove the conjectured formula using restriction to a onedimensional subtorus. This is a joint work with M. Zielenkiewicz. 
Tom Gannon: Quantization of the Ngô morphism We will discuss work, joint with Victor Ginzburg, which proves a conjecture of Nadler on the existence of a quantization (noncommutative deformation) of the Ngô morphism, a morphism of group schemes constructed by Ngô in his proof of the fundamental lemma in the Langlands program. We will first explain the construction of the Ngô morphism and discuss an extended example of this map for the group of invertible n x n complex matrices. Then, we will give a precise statement of our main theorem and discuss some of the tools used in proving this theorem, including a quantization of MooreTachikawa varieties. Time permitting, we will also discuss how the tools used to construct this morphism can be used to prove conjectures of BenZvi—Gunningham, which predict a “spectral decomposition” of DG categories with an action of a reductive group. 
2024 Spring
Date  Speaker  Affiliation  Title 

April 12  Syu Kato  Kyoto  Semiinfinite flag manifolds, space of maps, and quantum Kgroups of flag manifolds 
April 5  Syu Kato  Kyoto  Algebraic models of semiinfinite flag manifolds 
Mar 22  Thomas Haines  Maryland  Cellular pavings of convolution fibers and applications 
Feb 23  Jayce Getz  Duke  On the Poisson summation conjecture 
Feb 9  Luke Conners  UNC  RowColumn Mirror Symmetry for Colored Torus Knot Homology 
Syu Kato Talk 1: Algebraic models of semiinfinite flag manifolds Abstract: We first recall the classical BorelWeilBott theorem and examine its representationtheoretic meaning. Then, we exhibit affine analogue of their representationtheoretic meaning and form several schemes. Finally, we identify the resulting schemes with socalled semiinfinite flag manifolds settheoretically known from 1980s and its Schubert and Richardson varieties. This talk is mainly based on arXiv:1810.07106. 
Syu Kato Talk 2: Semiinfinite flag manifolds, space of maps, and quantum Kgroups of flag manifolds Abstract: We first recall the Plucker relations, that yield a description of a(n open) space of maps from a projective line to a (partial) flag manifolds. Then, we observe a close relationship between Richardson varieties of semiinfinite flag manifolds and the space of stable maps from a projective line to a flag manifold. This makes us possible to describe quantum Kgroups of flag manifolds with the Kgroup of semiinfinite flag manifolds. This talk is mainly based on arXiv:1805.01718. If time permits, we also explain a natural isomorphism between the Kgroups of semiinfinite flag manifolds and that of affine Grassmannians conjectured by LamLiShimozonoMihalcea in order to complete the picture. 
Tom Haines: Cellular pavings of convolution fibers and applications Abstract: A convolution morphism is the geometric analogue of a convolution of functions in a Hecke algebra. The properties of fibers of convolution morphisms are used in a variety of ways in the geometric Langlands program and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibers of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, rationality of the BBD Decomposition Theorem over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence. If time permits, I will describe a new combinatorial model for generalized MirkovicVilonen intersections and the branching to Levi subgroups. 
Jayce Getz: On the Poisson summation conjecture Abstract: Braverman, Kazhdan, Lafforgue, Ngo, and Sakellaridis have conjectured that Fourier analysis on a vector space is but the first example of a larger phenomenon. More generally, one should have Schwartz spaces, Fourier transforms, and, crucially, Poisson summation formulae for affine spherical varieties. I will give an account of the few cases in which the conjecture is known, and describe some techniques for proving new cases from old cases. 
Luke Conners: RowColumn Mirror Symmetry for Colored Torus Knot Homology Abstract: The HOMFLYPT polynomial is a 2variable link invariant generalizing the celebrated Jones polynomial and other Type A quantum link polynomials. Its construction passes through a Hecke algebra representation of the braid group, and by making use of certain idempotent elements in the Hecke algebra, one can extend the invariant to links with components labeled by arbitrary Young diagrams. The resulting invariant, called the colored HOMFLYPT polynomial, has a wellknown “mirror symmetry” property describing its behavior under exchanging each such Young diagram with its transpose. One categorical level up, Khovanov and Rozansky constructed a triplygraded homological link invariant that recovers the HOMFLYPT polynomial upon taking Euler characteristic. Various authors have constructed colored versions of triplygraded KhovanovRozansky homology, and these invariants are conjectured to satisfy a categorical lift of the polynomial mirror symmetry described above. In this talk, we will formulate this conjecture precisely and outline a recent proof in the special case of a positive torus knot colored by a single row or column of arbitrary length. 
2023 Fall
Date  Speaker  Affiliation  Title 

