The RTGC Research Seminar is held every Friday from 4:10 - 5:30 in 939 Evans (unless otherwise specified).
 Date Title Speaker 09/20 Categorification of small quantum groups (Friday, 4:10 - 5:30, 740 Evans) You Qi, UC Berkeley 09/27 Some Interesting Quivers and Why They're Interesting (Friday, 4:10 - 5:30, 740 Evans) Harold Williams, UC Berkeley 10/04 Instantons from residues of Liouville CFT (Friday, 4:10 - 5:30, 939 Evans) Shamil Shakirov, UC Berkeley 10/11 The mirror of a Legendrian knot (Friday, 4:10 - 5:30, 939 Evans) Vivek Shende , UC Berkeley 10/18 Geometric representations of rational Cherednik algebras (Friday, 4:10 - 5:30, 939 Evans) Zhiwei Yun , Stanford 10/25 Higher pentagram maps (Friday, 4:10 - 5:30, 939 Evans) Boris Khesin , University of Toronto 11/01 Mirkovic-Vilonen cycles and their coordinate rings (Friday, 4:10 - 5:30, 939 Evans) Allen Knutson , Cornell and UC Berkeley 11/08 Categorical Heisenberg actions on Hilbert schemes of points (Friday, 4:10 - 5:30, 939 Evans) Sabin Cautis , UBC and USC 11/15 Tautological relations via the orbifold C/Z_r (Friday, 4:10 - 5:30, 939 Evans) Emily Clader, University of Michigan 11/22 A geometric perspective on the piecewise polynomiality of double Hurwitz numbers (Friday, 4:10 - 5:30, 939 Evans) Steffen Marcus, University of Utah 12/06 TBA (Friday, 4:10 - 5:30, 939 Evans) Hiraku Nakajima, RIMS

Abstracts:

Categorification of small quantum groups, You Qi:

We propose an algebraic approach to categorification of quantum groups at a prime root of unity, in the scope of eventually categorifying Witten-Reshetikhin-Turaev three-manifold invariants. This is joint work with Mikhail Khovanov.

Some Interesting Quivers and Why They're Interesting, Harold Williams:

More precisely, we'll discuss some interesting sequences of mutations of some interesting quivers (and why those are interesting). To each such sequence of mutations is associated a corresponding sequence of cluster transformations (rational maps of a very specific form), which as we'll explain show up in a number of seemingly disparate contexts: the representation theory of quantum loop algebras, 4d N=2 gauge theories, and factorization problems in simple Lie groups. We'll explain our recent work developing this last connection, which in particular lets us prove the discrete integrability of these distinguished sequences.

Instantons from residues of Liouville CFT, Shamil Shakirov:

An interesting topic in modern mathematical physics is the so-called "2d\4d relation" between Liouville conformal blocks (hypergeometric type integrals over contours in CP^1) and instanton counting (sums over fixed points in the moduli space of instantons in C^2). We will tell about a simple way to prove this relation, using a natural q-deformation of the both sides. After the q-deformation, the Liouville integrals acquire poles and can be taken by residues. These residue sums are manifestly the sums that appear in instanton counting.

The mirror of a Legendrian knot, Vivek Shende :

This talk presents joint work with David Treumann and Eric Zaslow. To a Legendrian knot, which we view as sitting in the circle bundle at infinity in the cotangent bundle of the plane, we associate the subcategory of objects which end on the knot'' inside the (unwrapped) Fukaya category of T*R^2; this category is a Legendrian isotopy invariant of the knot and can be identified through the Nadler-Zaslow dictionary with the category of constructible sheaves on R^2 with singular support in the cone over the knot. This latter description allows for computations, which reveal that (1) for the planar closure of a positive braid, the weight polynomial of the moduli space of rank one objects (the 'mirror' of the knot) is the lowest order in a coefficient of the HOMFLY polynomial, (2) for the cylindrical closure of a positive braid, the moduli space of rank one objects is precisely the space which appears in the Webster-Williamson construction of HOMFLY homology, and the HOMFLY homology can be extracted from it, (3) in general we conjecture that the subcategory of rank one objects is equivalent to the augmentation category of the Chekanov-Eliashberg dga.

Geometric representations of rational Cherednik algebras, Zhiwei Yun :

Rational Cherednik algebras are (a degenerate version of) generalizations of Hecke algebras. Their finite-dimensional representations are extremely rigid. We shall construct finite-dimensional representations of these algebras from the cohomology of Hitchin fibers which have extra symmetry. This is joint work with A.Oblomkov.

Higher pentagram maps, Boris Khesin :

We define pentagram maps on polygons in any dimension, which extend R.Schwartz's definition of the 2D pentagram map. These maps turn out to be integrable for both closed and twisted polygons. The corresponding continuous limit of the pentagram map in dimension d is shown to be the (2,d+1)-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We discuss their Lax forms, properties of the 3D map, and various generalizations. This is a joint work with Fedor Soloviev (Univ. of Toronto).

Mirkovic-Vilonen cycles and their coordinate rings, Allen Knutson :

I'll recall the geometric Satake correspondence, which among other things says that each irrep V_lambda of a connected reductive complex algebraic group G can be obtained uniquely as the intersection homology of a cycle Gr^lambda in the affine Grassmannian Gr for the Langlands dual of G. If one uses a semiinfinite Borel of Loops(G-dual) to give a Morse decomposition of Gr^lambda, the resulting "Mirkovic-Vilonen cycles" give a basis for IH_*. Kamnitzer showed that these MV cycles can be indexed by their moment polytopes, and a direct description of those polytopes. For G simply-laced, Baumann-Kamnitzer gave a way to associate an MV polytope to a general point in Lusztig's nilpotent scheme (which I'll recall). I'll explain a conjectural extension of this to a construction of the actual MV cycle, and several consequences of this conjecture, joint with Joel Kamnitzer.

Categorical Heisenberg actions on Hilbert schemes of points, Sabin Cautis :

We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE spaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

Tautological relations via the orbifold C/Z_r, Emily Clader:

Tautological classes are certain elements of the cohomology or Chow ring of the moduli space of curves that are important in Gromov-Witten theory. We describe a method for deriving relations between these classes by studying the Gromov-Witten theory of the orbifold C/Z_r. Furthermore, we show that the quantum cohomology of C/Z_r is generically semisimple. Using recent ideas of Pandharipande-Pixton-Zvonkine, this semisimplicity may be useful for obtaining other tautological relations.

A geometric perspective on the piecewise polynomiality of double Hurwitz numbers, Steffen Marcus:

Hurwitz numbers count degree d branched covers of the Riemann sphere by a genus g Riemann surface with prescribed ramification over one branch point and simple ramification over the others. They are intimately related to the geometry of the moduli space of curves through the famous ELSV formula. Double Hurwitz numbers similarly count covers with prescribed ramification over two points. In this talk I'm going to explain how we can describe double Hurwitz numbers as intersection numbers on the moduli space of curves using the geometry of the moduli space of relative stable maps. This helps explains geometrically the chamber/wall-crossing piecewise polynomial structure of double Hurwitz numbers. This is joint work with Renzo Cavalieri.

TBA, Hiraku Nakajima:

To be announced.

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