2019
University of North Carolina at Chapel Hill
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 Shrawan Kumar.
2019 Fall
| Date | Speaker | Affiliation | Title |
|---|---|---|---|
| November 22 4:15-5:15pm | Gurbir Dhillon | Stanford | The tamely ramified Fundamental Local Equivalence |
| November 22 3-4pm | Vikraman Balaji | Chennai Mathematical Institute | Torsors on semistable curves and the problem of degenerations |
| November 15 | Alex Yong | University of Illinois at Urbana-Champaign | 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 D-modules 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 G-torsors on curves. The first one is to construct a at degeneration of the moduli stack G-torsors on a smooth projective curve when the curve degenerates to an irreducible nodal curve. Torsors for a generalization of the classical Bruhat-Tits group schemes to two-dimensional 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 G-torsor on an irreducible nodal curve. The absence of obvious analogues of torsion-free sheaves in the setting of G-torsors makes the question interesting. This also leads to the construction of a proper separated coarse space for G-torsors on an irreducible nodal curve. |
| Alex Yong The A.B.C.D’s of Schubert calculus We collect Atiyah-Bott 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. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory. 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, q-analog 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 so-called 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 K-theoretic 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 so-called 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 Chern-Schwartz-MacPherson class in H, motivic Chern class in K) extends to Ell, due to Borisov-Libgober. We will introduce a twisted version of the elliptic class, and show its relation to Tarasov-Varchenko 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 Khovanov-Rozansky 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 Hecke-type 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 K-theory of Grassmannians are generating functions for semistandard set-valued 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 (K-analogues of Demazure characters) by constructing a K-theoretic analogue of crystals with an appropriate analogue of a Demazure subcrystal. (Joint work with Cara Monical and Travis Scrimshaw.) |
| Matt Hogancamp Serre duality for Khovanov-Rozansky 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 Mazorchuk-Stroppel (2008) and Beilinson-Bezrukavnikov-Mirkovic (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 Khovanov-Rozansky link homology. |
| Arnav Tripathy A geometric model for complex analytic equivariant elliptic cohomology A long-standing 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. Berwick-Evans which turns Segal’s physically-inspired 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 Hecke-type algebras Abstract: We study various types of cohomological invariants of Schubert varieties in the generalized flag variety G/B. According to Bernstein-Gelfand-Gelfand the fundamental classes in cohomology can be computed via the action of the nil-Hecke algebra. It was shown by Aluffi and Mihalcea that the Chern-Schwartz-MacPherson classes are obtained by the action of the group ring of the Weyl group. Similarly, the motivic Chern classes in K-theory are related to the classical Hecke algebra, as announced by Aluffi-Mihalcea-Schürmann-Su. We will concentrate on the equivariant elliptic classes in the sense of Borisov-Libgober, which depend on an additional parameter – an auxiliary line bundle. We show that these classes are related to a Hecke-type elliptic algebra. The proof uses basic properties of the canonical divisor of the Schubert varieties and its Bott-Samelson resolutions. As a corollary we show that for G=GL_n the Borisov-Libgober elliptic classes are represented by Rimanyi-Tarasov-Varchenko elliptic weight function. |
| Spencer Leslie Parity sheaves and Smith theory Abstract: Parity sheaves are a topologically-defined 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 fixed-point 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 Frobenius-contraction 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 low-dimensional 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 4-manifolds 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 3-manifolds and smooth 4-manifolds.This talk is joint with physically inspired mathematics seminar |