ABSTRACT: The Verlinde formula is a celebrated explicit computation of the dimension of the space of sections of certain positive line bundles over the moduli space of semistable vector bundles over an algebraic curve. I will describe a recent generalization of this formula in which the moduli of vector bundles is replaced by the moduli of semistable Higgs bundles, a … Read More

ABSTRACT: In this talk, I will explain and generalize the following curious observation. First, two things: 1. The quiver Q with one node and g arrows has invariants which are conjecturally dimensions of parts of the middle dimensional cohomology of the twisted character variety of a genus-g surface. 2. There is a famous special-Lagrangian submanifold of six-space defined by Harvey … 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: Khovanov and Rozansky introduced a knot homology theory generalizing the HOMFLY polynomial. I will describe a conjectural relation between the Khovanov-Rozansky homology and the homology of sheaves on the flag Hilbert scheme of points on the plane. The talk is based on the joint work with Andrei Negut and Jacob Rasmussen.

I will discuss some mathematical aspects of instanton counting in two different physical theories- one with gauge group of rank N, the other of small fixed rank. It will be shown that instanton sectors of both theories are equivalent in the N to infinity limit.

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: Bow varieties were introduced by Cherkis as analog of ADHM type description of instanton moduli spaces on the Taub-NUT space ( $\mathbb C^2$ with a hyper-Kaehler metric, not an Euclidean one.) We study these varieties from more algebro-geometric point of view, and introduce their `multiplicative’ analog. Applications are their identifications with Coulomb branches of 3d and 4d gauge theories … 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