Talk

The iterated integrals from 01 to 10 in sequences of one-forms $dz/z$ and $dz/(z-1)$ are periods of mixed Tate motives over Spec Z. The obvious question is if in this way we shall get all periods of mixed Tate motives over Spec Z. We shall construct an l-adic Galois version of this problem and its generalization. We shall construct in the l-adic Galois setting all periods of mixed Tate motives (we call them coefficients) over Spec Z $[1/p]$, hence also over Spec Z, and also over Spec Z[i $\sqrt{p}$] (with $p=3(mod 4)$ in the last case).

We study the topological structure of non-isolated singularities of analytic functions of several complex variables. we proved that for some classes of non-isolated singularities Milnor fibre is homotopy equivalent to a bouquet of spheres. we give an explicit expression of the numbers of spheres in the bouquet.

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.