Affine Structures, Non-archimedean Analytic Geometryand Mirror Symmetry

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AbstractAward: DMS-0504048Principal Investigator: Yan SoibelmanThe project is devoted to the study of the relationship betweenmanifolds with integral affine structure and Calabi-Yau manifolds overnon-archimedean fields. It offers a new approach to HomologicalMIrror Symmetry based on the ideas of collapsing Calabi-Yau manifoldsdeveloped jointly by P.I. and Maxim Kontsevich. Part of the researchis devoted to an unexpected analogy between classical integrablesystems and Calabi-Yau manifolds over non-archimedean fields. Thisanalogy leads to different approaches to integral affine structures:one is via a kind of Liouville integrability and another one via thenotion of skeleton introduced by P.I. and Kontsevich. The problem ofreconstructing a non-archimedean space from the skeleton with(singular) integral affine structure is raised. The solution is givenfor K3 surfaces. Non-commutative version of the above analogy is alsoa part of the project.One of the most challenging problems of modern physics is theunification of all known forces under the roof of onetheory. Most promising candidate for such unification is StringTheory. It is far from being finished. Proposed research isdevoted to mathematical foundations of Mirror Symmetry, which isone of the better understood parts of String Theory. It is alsodevoted to unexpected mathematical analogies motivated by MirrorSymmetry. Hopefully this researh will give new theoretical toolsfor study microstructure of our world via better understanding ofthe underlying local geometry.