Geometric Methods in Representation Theory Seminar
University of North Carolina at Chapel Hill
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 Shrawan Kumar.
Zoom Meeting ID: 930 8567 8674, passcode: 2^10
2023 Spring
Date  Speaker  Affiliation  Mode  Title 

Apr 28  George Lusztig  MIT  Inperson  TBA 
Apr 14  Ádám Gyenge  Rényi Institute Budapest  Inperson  TBA 
March 24  Simon Riche  Clermont Auvergne  Inperson  TBA 
March 10  Lingfei Yi  Minnesota  Inperson  TBA 
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 
Ádám Gyenge: TBA Abstract: TBA 
Simon Riche: TBA Abstract: TBA 
Lingfei Yi: TBA Abstract: TBA 
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. 
2022 Fall
Date  Speaker  Affiliation  Mode  Title 

Dec 2  Mark Ebert  USC  Inperson  Derived Superequivalences for Spin Symmetric Groups and Odd sl2categorifications 
Nov 18  Daping Weng  UC Davis  Inperson  Cluster Structures on Double BottSamelson Cells 
Nov 11  Matt Hogancamp  Northeastern  Inperson  The dotted TemperleyLieb category and handleslides 
Nov 4  Peter Koroteev  Berkeley  Inperson  Opers and Integrable Systems 
Nov 3, Thursday 45 (“Special Geometry Seminar”)  Inkang Kim  KIAS/Stanford  Inperson  Signature, Toledo invariant, and the surface group representations in Hermitian semisimple Lie groups 
Oct 28  Junliang Shen  Yale  Inperson  The P=W conjecture for GL_n 
October 7  Xuqiang Qin  UNC Chapel Hill  Inperson  Birational geometry of BeauvilleMukai systems on K3 surfaces 
Sept 16  Till Wehrhan  University of Bonn and MPIM  Inperson  The rim hook rule in equivariant quantum Schubert calculus via Bethe and Clifford algebras 
Mark Ebert: Derived Superequivalences for Spin Symmetric Groups and Odd sl2categorifications Abstract: Since Chuang and Rouquier’s pioneering work showing that categorical sl(2)actions give rise to derived equivalences, the construction of derived equivalences has been one of the more prominent tools coming from higher representation theory. In this talk, we explain joint work with Aaron Lauda and Laurent Vera giving new super analogues of these derived equivalences stemming from the odd categorification of sl(2). Just as Chuang and Rouquier used their equivalences to prove Broué’s abelian defect conjecture for symmetric groups, we use our superequivalences to prove this long standing conjecture for spin symmetric groups. 
Daping Weng: Cluster Structures on Double BottSamelson Cells Abstract: By the Bruhat decomposition theorem, relative positions between flags in G/B are encoded by the corresponding Weyl group W. Given a pair of positive braids (b,d) associated with the Weyl group W, we define the double BottSamelson cell Conf^b_d(G) to be the moduli space of flags satisfying relative position conditions imposed by positive braid words of b and d. We construct cluster structures on double BottSamelson cells Conf^b_d(G) and describe their DonaldsonThomas transformation as a cyclic rotation on the circle of flags. As a consequence, this gives a new geometric proof of the Zamolodchikov periodicity conjecture. We also construct Deodhar stratifications on double BottSamelson cells and develop a formula for their F_q point counts. Moreover, in the case where G is of type A, the F_q point count of Conf^b_d(G) give a link invariant for rainbow closures of positive braids. This is based on joint work with L. Shen (1904.07992). 
Matt Hogancamp: The dotted TemperleyLieb category and handleslides Abstract: Khovanov homology can be upgraded to an invariant of pairs (K,V) where K is a framed knot and V is an object of the dotted TemperleyLieb category dTL. In this context, the pair (K,V) is called a colored knot, and its Khovanov invariant is called colored Khovanov homology. In my talk I will discuss recent joint work with David Rose and Paul Wedrich, in which we construct an object in dTL (more accurately, an indobject therein), called a Kirby color, whose associated colored Khovanov invariant satisfies the important handleslide relation from topology. I will also give a diagrammatic description of the Kirby color, extending the presentation which defines dTL. 
Peter Koroteev: Opers and Integrable Systems Abstract: I will explain how the geometric construction of opers (as well as its difference and elliptic generalizations) is related to quantum and classical integrable systems. Opers thereby provide a framework to study dualities between various types of integrable models of CalogeroRuijsenaars type and quantum spin chains (XXX, XXY, and XYZ). I will also mention connections between opers and cluster algebras. 
