Homological Mirror Symmetry and Functional Equations

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Abstract.Homological mirror symmetry is a conjecture, formulated by M.Kontsevich, which asserts the equivalence of certain categories associated to complex and symplectic structures on mirror dual Calabi-Yau manifolds. The investigator proposes to work on this conjecture in the case of elliptic curves. His previous results obtained in collaboration with E.Zaslow and D.Arinkin justify some part of this conjecture. He proposes to apply these results to the study of indefinite theta series. Another direction of research proposed here is related to a new class of functional equations associated to prehomogeneous vector spaces over local fields. Prehomogeneous vector spaces and their zeta-functions were studied extensively by M.Sato and his school. The investigator proposes to work on certain ``diagonalization'' of functional equations for Sato's zeta-functions. The next stage of this research would be to relate the constants in these functional equations to local L-factors. This would allow to find a new class of integrals for which the stationary phase approximation is exact.The first part of this project is aimed at proving a conjecture which originated from mathematical physics. This conjecture, which was proposed by M.Kontsevich in 1994, is expected to explain the phenomenon of mirror symmetry discovered by physicists about a decade ago. This discovery (along with other similar dualities in string theory) is an example of recent developments in theoretical physics which still lack solid mathematical foundation. The present work should be considered as a contribution to laying such a foundation. The second part of this project is devoted to some problems arising from number theory. It was known already in the 19-th century that some deep properties of numbers are encoded in certain functions of complex variable called zeta-functions. The proposed work is devoted to the study of a new class of functional equations satisfied by zeta-functions which arise in representation theory.