Abstract: I will explain how a recent “universal wall-crossing”framework of Joyce works in equivariant K-theory, which I view as amultiplicative refinement of equivariant cohomology. Enumerativeinvariants, possibly of strictly semistable objects living on the walls, are controlled by a certain (multiplicative version of) vertex lgebra structure on the K-homology groups of the ambient stack. In very special settings like refined Vafa-Witten … Read More
Ozsváth-Szabó’s Heegaard Floer homology is a holomorphic curve analogue of the Seiberg-Witten Floer homology of closed 3-manifolds. Bordered Heegaard Floer homology is an extension of (one version of) Heegaard Floer homology to 3-manifolds with boundary, developed jointly with Ozsváth and Thurston. This talk is an overview of bordered Heegaard Floer homology. We will start by describing the structure and aspects … Read More
Abstract: I will discuss a symplectic version of annular Khovanov homology, taking place in Fukaya-Seidel categories of `horizontal’ Hilbert schemes of type A Milnor fibres. This talk reports on joint work with Cheuk Yu Mak.
Abstract: I will explain how to define a t-structure on the wrapped Fukaya category of a complex conic symplectic manifold, whose heart is the global sections of a perverse sheaf of categories on the core of the symplectic manifold. Here, a perverse sheaf of a categories is just a sheaf of categories whose hom sheaves are (shifted) perverse. I will … Read More
Abstract: Many gauge theories in four dimensions are based on PDEs that involve a gauge connection coupled to other fields. The latter are usually a source of a major headache since they lead to non-compactness of the moduli spaces. Today we will discuss two aspects of this major problem and two ways of dealing with it. One will help us … Read More
Using brane quantization, we study the representation theory of the spherical double affine Hecke algebra of type in terms of the topological A-model on the moduli space of flat -connections on a once-punctured torus. In particular, we provide an explicit match between finite-dimensional representations and A-branes with compact support; one consequence is the discovery of new finite-dimensional indecomposable representations. We … Read More
Abstract:Homological mirror symmetry predicts an equivalence between the derived category of equivariant coherent sheaves on the additive Coulomb branch X and a version of the wrapped Fukaya category on multiplicative Coulomb branch Y with superpotential W. If one decategorifies both sides by taking K-theory, the construction still gives an interesting identification between well-known objects in the equivariant K-theory of X … Read More
Abstract: I will discuss algebraic structures associated to moduli of sheaves on elliptic surfaces, and describe their relation with other parts of mathematical physics. These algebraic structures control the enumerative geometry of these moduli spaces analogous to how quantum groups control enumerative invariants of quiver varieties. The main results discussed will include a description of the quantum differential equation in … Read More
In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into … Read More