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:

Aug 28 | Masahito Yamazaki | Integrability lattice models from four-dimensional gauge theory |

Sep 11 | Andrei Okounkov | Monodromy and quantizations |

Sep 18 | Tudor Dimofte | Koszul duality patterns in quantum field theory. |

Sep 25 | Andrey Smirnov | Double-elliptic Macdonald Polynomials. |

Oct 2 | Zijun Zhou | Quantum K-theory 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 | No Seminar | No Seminar |

Nov 27 | Vasily Pestun | Periodic Monopoles and Q-opers |

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 four-dimensional 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 K-theory 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 K-theory 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. Ben-Zvi and Z. Yun.)

Oct 23

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).

Oct 30

Y-algebras form a four parameter family of vertex operator algebras
associated to Y-shaped junctions of interfaces in N=4 super Yang-Mills theory.
One can glue such Y-shaped junctions into the more complicated webs of interfaces.
Corresponding vertex operator algebras can be identified with conformal extensions
of tensor products of Y-algebras associated to the trivalent junctions of the web by
fusions of Y-algebras bi-modules associated to the finite interfaces. At the level of
characters, the construction is analogous to the topological vertex like counting of
D0-D2-D4 bound states in toric Calabi-Yau 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 Labastida-Marino-Ooguri-Vafa
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 well-defined only away from walls of marginal stability, where wall-crossing phenomena take place.
Surprisingly, while BPS spectra are ill-defined, 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 wall-crossing invariant of Kontsevich and Soibelman, a.k.a the BPS monodromy, based entirely on topological data of BPS graphs, and derived by wall-crossing in presence of surface defects.

Nov 27

I will discuss quantization of integrable system of periodic monopoles, or equivalently group valued Higgs bundles on a curve, and construction of q-oper Lagrangian variety in the special coordinates.