Inkang Kim: Signature, Toledo invariant, and the surface group representations in Hermitian semisimple Lie groups Abstract: People study higher Teichmuller theory and using several invariants, one tries to characterize the representations. We give a unifying formula between (AtiyahPatodiSinger) signature, Toledo invariant, and rho invariant that we invented for the surface group (with boundary) representations into Hermitian semisimple Lie groups. This is a joint work with P. Pansu and X. Wan. 
Junliang Shen: The P=W conjecture for GL_n Abstract: In 2010, de CataldoHauselMigliorini proposed a conjecture connecting topology of the Hitchin system and Hodge theory of the corresponding character variety via the nonabelian Hodge theory. This conjecture is now referred to as the P=W conjecture. The purpose of this talk is to explain a recent proof of this conjecture (for GL_n) in joint work with Davesh Maulik for any rank and genus, where we combine tools from algebraic geometry and representation theory. 
Xuqiang Qin: Birational geometry of BeauvilleMukai systems on K3 surfaces Abstract: A BeauvilleMukai system on a K3 surface is a moduli space of stable torsion sheaves, which admits a Lagrangian fibration given by mapping each sheaf to its support. In this talk, we will focus on a class of BeauvilleMukai systems which are birational to Hilbert schemes of points on the surface. Using wallcrossing techniques from Bridgeland stability, we decompose the birational map into a sequence of flops, whose exceptional loci are BrillNoether type subsets. As a result, we give full description of the birational geometry of such systems. This is based on joint work with Justin Sawon. 
Till Wehrhan: The rim hook rule in equivariant quantum Schubert calculus via Bethe and Clifford algebras Abstract: It was proven by Bertiger, Milicevic and Taipale that the the strucutre coefficients in the equivariant quantum cohomology of a Grassmannian can be determined by computing the usual cup product in a certain larger Grassmannian and then applying a combinatorial rim hook algorithm. In this talk, we discuss a generalization of this result using the realization of the equivariant quantum cohomology of Grassmannians as Bethe algebras of a specific integrable model established by Gorbounov, Korff and Stroppel. If time permits, we also discuss further possible generalizations to equivairant quantum Ktheory. 
2022 spring
Date  Speaker  Affiliation  Mode  Title 

April 29  Elijah Bodish  Oregon  Inperson  Semisimpliflying type C tilting modules in mediumlow quantum characteristic 
April 29 2:45pm  David Rose  UNC  Inperson  Type C Webs 
April 22  Dinakar Muthiah  Glasgow  Zoom  Fundamental monopole operators and affine Grassmannian slices 
April 1  Tommaso Botta  ETH Zurich  Inperson  Shuffle products for stable envelopes of Nakajima varieties 
March 25  Ivan Loseu  Yale  Inperson  HarishChandra modules over quantizations of nilpotent orbits 
March 4  Joshua Kiers  Ohio State  Inperson  Demazure polytopes and saturation 
Feb 18  Shiliang Gao  UIUC  Inperson  NewellLittlewood numbers 
Feb 11  Cristian Lenart  SUNY Albany  Zoom  A combinatorial Chevalley formula for semiinfinite flag manifolds and related topics 
Jan 21  Prakash Belkale  UNC  Hybrid  Rigid local systems and the multiplicative eigenvalue problem 
Elijah Bodish Abstract: Given a semisimple Lie algebra and a positive integer L, one obtains the monoidal category of tilting modules for the associated quantum group at a primitive Lth root of unity. The category of tilting modules has a unique semisimple monoidal quotient. The images of indecomposable tilting modules with nonzero dimension exhaust the isomorphism classes of simple objects in the quotient category. If L is larger than the Coxeter number of the Lie algebra, then the indecomposable tilting modules with nonzero dimension are exactly those with highest weight in the fundamental alcove (with respect to the L dilated affine Weyl group). When L is less than the Coxeter number much less is known. The state of the art is that in early 2020 BrundanEntovaEtingofOstrik determined which tilting modules have nonzero dimension for gl_n when L < n+1 (note that the Coxeter number for gl_n is n). The Coxeter number for sp_{2n} is 2n. We will propose a method to determine which indecomposable tilting modules for sp_{2n} have nonzero dimension, when n< L< 2n+1. The answer appears to be related to KazhdanLusztig cells in the antispherical Hecke module, which we hope will indicate what to expect in the much more complicated case when L< n+1. 
