Homological Mirror Symmetry and Functional Equations

同调镜像对称是M.Kontsevich提出的一个猜想，它断言与镜像对偶Calabi-Yau流形上的复结构和辛结构相关的某些范畴等价。研究人员建议在椭圆曲线的情况下研究这一猜想。他之前与E.Zaslow和D.Arinkin合作获得的结果证明了这一猜测的部分合理性。他建议将这些结果应用于不定的theta级数的研究。本文提出的另一个研究方向是与局部域上的非齐次向量空间相关的一类新的泛函方程。准齐次向量空间及其Zeta-函数是M.Sato及其学派广泛研究的结果。这位研究人员建议对佐藤Zeta函数的函数方程的某些对角化进行研究。这项研究的下一步将是将这些函数方程中的常量与当地的L因子联系起来。这将允许找到一类新的积分，它的驻相近似是精确的。这个项目的第一部分旨在证明一个源于数学物理的猜想。这一猜想由M.Kontsevich于1994年提出，有望解释大约十年前物理学家发现的镜像对称现象。这一发现(以及弦理论中的其他类似对偶)是理论物理学最近发展的一个例子，理论物理学仍然缺乏坚实的数学基础。目前的工作应被视为对奠定这一基础的贡献。这个项目的第二部分是关于数论产生的一些问题。人们早在19世纪就知道，数的一些深层性质被编码在称为Zeta函数的某些复变量函数中。本文致力于研究表象理论中出现的一类新的由Zeta函数满足的泛函方程。

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.