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.
2021 Spring–Zoom Meeting ID: 975 6220 3148
Date  Speaker  Affiliation  Title 

March 26  Alex Weeks  University of British Columbia  TBA 
March 19  TsaoHsien Chen  University of Minnesota  TBA 
March 5  Anne Dranowski  IAS  TBA 
Feb 19  Charlotte Chan  MIT  Flag varieties and representations of padic groups 
Jan 29  Yau Wing Li  MIT  Endoscopy for affine Hecke categories 
Charlotte Chan Flag varieties and representations of padic groups Abstract: Deligne–Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to socalled “depthzero” supercuspidal representations of padic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint work with Masao Oi. 
Yau Wing Li Endoscopy for affine Hecke categories Abstract: Affine Hecke categories are categorifications of IwahoriHecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahorifixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H. It is consistent with the Langlands functoriality conjecture. 
2020 Spring
Date  Speaker  Affiliation  Title 

Feb 21  Dennis Tseng  Harvard  Equivariant degenerations of plane curve orbits 
Feb 14  Rekha Biswal  MPIM  Macdonald polynomials and level two Demazure modules for affine sl_{n+1} 
Rekha Biswal Macdonald polynomials and level two Demazure modules for affine sl_{n+1} Abstract: An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomialsindexed by a pair of dominant weights of sl_{n+1} which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules. Using representation theory, we will see that these new family of polynomials interpolate between characters of level one and level two Demazure modules for affine sl_{n+1} and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand. 
Dennis Tseng Equivariant degenerations of plane curve orbits Abstract: In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel. 
2019 Fall
Date  Speaker  Affiliation  Title 

November 22 4:155:15pm  Gurbir Dhillon  Stanford  The tamely ramified Fundamental Local Equivalence 
November 22 34pm  Vikraman Balaji  Chennai Mathematical Institute  Torsors on semistable curves and the problem of degenerations 
November 15  Alex Yong  University of Illinois at UrbanaChampaign  The A.B.C.D’s of Schubert calculus 
November 1  Cris Negron  UNC Chapel Hill  Modularization of quantum groups and some conformal field theory 
October 25  Andrey Smirnov  UNC Chapel Hill  Elliptic stable envelope for Hilbert scheme of points on C^2 
September 27  Richard Rimanyi  UNC Chapel Hill  Elliptic classes of Schubert varieties 
Gurbir Dhillon The tamely ramified Fundamental Local Equivalence Let G be an almost simple algebraic group with Langlands dual G’. Gaitsgory conjectured that affine Category O for G at a noncritical level should be equivalent to Whittaker Dmodules on the affine flag variety of G’ at the dual level. We will provide motivation and background for this conjecture, which is some form of geometric Satake for quantum groups. We have proven this conjecture when the level is appropriately integral with Justin Campbell, and the general case is work in progress with Sam Raskin. 
Vikraman Balaji Torsors on semistable curves and the problem of degenerations Let G be an almost simple, simply connected algebraic group G over the field of complex numbers. In this talk I answer two basic questions in the classification of Gtorsors on curves. The first one is to construct a at degeneration of the moduli stack Gtorsors on a smooth projective curve when the curve degenerates to an irreducible nodal curve. Torsors for a generalization of the classical BruhatTits group schemes to twodimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools. The second question is to give an intrinsic definition of (semi)stability for a Gtorsor on an irreducible nodal curve. The absence of obvious analogues of torsionfree sheaves in the setting of Gtorsors makes the question interesting. This also leads to the construction of a proper separated coarse space for Gtorsors on an irreducible nodal curve. 
Alex Yong The A.B.C.D’s of Schubert calculus We collect AtiyahBott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. BergeronF. Sottile, S. BilleyM. Haiman, P. Pragacz, and T. IkedaL. MihalceaI. Naruse. The relation complements a theorem of A. KreschH. Tamvakis in quantum cohomology. Results of A. BuchV. Ravikumar rule out a similar correspondence in Ktheory. This is joint work with Colleen Robichaux and Harshit Yadav. 
Cris Negron Modularization of quantum groups and some conformal field theory I will discuss recent work on constructing small quantum groups at even order roots of unity. (Recall that the small quantum group for a given simple Lie algebra is a characteristic 0, qanalog of its corresponding restricted enveloping algebra.) Our investigations are inspired by a conjectured equivalence of categories between representations for small quantum sl_2, at a certain even order parameter q, and representations for the socalled triplet conformal field theory. I will elaborate on this conjecture and explain how its resolution necessitates the introduction of certain “new” quantum groups, which are obtained from “old” quantum groups via deformation. No familiarity with quantum groups or conformal field theory will be assumed, and all relevant notions will be defined in the talk. 
Andrey Smirnov Elliptic stable envelope for Hilbert scheme of points on C^2 In this talk I describe an explicit formula for elliptic stable envelope of torus fixed points on the Hilbert scheme of points in C^2. In Ktheoretic limit we obtain new combinatorial formulas for Schur, rational Schur and Macdonald polynomials. In particular, we obtain explicit combinatorial formula for the coefficients of the Kostka matrix. 
Richard Rimanyi Elliptic classes of Schubert varieties Assigning characteristic classes to singular varieties is an effective way of studying the enumerative properties of the singularities. Initially one wants to consider the socalled fundamental class in H, K, or Ell, but it turns out that in Ell such class is not well defined. However, a deformation of the notion of fundamental class (under the name of ChernSchwartzMacPherson class in H, motivic Chern class in K) extends to Ell, due to BorisovLibgober. We will introduce a twisted version of the elliptic class, and show its relation to TarasovVarchenko weight functions and Okounkov’s stable envelopes. (Based on results with A. Weber, and on a work in progress with S. Kumar and A. Weber.) 
2019 Spring
Date  Speaker  Affiliation  Title 

