Feb 1. Hyenho Lho. I will discuss the holomorphic anomaly equation for GW invariants of several Calabi-Yau geometries. Holomorphic anomaly equation was originated from Physics B-model which recursively determine higher genus GW invariants in terms of lower genus invariants. I will explain how one can study holomorphic anomaly equation in mathematics for several examples. This talk is based on joint work with Rahul Pandharipande.
Feb 11. Eugene Gorsky. Khovanov and Rozansky introduced a link homology theory which categorifies the HOMFLY polynomial. This invariant has a lot of interesting properties, but it is notoriously hard to compute. I will introduce HOMFLY homology and discuss its conjectural relation to algebraic geometry of the Hilbert scheme of points on the plane. In particular, I will compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme. The talk is based on joint works with Matt Hogancamp, Andrei Negut and Jacob Rasmussen.
Feb 21. Nora Ganter. I will give an introduction to the construction of equivariant elliptic cohomology and then talk about some recent geometric approaches to the subject.
Feb 26. Matthew Krauel. We will briefly explain how differential operators of Jacobi forms arise in the theory of vertex operator algebras (VOAs). We then introduce a family of differential operators that arise for a certain class of VOAs and explain some possible applications. Time permitting, we will also mention how the theory of VOAs can be used to show how certain polynomials of quasi-Jacobi forms are Jacobi forms.
Mar 5. Katrin Wehrheim. Starting from string diagrams for 2-categories, I will introduce the basic approach of my joint works (in progress) with Bottman / Ma'u / Woodward: Translate string diagrams into moduli spaces of pseudoholomorphic quilts, prove the algebraic axioms by adiabatic analysis, and get 2-categorical structures. In particular, this associates to any Lagrangian relation L\subset M \times N a functor Fuk(M) \to Fuk(N) between (some version of) Fukaya categories. (Depending on the symplectic manifolds, this is a result with Ma'u-Woodward or work in progress with Bottman.)