This project is concerned with the mathematical underpinnings of the Strominger-Yau-Zaslow conjecture originating in the physical field of mirror symmetry. Mirror symmetric is a phenomenon in high energy physics when two theories which explain interaction of elementary particles of high energies complement each other in a certain way. Each such physical theory is based on a mathematical object, a family of complex Calabi-Yau manifolds, an when two such families correspond to the complementary theories, they are called mirror partners. The Strominger-Yau-Zaslow (SYZ) conjecture seeks to give a mathematical explanation to the phenomen of mirror symmetry, but mathematical questions that arise in it are interesting in their own right, from a purel mathematics point of view, and they make sense even for families of Calabi-Yau manifolds that do not have physical significance.
In the mid-2000s two approaches to the SYZ conjecture --- metric and non-archimedean --- were put forward by Kontsevich and Soibelman. Each approach proposed certain mathematical construction that would take a family of Calabi-Yau manifolds of dimension 2n and produce an n-dimensional sphere ("base of the SYZ fibration") endowed with additional mathematical structures. The structures produced by the two approaches are different, but are both underlied by a common geometric strucuters called singular integral affine structure. Both approaches have been extensively studied each on its own, but the relationship between the two remains mysterious. Konstevich and Soibelman conjectured that the singular affine structures arising from both approaches should be related in a certain way. The aim of this project is to develop mathematical tools that would allow to describe this relationship and study the interaction of mathematical structures in both approaches via the singular integral affine structure that underlies them. Development of these tools will have impact on several fields of pure mathematics that intersect in the mathematical treatment of mirror symmetry.
