Organized by Mina Aganagic, Dima Galakhov, Vivek Shende and Andrey Smirnov

Mondays, 2:00-3:00, 402 Le Conte Hall.

Schedule of talks for Spring 2017:

A note to the speakers:
This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas.
The audience may fail to appreciate the glory of the result, otherwise.

The seminar archive can be found: here

Abstracts

Jan 11

The Verlinde formula is a celebrated explicit computation of the dimension of the space of sections
of certain positive line bundles over the moduli space of semistable vector bundles over an algebraic curve.
I will describe a recent generalization of this formula in which the moduli of vector bundles is replaced by
the moduli of semistable Higgs bundles, a moduli space of great interest in geometric representation theory
and mathematical physics. A key part of the proof is a new ``virtual non-abelian localization formula" in K-theory,
which has broader applications in enumerative geometry. The localization formula is an application of the nascent theory of
Theta-stratifications,
and it serves as a new source of applications of derived algebraic geometry to more classical questions.

Jan 16

In this talk, I will explain and generalize the following curious observation.
First, two things:
1. The quiver Q with one node and g arrows has invariants which are
conjecturally dimensions of parts of the middle dimensional
cohomology of the twisted character variety of a genus-g surface.
2. There is a famous special-Lagrangian submanifold of six-space
defined by Harvey and Lawson, and used by Aganagic, Klemm and Vafa in the
context of string theory. The open Gromov-Witten invariants of this
submanifold depend on an integer-valued "framing" at infinity.
Their generating function (superpotential) can be written as a
power series in dilogarithms with integer coefficients, the BPS numbers.
If we choose framing g, then the BPS numbers are the quiver invariants of Q.
Note that g does *not* represent any kind of genus in the physics story!?
In general, the invariants of *all* symmetric quivers with n nodes are
controlled by a single moduli space associated to a genus-n surface. Go figure.
(This is based on joint work with Linhui Shen and David Treumann.)

Jan 30

We discuss recent work showing that in type A_n the category of
equivariant perverse coherent sheaves on the affine Grassmannian categorifies the
cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically,
this can be understood as a statement about line operators in this theory, following
ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC
sheaves are the precise algebro-geometric avatars of Wilson-'t Hooft line operators.
The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras.
A key moral is that the appearance of cluster structures is in large part forced by the compatibility
between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory).
This is joint work with Sabin Cautis.

Feb 6

Khovanov and Rozansky introduced a knot homology theory generalizing the HOMFLY polynomial.
I will describe a conjectural relation between the Khovanov-Rozansky homology and the homology of
sheaves on the flag Hilbert scheme of points on the plane. The talk is based on the joint work with Andrei Negut
and Jacob Rasmussen.

Feb 27

I will discuss some mathematical aspects of instanton counting in two different physical theories-
one with gauge group of rank N, the other of small fixed rank.
It will be shown that instanton sectors of both theories are equivalent in the N to infinity limit.

Mar 13

I will start the talk by reviewing our recent work with M.Finkelberg
and H.Nakajima on the mathematical construction of Coulomb branches of
3-dimensional N=4 super-symmetric gauge theories as affine complex
(possibly singular) symplectic algebraic varieties admitting a
canonical quantization (no physics background will be assumed). I
will then proceed to the discussion of the generalization of this
construction to Coulomb branches of 4-dimensional gauge theories
compactified on a circle. I will present a set of conjectures about
the resulting varieties and give some examples

Mar 20

Bow varieties were introduced by Cherkis as analog of ADHM
type description of instanton moduli spaces on the Taub-NUT space (
$\mathbb C^2$ with a hyper-Kaehler metric, not an Euclidean one.) We
study these varieties from more algebro-geometric point of view, and
introduce their `multiplicative' analog. Applications are their
identifications with Coulomb branches of 3d and 4d gauge theories
respectively. The first part is joint work with Y.Takayama, while the second is with
D.Yamakawa.

Apr 3

The Reshetikhin-Turaev construction associated knot
invariants to the data of a simple Lie algebra and a choice of
irreducible representation. The Jones polynomial is the most famous
example coming from the Lie algebra sl(2) and its two-dimensional
representation. In this talk we will explain
Cautis-Kamnitzer-Morrison's novel new approach to studying RT
invariants associated to the Lie algebra sl(n). Rather than delving
into a morass of representation theory, we will show how two
relatively simple Lie theoretic ingredients can be combined with a
powerful duality (skew Howe) to give an elementary and diagrammatic
construction of these invariants. We will explain how this new
framework solved an important open problem in representation theory,
proves the existence of an (a,q)-super polynomial conjectured by
physicists (joint with Garoufalidis and Lê), and leads to a new
elementary approach to Khovanov homology and its sl(n) analogs (joint
with Queffelec and Rose).

Apr 10

Kramers-Wannier duality is a symmetry relating the high-and low-temperature phases of
the 2-dimensional lattice Ising model. Electric-Magnetic duality is a 3-dimensional
duality between abelian (flat) gauge theories for Pontryagin dual abelian groups. Both
dualities generalize to higher-dimensional manifolds. We describe the relation between
them using the notion of relative field theory. The order and disorder operators of the
Ising model are endpoints of Wilson and t’Hooft electro-magnetic loops, respectively.
There is a higher-dimensional generalization to finite homotopy types.
This is based on joint work with Dan Freed.

Apr 17

I will describe several recently posed conjectures that relate BPS states in four-dimensional N=2 quantum
field theories to representation theory of non-unitary chiral algebras. These conjectures construct wall-crossing
invariant generating functions of refined BPS indices which surprisingly are equal to characters of chiral algebras.
These characters frequently enjoy nice modular properties.
I will also discuss extensions to BPS states in the presence of defects which further enrich the correspondence.

May 1

The original "shuffle conjecture" of Haglund, Haiman, Loehr, Ulyanov, and Remmel
predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A,
in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the
ideas that went into my recent proof of this conjecture with Anton Mellit, namely the construction of a new algebra
which has many elements in common with DAHA's, and which has been expected to have a geometric construction.
I will then explain a new result with Eugene Gorsky and Mellit, in which we construct this algebra using the torus-equivariant
K theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals in C[x,y] of finite codimension.

May 8

TIt is known that Chern-Simons topological quantum field theory admits a one-parameter deformation -- refinement -- in the genus 1 sector.
I will tell about a genus 2 generalization of this fact. Just like in the torus case, the crucial role is played by an interesting quantum-mechanical
integrable system, which is a genus 2 generalization of Ruijsenaars-Schneider-Macdonald system. The knowledge of this quantum mechanics allows us
to prove mathematically the corresponding mapping class group action, which is a genus 2 generalization of SL(2,Z)