Nov 20  Jianqiao Xia  Harvard  Equivalence of Hecke Categories with Deeper Level Structures 
Nov 1719  –  –  Workshop on geometric representation theory and moduli spaces 
Nov 3  Thomas Lam  Michigan  Monotone links in the DAHA and EHA 
Oct 25  Szilárd Szabó  Budapest University of Technology and Economics  Hitchin WKB problem and Geometric P=W conjecture in rank 2 
Sep 8  Mikhail Kapranov  IPMU  Nspherical functors and categorification of Euler’s continuants 
Aug 25  Joseph Landsberg  Texas A&M  Linear spaces of matrices of bounded rank 
Jianqiao Xia: Equivalence of Hecke Categories with Deeper Level Structures Inspired by the theory of positive depth representations of padic reductive groups, we study Hecke categories associated to certain open compact subgroups smaller than the Iwahori subgroup. In this talk, I will prove that in some cases these Hecke categories are monoidally equivalent to depth 0 Hecke categories of smaller groups. On the function level, our result recovers a family of Hecke algebra isomorphisms already proven by JuLee Kim. The categorical equivalences should be an important ingredient of the local Geometric Langlands conjecture. 
Szilard Szabo: Hitchin WKB problem and Geometric P=W conjecture in rank 2 After reviewing nonabelian Hodge and Riemann–Hilbert correspondences over curves, I state two closely related open problems concerning their largescale behavior. I then propose an answer to these questions in some particular cases. The talk will mainly use tools from complex geometry and analysis. 
Thomas Lam: Monotone links in the DAHA and EHA Morton and Samuelson related certain skein algebras on the torus with the double affine Hecke algebra (DAHA) and the elliptic Hall algebra (EHA). We use this construction to study link homology of a class of “monotone links” on the torus, closely related to the Coxeter links of Oblomkov and Rozansky and to the positroid links in our earlier work. This is joint work with Pavel Galashin. 
Mikhail Kapranov: Nspherical functors and categorification of Euler’s continuants Abstract: Euler’s continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction in terms of its entries. Remarkably, they make an appearance in the very foundations of category theory: in the formalism of adjoint functors. More precisely, they upgrade to natural complexes of functors built out of a given functor and its iterated adjoints. Requiring exactness of some of these complexes leads to the concept of an Nspherical functor which specializes to that of an ordinary spherical functor for N=4. Such functors describe Nperiodic semiorthogonal decompositions of (enhanced) triangulated categories. Like ordinary spherical functors, they give rise to interesting selfequivalences. Conceptually, they can be seen as categorification of certain irregular differential equations (polynomial Schroedinger) in the complex plane. Joint work with T. Dyckerhoff, V. Schechtman. 
Joseph Landsberg: Linear spaces of matrices of bounded rank Abstract: A classical problem in linear algebra is to classify linear spaces of matrices such that no element of the space has full rank. Work of Eisenbud and Harris showed that the problem may be rephrased in terms of classifying sheaves on projective space with certain properties. 40 years ago spaces of bounded rank at most three were classified and there have been interesting, isolated examples of spaces discovered that are related to wellstudied objects in algebraic geometry such as instanton bundles, but there had been no progress on the classification problem. Motivated by questions in theoretical computer science and quantum information theory, H. Huang and myself revisited this problem. Using methods from algebraic geometry and commutative algebra, we classified spaces of bounded rank four. 
2023 Spring
Date  Speaker  Affiliation  Mode  Title 

