Organized by Mina Aganagic, Dima Galakhov, Vivek Shende and Andrey Smirnov
Mondays, 2:003:00, 402 Le Conte Hall.
Schedule of talks for Spring 2017:
Aug 28  Masahito Yamazaki  Integrability lattice models from fourdimensional gauge theory 
Sep 11  Andrei Okounkov  Monodromy and quantizations 
Sep 18  Tudor Dimofte  Koszul duality patterns in quantum field theory. 
Sep 25  Andrey Smirnov  Doubleelliptic Macdonald Polynomials. 
Oct 2  Zijun Zhou  Quantum Ktheory of hypertoric varieties 
Oct 9  No Seminar  No Seminar 
Oct 16  David Nadler  Betti Geometric Langlands 
Oct 23  Aaron Lauda  A new look at the quantum knot invariants 
Oct 30 
Miroslav Rapcak  Vertex operator algebras via topological vertex like construction 
Nov 6  Poitr Sulkowski  Knots quivers correspondence 
Nov 13  Pietro Longhi  BPS Graphs of Class S Theories 
Nov 20  TBA  TBA 
Nov 27  Vasily Pestun  
Dec 4  TBA 
TBA

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
Sep 11
TBA
Aug 28
I will describe my recent work with Kevin Costello and Edward Witten, on explaining integrable models from a fourdimensional gauge theory.
Sep 11
For a general symplectic resolution X, Bezrukavnikov and Kaledin used quantization in prime characteristic to construct
certain very interesting derived automorphisms of X. Their action on K(X) has been since conjectured to match the monodromy of the
quantum differential equation for X. This talk will be about our joint work in progress
with Bezrukavnikov in which we prove this conjecture for the infinite series of symplectic resolutions.
Sep 18
"Koszul duality" is a fundamental idea
spanning several branches of mathematics, with origins in representation theory and rational homotopy theory.
In its simplest incarnation, it relates pairs of algebras (such as symmetric and exterior algebras) that have
equivalent categories of representations. Koszul duality also turns out to play a fundamental role in physics,
governing the structure of boundary conditions in QFT  in various dimensions, with various amounts of supersymmetry.
I will explain the general idea, and give a simple example that leads to a new understanding of gauge/global symmetry in supersymmetric quantum mechanics.
Sep 25
I will describe a new geometric way to think about symmetric polynomials.
We will consider some special classes in the equivariant elliptic cohomology of Hilbert scheme
of points on the complex plane (elliptic stable envelopes). It is natural to think about these classes
as two parametric elliptic generalization of Macdonald polynomials. All other important symmetric
polynomials such as Macdonald, Jack and Schur polynomials appear as different limits of these
most general objects.
Oct 2
Okounkov's quantum Ktheory is defined via virtual counting of parameterized quasimaps. In this talk I will consider explicit computations in the case of hypertoric varieties,
where the quantum Ktheory relation will arise from analysis of the bare vertex function.
Oct 16
I’ll describe a variant of the geometric Langlands program that has more of the topological flavor of some physical accounts.
I’ll explain how it fits into broader patterns in mirror symmetry, and also the form it takes in some examples.
A key quest is for a “categorical Verlinde formula” to reduce the case of
high genus curves to nodal configurations. (Joint work in parts with D. BenZvi and Z. Yun.)
Oct 23
The ReshetikhinTuraev 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 twodimensional representation. In
this talk we will explain CautisKamnitzerMorrison’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).
Oct 30
Yalgebras form a four parameter family of vertex operator algebras
associated to Yshaped junctions of interfaces in N=4 super YangMills theory.
One can glue such Yshaped junctions into the more complicated webs of interfaces.
Corresponding vertex operator algebras can be identified with conformal extensions
of tensor products of Yalgebras associated to the trivalent junctions of the web by
fusions of Yalgebras bimodules associated to the finite interfaces. At the level of
characters, the construction is analogous to the topological vertex like counting of
D0D2D4 bound states in toric CalabiYau manifolds. Gluing construction sheds new
light on the structure of vertex operator algebras conventionally constructed by BRST
reductions and provides us with a way to construct new algebras.
Nov 6
I will present a surprising relation between knot invariants
and quiver representation theory, motivated by various string theory
constructions involving BPS states. Consequences of this relation
include the proof of the famous LabastidaMarinoOoguriVafa
conjecture (at least for symmetric representations), explicit (and
unknown before) formulas for colored HOMFLY polynomials for various
knots, new viewpoint on knot homologies, a novel type of
categorification, new dualities between quivers, and many others.
Nov 13
BPS quivers and Spectral Networks are two powerful tools for computing BPS spectra in 4d N=2 theories. On the Coulomb branches of these theories, the BPS spectrum is welldefined only away from walls of marginal stability, where wallcrossing phenomena take place.
Surprisingly, while BPS spectra are illdefined, there is a lot of information hidden in spectral networks at maximal intersections of MS walls. In this talk I will describe how they give rise to BPS graphs, and how the relation of the latter to BPS quivers emerges naturally. I will also present a novel construction of the wallcrossing invariant of Kontsevich and Soibelman, a.k.a the BPS monodromy, based entirely on topological data of BPS graphs, and derived by wallcrossing in presence of surface defects.