Mathematical Physics

Connections between representation categories of quantum groups and vertex operator algebras (VOAs) have been studied since the 1990s starting with the pioneering work of Kazhdan and Lusztig. Recently, connections have been found between unrolled quantum groups and certain families of VOAs. In this talk, I will introduce unrolled quantum groups and describe their connections to the Singlet, Triplet, and Bp vertex operator algebras.

Factorization algebras provide a flexible language for describing the observables of a perturbative QFT, as shown in joint work with Kevin Costello. In joint work with Eugene Rabinovich and Brian Williams, we extend those constructions to a manifold with boundary for a special class of theories that includes, as an example, a perturbative version of the correspondence between chiral U(1) currents on a Riemann surface and abelian Chern-Simons theory on a bulk 3-manifold. Given time, I'll sketch a systematic higher dimensional version for higher abelian CS theory on an oriented smooth manifold of dimension 4n+3 with boundary a complex manifold of complex dimension 2n+1. (This talk will be very informal.)

Absolute Gromov-Witten theory is known to have many nice structural properties, such as quantum cohomology, WDVV equation, Givental's formalism, mirror theorem, CohFT etc.. In this talk, I will explain how to obtain parallel structures for relative Gromov-Witten theory via the relation between relative and orbifold Gromov-Witten invariants. This is based on joint works with Honglu Fan, Hsian-Hua Tseng and Longting Wu. 

Bergman's Diamond Lemma for ring theory gives an algorithm to produce a (non-canonical) basis for a ring presented by generators and relations. After demonstrating this algorithm in concrete, geometrically-minded examples, I'll turn to preprojective algebras and their multiplicative counterparts. Using the Diamond Lemma, I'll reprove a few classical results for preprojective algebras. Then I'll propose a conjectural basis for multiplicative preprojective algebras. Finally I'll explain why the set is a basis in the case of multiplicative preprojective algebras for quivers containing a cycle, the subject of joint work with Travis Schedler.

In this talk I will present a construction of relative Bridgeland stability conditions that appeared in my work on stability of topological Fukaya categories of surfaces. This construction gives a local-to-global tool for constructing stability conditions. I will explain what technical features of such Fukaya categories render this construction useful in their context, and time allowing discuss the challenges involved in applying these ideas to other contexts such as Bridgeland-Smith type stability conditions and topological Fukaya categories with coefficients.

A recent construction of HOMFLY-PT knot homology by Oblomkov-Rozansky has its physical origin in “B-twisted” 3D N=4 gauge theory, with adjoint and fundamental matter. Mathematically, the construction uses certain categories of matrix factorization. We apply 3D Mirror Symmetry to identify an A-twisted mirror of this construction. In the case of algebraic knots, we find that knot homology on the A side gets expressed as cohomology of affine Springer fibers (related but not identical to work if Gorsky-Oblomkov-Rasmussen-Shende). More generally, we propose a Fukaya-Seidel category mirror to the Oblomkov-Rozansky matrix factorization. Joint work with N Garner, J Hilburn, A Oblomkov, and L Rozansky.

There is a close relationship between derived loop spaces, a geometric object, and Hochschild homology, a categorical invariant, made possible by derived algebraic geometry, thus allowing for both intuitive insights and new computational tools.  In the case of a quotient stack, we discuss a "Jordan decomposition" of loops which is made precise by an equivariant localization result.  We also discuss an Atiyah-Segal completion theorem which relates completed periodic cyclic homology to Betti cohomology.