Principal Investigator: Yan Soibelman
The project is devoted to the study of the relationship between manifolds with integral affine structure and Calabi-Yau manifolds over non-archimedean fields. It offers a new approach to Homological MIrror Symmetry based on the ideas of collapsing Calabi-Yau manifolds developed jointly by P.I. and Maxim Kontsevich. Part of the research is devoted to an unexpected analogy between classical integrable systems and Calabi-Yau manifolds over non-archimedean fields. This analogy leads to different approaches to integral affine structures: one is via a kind of Liouville integrability and another one via the notion of skeleton introduced by P.I. and Kontsevich. The problem of reconstructing a non-archimedean space from the skeleton with (singular) integral affine structure is raised. The solution is given for K3 surfaces. Non-commutative version of the above analogy is also a part of the project.
One of the most challenging problems of modern physics is the unification of all known forces under the roof of one theory. Most promising candidate for such unification is String Theory. It is far from being finished. Proposed research is devoted to mathematical foundations of Mirror Symmetry, which is one of the better understood parts of String Theory. It is also devoted to unexpected mathematical analogies motivated by Mirror Symmetry. Hopefully this researh will give new theoretical tools for study microstructure of our world via better understanding of the underlying local geometry.
