Kahler几何与辛拓扑中若干问题的研究

The main objective of this research proposal is to study several problems in Kahler geometry and symplectic topology that originated from understanding mirror symmetry. More concretely, we will do research on the following two aspects. First, we will study the geometric properties of Kahler-Einstein and Calabi-Yau manifolds, including studying the convergence of these canonical metrics as complex structures degenerate, and how Calabi-Yau metrics are related to Lagrangian and special Lagrangian fibration structures of Calabi-Yau manifolds. Second, we plan to study how to understand and effectively compute Fukaya category and other symplectic invariants. Results from research on these problems will not only be of interest in their field, but also further our understanding of the phenomena and mechanism of mirror symmetry.

本研究课题的主题是Kahler几何和辛拓扑方面的若干问题。 更具体的内容主要是在两个方面进行深入研究。一方面是Calabi-Yau及Kahler-Einstein流形的几何性质，包括复结构退化时度量的收敛性，以及Calabi-Yau度量与(special) Lagrangian纤维化的关系；另一方面研究如何理解辛流形上的Fukaya 范畴以及有关的辛拓扑结构， 如何有效地计算辛拓扑不变量。这些方面的结果除了在Kahler几何和辛拓扑方面的意义之外，也会促进对镜像对称的深入理解。

本课题的主题是Kahler几何与辛几何方面若干课题的研究。具体的问题包括研究Kahler流形，特别是Calabi-Yau流形的度量的收敛，构造和计算辛拓扑不变量， 并研究这些不变量在镜对称中的应用。我们在这些方面取得了一定的进展， 还进行了通过Yang-Yang函数计算扭结不变量的研究。本课题除了在几何和拓扑方面的学术意义外， 也促进了博士后、研究生及本科生的培养和教育。