David Rose Abstract: In his seminal 1996 paper, Kuperberg gives presentations for the categories of finitedimensional representations of quantum groups associated to rank 2 simple complex Lie algebras (as braided pivotal categories). Such presentations underly various invariants in lowdimensional topology; in particular, they serve as a “foundation” for link homology theories. Kuperberg also poses the following problem: to find analogous descriptions of these categories for quantum groups associated with higher rank Lie algebras. In 2012, CautisKamnitzerMorrison solved this problem in type A using skew Howe duality, a technique that does not immediately extend to give a solution in other types. In this talk, we will present a solution to Kuperberg’s problem in type C. Our proof combines results on pivotal categories and quantum group representations with diagrammatic/topological analogues of theorems concerning reduced expressions in the symmetric group. Time permitting, we’ll discuss some future directions. This work is joint with Elijah Bodish, Ben Elias, and Logan Tatham. 
Dinakar Muthiah Abstract: Affine Grassmannians are objects of central interest in geometric representation theory. For example, the geometric Satake correspondence tells us that their singularities carry representation theoretic information. In fact, it suffices to work with affine Grassmannian slices, which retain all of this information. Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in KacMoody type as well. Their work opens the door to studying affine Grassmannians for KacMoody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators. This is joint work in progress with Alex Weekes. 
Tommaso Botta Abstract: The notion of stable envelopes of a symplectic resolution, developed by Okounkov and his coauthors in the last decade, lies at the heart of the geometric approach to the representation theory of quantum groups and qdifference equations. Nakajima quiver varieties form a rich family of symplectic resolutions, whose geometry governs the representation theory of KacMoody Lie algebras and, via stable envelopes, their qdeformations. In this talk, I will introduce an inductive formula that produces the stable envelopes of an arbitrary Nakajima variety, taking as input the stable envelopes of two other Nakajima varieties with smaller dimension and framing vectors. Some explicit examples will be also discussed. This formula is a wide generalisation earlier results inherited form the theory of weight functions. Time permitting, I will also discuss some connections with cohomological Hall algebras (CoHa) and Cherkis bow varieties, which are object of ongoing research. 
Ivan Loseu Abstract: Let O be a nilpotent orbit in a semisimple Lie algebra over the complex numbers. Then it makes sense to talk about filtered quantizations of O, these are certain associative algebras that necessarily come with a preferred homomorphism from the universal enveloping algebra. Assume that the codimension of the boundary of O is at least 4, this is the case for all birationally rigid orbits (but six in the exceptional type), for example. In my talk I will explain a geometric classification of faithful irreducible HarishChandra modules over quantizations of O, concentrating on the case of canonical quantizations — this gives rise to modules that could be called unipotent. The talk is based on a joint paper with Shilin Yu (in preparation). 
Josh Kiers Abstract: Demazure modules are certain Bsubmodules of irreducible Gmodules, where B is a Borel subgroup of a semisimple Lie group G. They arise as cohomology groups of line bundles on Schubert varieties, and their characters are described by the Demazure character formula. We will consider the weight polytopes associated to these characters, presenting basic formulas and results on them. We will show how the polytopes shed light on the supports of the characters, at least in classical types. This gives an alternate proof of a conjecture of Monical, Tokcan, and Yong, first proven by Fink, Meszaros, and St. Dizier. 
Shiliang Gao Abstract: The NewellLittlewood numbers are defined in terms of the LittlewoodRichardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LRcone and established defining linear inequalities. We prove analogues for the saturated NLcone. This is based on work with Gidon Orelowitz, Nicolas Ressayre and Alexander Yong; see arxiv.org/abs/2005.09012, arxiv.org/abs/2009.09904, and arxiv.org/abs/2107.03152. 
Cristian Lenart Abstract: I present a combinatorial Chevalley formula for an arbitrary weight in the equivariant Ktheory of semiinfinite flag manifolds, which are certain affine versions of finite flag manifolds G/B. The formula is expressed in terms of the socalled quantum alcove model. One application is a Chevalley formula in the equivariant quantum Ktheory of G/B. Another application is that the socalled quantum Grothendieck polynomials represent Schubert classes in the (nonequivariant) quantum Ktheory of the type A flag manifold. Both applications solve longstanding conjectures. Other results include the Chevalley formula for partial flag manifolds G/P and related combinatorics of the quantum alcove model. This is joint work with Takafumi Kouno, Satoshi Naito, and Daisuke Sagaki. The talk will be largely selfcontained. 
Prakash Belkale Abstract: Local systems are sheaves which describe the behavior of solutions of differential equations. A local system is rigid if local monodromy determines global monodromy. We give a construction which produces irreducible complex rigid local systems on a punctured Riemann sphere via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices (which can be inductively determined) of these polytopes give (all) possible unitary irreducible rigid local systems. We note that these polytopes are generalizations of the classical LittlewoodRichardson cones of algebraic combinatorics. Answering a question of Nicholas Katz, we show that there are no irreducible rigid local systems on a punctured Riemann sphere of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing n, when n is a prime number. 