Seminars & Colloquia Calendar

Strominger, Yau, and Zaslow proposed a geometric explanation for mirror symmetry via a dualization procedure relating symplectic manifolds equipped with Lagrangian torus fibration with complex manifolds equipped with totally real torus fibrations. By considering the family of symplectic manifolds obtained by rescaling the symplectic form, one obtains a degenerating family of complex manifolds, which is expected to be the mirror.

Because of convergence problems with Floer theoretic constructions, it is difficult to make this procedure completely rigorous. Kontsevich and Soibelman thus proposed to consider the mirror as a rigid analytic space, defined over the field C((t)), equipped with the non-archimedean t-adic valuation, or more generally over the Novikov field. This is natural because the Floer theory of a symplectic manifold is defined over the Novikov field.

After explaining this background, I will give some indication of the tools that enter in the proof of homological mirror symmetry in the simplest class of examples which arise from these considerations, namely Lagrangian torus fibrations without singularities.