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|
||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
I will describe my recent work with Kevin Costello and Edward Witten, on explaining integrable models from a four-dimensional gauge theory.
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.
"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.
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.
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.
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.)
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).
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.
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.
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.
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.