April 12  Oliver Pechenik  University of Michigan  Crystal structures for symmetric Grothendieck polynomials 
March 29  Matt Hogancamp  USC  Serre duality for KhovanovRozansky homology 
March 22  Arnav Tripathy  Harvard  A geometric model for complex analytic equivariant elliptic cohomology 
March 15  Justin Allman  USNA Annapolis  Interpolating quantum dilogarithm identities, the topological viewpoint 
March 1  Andrzej Weber  University of Warsaw, Poland  Characteristic classes of Schubert varieties and Hecketype algebras 
Jan 25  Spencer Leslie  Duke  Parity sheaves and Smith theory 
Jan 18 3:30pm  Sergei Gukov  Caltech  Hidden algebraic structures in topology 
Oliver Pechenik Crystal structures for symmetric Grothendieck polynomials The symmetric Grothendieck polynomials representing Schubert classes in the Ktheory of Grassmannians are generating functions for semistandard setvalued tableaux. We construct a type A crystal structure on these tableaux. Applications include a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For rectangular shapes, we give a new interpretation of Lascoux polynomials (Kanalogues of Demazure characters) by constructing a Ktheoretic analogue of crystals with an appropriate analogue of a Demazure subcrystal. (Joint work with Cara Monical and Travis Scrimshaw.) 
Matt Hogancamp Serre duality for KhovanovRozansky homology I will discuss recent joint work with Gorsky, Mellit, and Nagane in which we consider a monoidal version of Serre duality for the category of Soergel bimodules in type A, in which the role of the Serre functor is played by the Rouquier complex associated to the full twist braid. This is a lift of (the type A special case of) a result of MazorchukStroppel (2008) and BeilinsonBezrukavnikovMirkovic (2004), which states that the action of the full twist is the Serre functor on the BGG category O, and as a result we obtain a new “topological” proof of this fact. I will conclude by discussing consequences for KhovanovRozansky link homology. 
Arnav Tripathy A geometric model for complex analytic equivariant elliptic cohomology A longstanding question in the study of elliptic cohomology or topological modular forms has been the search for geometric cocycles. Such cocycles are crucial for applications in both geometry and, provocatively, for the elliptic frontier in representation theory. I will explain joint work with D. BerwickEvans which turns Segal’s physicallyinspired suggestions into rigorous cocycles for the case of equivariant elliptic cohomology over the complex numbers, with some focus on the role of supersymmetry on allowing for the possibility of rigorous mathematical definition. As time permits, I hope to indicate towards the end how one might naturally extend these ideas to higher genus.This talk is joint with the Physically Inspired Mathematics Seminar 
Justin Allman Interpolating quantum dilogarithm identities, the topological viewpoint Quantum dilogarithm identities have a rich history and connections to partition counting, BPS spectra in quantum field theories, stability conditions for quiver representations, Poincare series of cohomological Hall algebras, cluster algebras/categories, and Donaldson–Thomas invariants. In this talk we describe a family of factorization formulas for the combinatorial Donaldson–Thomas invariant for an acyclic quiver. A quantum dilogarithm identity originally due to Reineke, and again established by Rimanyi by counting dimensions of quiver loci, gives two extremal cases of our formulation in the Dynkin case. We establish interpolating factorizations explicitly with a dimension counting argument by defining stratifications of the space of representations for the quiver and calculating Betti numbers of strata in the corresponding equivariant cohomology algebras. 
Adrzej Weber Characteristic classes of Schubert varieties and Hecketype algebras Abstract: We study various types of cohomological invariants of Schubert varieties in the generalized flag variety G/B. According to BernsteinGelfandGelfand the fundamental classes in cohomology can be computed via the action of the nilHecke algebra. It was shown by Aluffi and Mihalcea that the ChernSchwartzMacPherson classes are obtained by the action of the group ring of the Weyl group. Similarly, the motivic Chern classes in Ktheory are related to the classical Hecke algebra, as announced by AluffiMihalceaSchürmannSu. We will concentrate on the equivariant elliptic classes in the sense of BorisovLibgober, which depend on an additional parameter – an auxiliary line bundle. We show that these classes are related to a Hecketype elliptic algebra. The proof uses basic properties of the canonical divisor of the Schubert varieties and its BottSamelson resolutions. As a corollary we show that for G=GL_n the BorisovLibgober elliptic classes are represented by RimanyiTarasovVarchenko elliptic weight function. 
Spencer Leslie Parity sheaves and Smith theory Abstract: Parity sheaves are a topologicallydefined class of sheaves on a variety with many important connections to modular representation theory. In this talk, we discuss connections between parity sheaves on a variety X endowed with the action of a finite cyclic group of order p and parity sheaves on the fixedpoint set. For this we use Smith theory, which gives a localization functor for actions of finite cyclic groups. The key idea is to define a good notion of parity sheaf in an intermediate localized category. In the context of the geometric Satake equivalence, this gives a geometric construction of the Frobeniuscontraction functor on tilting modules. 
Sergei Gukov Hidden algebraic structures in topology Abstract: The goal of this talk is to give an exposition, from several different angles, of new connections that seem to emerge between lowdimensional topology and abstract algebra. While the general picture for such connections — which will be presented toward the end — is rooted in physics, each particular connection can be formulated as a concrete and verifiable statement or, in some cases, even as a theorem. For example, we will see how Kirby moves of 4manifolds can be realized as equivalences of vertex operator algebras, some of which were known previously and some are new. In the opposite direction, we will see how starting with purely algebraic questions that involve, say, classification of modular tensor categories and the Witt group, we will be naturally led to questions in topology of 3manifolds and smooth 4manifolds.This talk is joint with physically inspired mathematics seminar 
2018 Fall
Date  Speaker  Affiliation  Title 

