Frobenius manifolds in algebraic geometry and singularity theory

复流形上的Frobenius结构由切丛上的乘法和满足几个相容条件的度量组成。这些结构出现在完全不同的数学分支中，如奇点理论、代数拓扑学或微分几何。镜像对称是数学和理论物理的中心研究领域之一，可以说是Frobenius流形的等价。该提案旨在扩展已知的等价Frobenius结构类。特别地，我们计划研究半单的Frobenius流形、Stokes现象和Orbiolds(模为有限群作用的流形)的量子上同调。我们还试图理解在镜像的另一边(例如奇点理论)的一种类型的Frobenius流形(就像Gromov-Witten理论中的引力后代)中已知的结构的含义。

A Frobenius structure on a complex manifold consists of a multiplication on the tangent bundle and a metric subject to several compatibility conditions. These structures appear in quite different branches of mathematics like singularity theory, algebraic topology or differential geometry. Mirror symmetry, one of the central research areas in mathematics and theoretical physics can be stated as equivalence of Frobenius manifolds. The proposal aims at extending the known classes of equivalent Frobenius structures. In particular, we plan to study semi-simple Frobenius manifolds, the Stokes phenomena and quantum cohomology for orbifolds (manifolds modulo finite group actions). We also seek to understand the meaning of structures known in one type of Frobenius manifolds (like gravitational descendants in Gromov-Witten-theory) on the other side of the mirror picture (e.g. in singularity theory).