Organized by Mina Aganagic, Dima Galakhov, Vivek Shende and Andrey Smirnov
Mondays, 2:00-3:00, 402 Le Conte Hall.
Schedule of talks for Fall 2016:
|Sep 12||Mina Aganagic||Two mathematical applications of little string theory|
|Sep 19||Vivek Shende||Categories and moduli spaces associated to singular Lagrangians|
|Sep 26||Andrei Negut||q-deformed W-algebras, Ext operators and the AGT-W relations|
|Oct 3||Dmitry Galakhov||The two-dimensional Landau-Ginzburg approach to link homology|
|Oct 10||No Seminar|
|Oct 17||Lotte Hollands||Opers, spectral networks and the T theory|
|Oct 24||Michael Viscardi||Equivariant quantum cohomology and the geometric Satake equivalence|
||Pavel Putrov||Resurgence in Chern-Simons theory|
|Nov 7||Nikita Nekrasov||How I learned to stop worrying and to love both instantons and anti-instantons|
|Nov 14||Andrey Smirnov||Quantum K-theory and Nekrasov-Shatashvili conjecture|
|Nov 21||No Seminar|
|Nov 28||No Seminar|
|Dec 5||Tudor Dimofte (2:30-4) and Alexander Braverman (4-5)
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 two mathematical applications of little string theory. The first leads to a variant of AGT correspondence that relates q-deformed W algebra conformal blocks to K-theoretic instanton counting. This correspondence can be proven for any simply laced Lie algebra. The second leads to a variant of quantum Langlands correspondence which relates q-deformed conformal blocks of an affine Lie algebra and a W algebra, associated to a Langlands dual pair of Lie algebras. The proof of the correspondence for simply laced Lie algebras involves, in a crucial way, the recently discovered elliptic stable envelopes. This is based on joint works with Nathan Haouzi and with Edward Frenkel and Andrei Okounkov.
I describe certain categories which arise from the consideration of singular Lagrangian geometries in symplectic manifolds, in 4 and 6 dimensions. These are (mathematically) the Fukaya category of a neighborhood of the singular space; presumably in physics, some category of branes. I will explain how moduli spaces usually associated with irregular singularities, cluster varieties, knot homology, and various other subjects arise in this manner, and then pose some questions regarding the connection to brane-tiling, higher genus effects, and quantization.
I will present a construction of the q-deformed W-algebra of gl_r and its Verma module, that does not use the free field realization or screening charges. The upshot is that our method allows us to directly compute the commutation relations between the Carlsson-Okounkov Ext operator on the moduli space of rank r sheaves and the defining currents of the q-deformed W-algebra. This implies a geometric representation theory interpretation of the AGT-W relations.
We describe rules for computing a homology theory of knots and links in R^3. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of SU(2) magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.
In this seminar I will explain how certain partition functions of four-dimensional quantum field theories, such as the non-Lagrangian T theory, have a geometric interpretation as generating functions of so-called opers. This will reveal close ties to spectral networks and the exact WKB method. (This is joint work with Andy Neitzke.)