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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 a 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.

We present a deformed star product for a particle in the presence of magnetic monopole. The product is obtained within a Weyl quantization-dequantization scheme, with a correspondence between classical observables and operators on a quaternionic Hilbert space following work of Emch and Jadczyk. The resulting product among complex valued functions is well defined for a large class of functions and reproduces (at first order in h bar) the Poisson structure of the particle in the monopole field.

In the talk, I explain a method of finding non-trivial elements in homotopy groups of groups of symplectomorphisms via constructing suitable hamiltonian fiber bundles. This circle of ideas appared in works of Reznikov and was developed by McDuff. I apply this method to symplectic homogeneous spaces, and relate cohomology of the classifying space of the group of hamiltonian symplectomorphisms, to cohomology of lattices in semisimple Lie groups of non-compact type.