ABSTRACT: “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 … Read More

ABSTRACT: 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, … Read More

ABSTRACT: 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.

ABSTRACT: 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 … Read More

ABSTRACT: 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 … Read More

ABSTRACT: 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 … Read More

ABSTRACT: 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 … Read More

ABSTRACT: 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 … Read More

ABSTRACT: 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.

Abstract: Several deep mathematical and physical results such as Kontsevich’s deformation-quantization, Drinfeld’s associators, and the Deligne hypothesis are controlled by the vanishing of certain obstruction classes in the theory of differential graded operads. I will talk about a way to obtain such vanishing results, as well as higher-genus analogues, using a weight theory implied by a new motivic point of … Read More