Mathematical Sciences: Geometry and Topology

Libgober 9503616 本文的研究方向是复微分几何与代数几何的结合点。研究者对镜像对称现象提出了一些问题。这包括一个研究的超几何微分方程，满足家庭的周期的卡-丘流形。镜像对称猜想，首先由物理学家观察到，断言每个卡-丘3重都有一个“镜像”（也是卡-丘3重），卡-丘流形的某些结构与其镜像流形的看似非常不同的结构紧密相关。 镜像对称现象与理论物理、几何和代数中出现的不同结构有着本质的联系。粗略的想法是，理论物理中的结构反映在纯数学的结构中。这整个现象，在特殊情况下得到验证，仍然是一个有趣的数学难题。

Libgober 9503616 The proposed research lies in the interface of complex differential geometry and algebraic geometry. The investigator poses a number of problems on the phenomenon of mirror symmetry. This includes a study of the hypergeometric differential equations that are satisfied by the periods of families of Calabi-Yau manifolds. The mirror symmetry conjecture, first observed by physicists, asserts that each Calabi-Yau 3-fold has a 'mirror' (also a Calabi-Yau 3-fold) and that certain structures of a Calabi-Yau manifold are tightly related to seemingly very different structures of its mirror manifold. The phenomenon of mirror symmetry relates qualitatively different structures arising in theoretical physics, geometry and algebra. The rough idea is that there are structures in theoretical physics which are mirrored in structures in pure mathematics. This entire phenomenon, verified in special cases, remains an intriguing mathematical puzzle.