SR1

The Katz-Klemm-Vafa formula is a conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. In genus 0 it reduces to the (proved) Yau-Zaslow formula. I will explain how the correspondence between stable pairs and Gromov-Witten theory for toric 3-folds (proved by Maulik-Oblomkov-Okounkov-Pandharipande), some calculations with stable pairs (due to Kawai-Yoshioka) and some deformation theory lead to a proof of the KKV formula.
(This is joint work with Davesh Maulik and Rahul Pandharipande. Only they understand the actual formulae. People who like modular forms are not encouraged to come to this talk.)

We will begin by reviewing the construction of the symplectic quotient and the definition of the Kirwan map. Then we will give an overview of Kirwan's original proof of the surjectivity of this map and some generalizations of this result. Finally we will talk about the techniques that are being developed to construct right inverses for the Kirwan map.