Dec 07 3:15pm  Peter Fiebig  University of Erlangens  Sheaves on the alcoves and modular representations 
Nov 16  Alejandro Ginory  Rutgers  The Verlinde Formula and Twisted Affine Lie Algebras 
Nov 09 3:30pm  Changjian Su  Toronto  Motivic Chern classes, Ktheoretic stable basis and Iwahori invariants of principal series 
Oct 5  Michael Strayer  UNC  Minuscule KacMoody settings unified by new poset coloring properties 
Sep 21  Yiqiang Li  SUNY at Buffalo  Quiver varieties and symmetric pairs 
Sep 07  Sami Assaf  University of Southern California  Nonsymmetric Macdonald polynomials and Demazure characters 
Aug 31  Richárd Rimányi  UNC Chapel Hill  Motivic characteristic classes, Hall algebras, and DT type identities 
Peter Fiebig Sheaves on the alcoves and modular representations Abstract: The main topic is the problem of determining the characters of reductive algebraic groups over a field of positive characteristic. Since Williamson gave countless (unexpected) examples of characters that differ from the character formula given by George Lusztig in 1980 (which holds for large enough characteristics), the community is hoping for a revised formula, or an effective algorithm, that provides a solution in the general case. For this, one surely needs a much better understanding on the torsion phenomena occurring for small primes. In my talk I first want to give an understandable introduction to the problem, and then I want to introduce a new category related to modular representation theory that hopefully will help us understand the problem in greater depth.
2018 Spring
