ABSTRACT: I will describe several recently posed conjectures that relate BPS states in four-dimensional N=2 quantum field theories to representation theory of non-unitary chiral algebras. These conjectures construct wall-crossing invariant generating functions of refined BPS indices which surprisingly are equal to characters of chiral algebras. These characters frequently enjoy nice modular properties. I will also discuss extensions to BPS states … Read More

ABSTRACT: The original “shuffle conjecture” of Haglund, Haiman, Loehr, Ulyanov, and Remmel predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A, in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the ideas that went into my recent proof of this conjecture with Anton … Read More

ABSTRACT: It is known that Chern-Simons topological quantum field theory admits a one-parameter deformation — refinement — in the genus 1 sector. I will tell about a genus 2 generalization of this fact. Just like in the torus case, the crucial role is played by an interesting quantum-mechanical integrable system, which is a genus 2 generalization of Ruijsenaars-Schneider-Macdonald system. The … Read More

ABSTRACT: In a celebrated paper in 1989, E. Witten discovered a beautiful connection between knot invariants (such as the Jones polynomial) and the three-dimensional Chern-Simons theory. Since there are similarities between knot theory and integrable models, it is natural to ask if there is also a gauge-theory explanation for integrable models. The answer to this long-standing question was given only … Read More

ABSTRACT: 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 … Read More

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: 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