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Department of Mathematics

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 Richard Rimanyi.

2024 Spring

Date Speaker Affiliation Title
April 12 Syu Kato Kyoto Semi-infinite flag manifolds, space of maps, and quantum K-groups of flag manifolds
April 5 Syu Kato Kyoto Algebraic models of semi-infinite 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 Row-Column Mirror Symmetry for Colored Torus Knot Homology

Syu Kato Talk 1: Algebraic models of semi-infinite flag manifolds
Abstract:We first recall the classical Borel-Weil-Bott theorem and examine its representation-theoretic meaning. Then, we exhibit affine analogue of their representation-theoretic meaning and form several schemes. Finally, we identify the resulting schemes with so-called semi-infinite flag manifolds set-theoretically known from 1980s and its Schubert and Richardson varieties. This talk is mainly based on arXiv:1810.07106.

Syu Kato Talk 2: Semi-infinite flag manifolds, space of maps, and quantum K-groups 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 semi-infinite flag manifolds and the space of stable maps from a projective line to a flag manifold. This makes us possible to describe quantum K-groups of flag manifolds with the K-group of semi-infinite flag manifolds. This talk is mainly based on arXiv:1805.01718. If time permits, we also explain a natural isomorphism between the K-groups of semi-infinite flag manifolds and that of affine Grassmannians conjectured by Lam-Li-Shimozono-Mihalcea 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 Mirkovic-Vilonen 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: Row-Column Mirror Symmetry for Colored Torus Knot Homology
Abstract: The HOMFLYPT polynomial is a 2-variable 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 well-known “mirror symmetry” property describing its behavior under exchanging each such Young diagram with its transpose.

One categorical level up, Khovanov and Rozansky constructed a triply-graded homological link invariant that recovers the HOMFLYPT polynomial upon taking Euler characteristic. Various authors have constructed colored versions of triply-graded Khovanov-Rozansky 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 17-19 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 N-spherical 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 p-adic 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 Ju-Lee 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 large-scale 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: N-spherical 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 N-spherical functor which specializes to that of an ordinary spherical functor for N=4. Such functors describe N-periodic semi-orthogonal decompositions of (enhanced) triangulated categories. Like ordinary spherical functors, they give rise to interesting self-equivalences. 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 well-studied 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 In-person Strata in reductive groups
Apr 14 Ádám Gyenge Rényi Institute Budapest In-person Blow-ups and the quantum spectrum of surfaces
Apr 11 Mikhail Kapranov IPMU In-person Perverse sheaves and Hopf algebras
March 31 Anders Buch Rutgers In-person Pieri formulas for the quantum K-theory of cominuscule Grassmannians
March 24 Simon Riche Clermont Auvergne In-person 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 In-person Physical rigidity of Frenkel-Gross connection
Jan 27 Pavel Etingof’s lecture in the sister (Physically Inspired Mathematics) seminar
Jan 20 Tommaso Botta ETH Zurich In-person Solution of qKZB equations from the geometry of Nakajima quiver varieties
Jan 17 Dima Arinkin University of Wisconsin In-person 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: Blow-ups and the quantum spectrum of surfaces
Abstract: : The cup product of ordinary cohomology describes how submanifolds of a manifold intersect each other. Gromov-Witten 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 blow-ups 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 self-dual category. On the other hand, Hopf algebras, or bialgebras provide an example of a self-dual 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 K-theory of cominuscule Grassmannians
Abstract: The quantum K-theory ring QK(X) of a flag variety X encodes the K-theoretic Gromov-Witten invariants of X, defined as arithmetic genera of Gromov-Witten 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 G-connection over a smooth complex curve is called physically rigid if it is determined by its local monodromies. We show that the Frenkel-Gross connection is physically rigid, thus confirming the de Rham version of a conjecture of Heinloth-Ng^o-Yun. 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 K-theory.
The goal of this talk is to extend some of the above ideas to the elliptic setting. Firstly, I will exploit Aganagic-Okounkov’s theory of elliptic stable envelopes of Nakajima varieties to define a system of elliptic difference equations— the Knizhnik-Zamolodchikov-Bernard (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.