Apr 28  George Lusztig  MIT  Inperson  Strata in reductive groups 
Apr 14  Ádám Gyenge  Rényi Institute Budapest  Inperson  Blowups and the quantum spectrum of surfaces 
Apr 11  Mikhail Kapranov  IPMU  Inperson  Perverse sheaves and Hopf algebras 
March 31  Anders Buch  Rutgers  Inperson  Pieri formulas for the quantum Ktheory of cominuscule Grassmannians 
March 24  Simon Riche  Clermont Auvergne  Inperson  Characters of modular representations of reductive algebraic groups 
March 10  –  –  –  Hitoshi Konno’s lecture in the sister (Physically Inspired Mathematics) seminar 
March 10  Lingfei Yi  Minnesota  Inperson  Physical rigidity of FrenkelGross connection 
Jan 27  –  –  –  Pavel Etingof’s lecture in the sister (Physically Inspired Mathematics) seminar 
Jan 20  Tommaso Botta  ETH Zurich  Inperson  Solution of qKZB equations from the geometry of Nakajima quiver varieties 
Jan 17  Dima Arinkin  University of Wisconsin  Inperson  Integrating symplectic stacks 
George Lusztig: Strata in reductive groups Abstract: : Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata each of which is a union of conjugacy classes of fixed dimension. The strata are indexed by a set independent of the characteristc. The strata can be described purely in terms of the Weyl group. 
Ádám Gyenge: Blowups and the quantum spectrum of surfaces Abstract: : The cup product of ordinary cohomology describes how submanifolds of a manifold intersect each other. GromovWitten invariants give rise to quantum product and quantum cohomology, which describe how subspaces intersect in a ”fuzzy”, ”quantum” way. Dubrovin observed that quantum cohomology can be used to define a flat connection on a certain vector bundle called the quantum connection. We verify a conjecture of Kontsevich on the behaviour of the spectrum of the quantum connection under blowups for smooth projective surfaces. Joint work with Szilard Szabo. 
Mikhail Kapranov: Perverse sheaves and Hopf algebras Abstract: Perverse sheaves were originally introduced as a conceptual framework for intersection homology, a (co)homology theory for singular spaces that satisfies Poincare duality. As such, they occupy an intermediate position between sheaves (coefficients for cohomology) and cosheaves (coefficients for homology), forming a selfdual category. On the other hand, Hopf algebras, or bialgebras provide an example of a selfdual structure in a purely algebraic context, being equipped both with a multiplication and a comultiplication. The talk, based on joint work with V. Schechtman, will explain a connection between these two type of structures so that universal identities among various composite (co)operations turn out to give the relations in the quivers describing perverse sheaves on configuration spaces of the complex line. In particular, this gives a relation between graded bialgebras and factorizing systems of perverse sheaves, whose instances are known in the theory of quantum groups. 
Anders Buch: Pieri formulas for the quantum Ktheory of cominuscule Grassmannians Abstract: The quantum Ktheory ring QK(X) of a flag variety X encodes the Ktheoretic GromovWitten invariants of X, defined as arithmetic genera of GromovWitten varieties parametrizing curves meeting fixed Schubert varieties. A Pieri formula means a formula for multiplication with a set of generators of QK(X). Such a formula makes it possible to compute efficiently in this ring. I will speak about a Pieri formula for QK(X) when X is a cominuscule Grassmannian, that is, an ordinary Grassmannian, a maximal orthogonal Grassmannian, or a Lagrangian Grassmannian. This formula has a simple statement in terms of order ideals in a partially ordered set that encodes the degree distance between opposite Schubert varieties. This set generalizes both Postnikov’s cylinder and Proctor’s description of the Bruhat order of X. This is joint work with P.E. Chaput, L. Mihalcea, and N. Perrin.

Simon Riche: Characters of modular representations of reductive algebraic groups Abstract: One of the main questions in the representation theory of reductive algebraic groups is the computation of characters of simple modules. A conjectural solution to this problem was proposed by G. Lusztig in 1980, and later shown to be correct assuming the base field has large characteristic. However in 2013 G. Williamson found (counter)examples showing that this answer is not correct without this assumption. In this talk I will explain a new solution to this problem, obtained in a combination of works involving (among others) P. Achar and G. Williamson, which is less explicit but has the advantage of being valid in all characteristics. 
Lingfei Yi: TBA Abstract: A Gconnection over a smooth complex curve is called physically rigid if it is determined by its local monodromies. We show that the FrenkelGross connection is physically rigid, thus confirming the de Rham version of a conjecture of HeinlothNg^oYun. The proof is based on the construction of the Hecke eigensheaf of a connection with only generic oper structure, using the localization of Weyl modules. We will review the notion of opers and give the sketch of the proof. Time permitting, we will describe a conjectural generalization of the result relating theta connections to Langlands parameters of Epipelagic representations. 
Tommaso Botta: Solution of qKZB equations from the geometry of Nakajima quiver varieties Abstract: The quantum Knizhnik–Zamolodchikov (qKZ) equations are an important family of difference equations, deeply related to the representation theory of affine quantum enveloping algebras (trigonometric quantum groups). Over the past years, Okounkov, Smirnov and their coauthors have succeeded in studying the qKZ equations via the geometry of Nakajima varieties and producing integral solutions through enumerative counts in Ktheory. The goal of this talk is to extend some of the above ideas to the elliptic setting. Firstly, I will exploit AganagicOkounkov’s theory of elliptic stable envelopes of Nakajima varieties to define a system of elliptic difference equations— the KnizhnikZamolodchikovBernard (qKZB) equations — for arbitrary quiver varieties. Then I will discuss how to produce integral presentations of their solutions. In this context, a Cohomological Hall algebra (CoHA) interpretation of the stable envelopes will replace Okounkov’s technology of enumerative counts. This talk is based on joint work in preparation with Felder and Wang.

Dima Arinkin: Integrating symplectic stacks Abstract: Shifted symplectic stacks, introduced by Pantev, Toën, Vaquie, and Vezzosi, are a natural generalization of symplectic manifolds in derived algebraic geometry. The word `shifted’ here refers to cohomological shift, which can naturally occur in the derived setting: after all, the tangent space is now not a vector space, but a complex. Several classes of interesting moduli stacks carry shifted simplectic structures. In my talk (based on a joint project with T.Pantev and B.Toën), I will present a way to generate shifted symplectic stacks. Informally, it involves integration along a (compact oriented) topological manifold X: starting with a family of shifted symplectic stacks over X, we produce a new stack of sections of this family, and equip it with a symplectic structure via an appropriate version of the Poincaré duality. 