Generalized Homological Mirror Symmetry

拓扑量子场论研究从黎曼曲面到目标的映射-代数叠加。 同调镜像对称使我们能够更深入地研究这些堆栈和代数簇的几何。本课题主要研究两个问题：1.代数簇的合理性与单合理性。该方向的最大程序是理解一般四维三次曲面的非有理性和一般三维四次曲面的非单有理性.代数周期理论-周和格里菲斯群的计算，积分和霍奇周期。在这个方向上的最小程序是理解一般类型曲面的Chow群和所谓Fano - Calabi-Yau流形的Griffiths群的Bloch猜想。到目前为止，拓扑量子场论和同调镜像对称已经建立为封闭的， 弦理论中美丽的数学部分不管欧洲核子研究中心的高能对撞机实验成功与否，也不管弦理论作为万有理论的确认与否，人们仍然可以使用拓扑量子场论和同调镜像对称来得到很好的数学结果。特别是人们可以用它们来得到一些长期存在的问题的答案代数几何。

Topological Quantum Field Theory studies maps from Riemann surfaces to targets - algebraic stacks. Homological Mirror Symmetry allows us to look deeper into geometry of these stacks and algebraic varieties. In particular this project studies two main questions:1. Rationality and unirationality of algebraic varieties. Program maximum in this direction is understanding non-rationality of generic four dimensional cubic and non-unirationality of generic three dimensional quartic.2. Theory of algebraic cycles - computations of Chow and Griffiths groups, integral and Hodge cycles. Program minimum in this direction is understanding Bloch's conjecture for Chow groups of surfaces of general type and Griffiths groups of so called Fano - Calabi Yau manifolds.By now Topological Quantum Field Theory and Homological Mirror Symmetryhave been established as closed and beautiful mathematical parts ofString theory. Regardless of the success of the experiments in the high energy collider in CERN and confirmation of String Theory as a theory of everything, one can still use Topological Quantum Field Theory and Homological Mirror Symmetry in order to get nice mathematical consequences. In particular one can use them to get answers of some long standing questions in Algebraic geometry.