ABSTRACT: 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 … Read More

ABSTRACT: We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina — in short, coherent … Read More

ABSTRACT: I will start the talk by reviewing our recent work with M.Finkelberg and H.Nakajima on the mathematical construction of Coulomb branches of 3-dimensional N=4 super-symmetric gauge theories as affine complex (possibly singular) symplectic algebraic varieties admitting a canonical quantization (no physics background will be assumed). I will then proceed to the discussion of the generalization of this construction to … Read More

ABSTRACT: Kramers-Wannier duality is a symmetry relating the high-and low-temperature phases of the 2-dimensional lattice Ising model. Electric-Magnetic duality is a 3-dimensional duality between abelian (flat) gauge theories for Pontryagin dual abelian groups. Both dualities generalize to higher-dimensional manifolds. We describe the relation between them using the notion of relative field theory. The order and disorder operators of the Ising … Read More

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