Abouzaid Abstract: 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.