Heegaard 分解是低维流形研究中的经典课题，相关问题的研究一直是几何拓扑中活跃的方向。本项目试图通过Heegaard 图的表示给出Heegaard分解的一种具体的刻画方式，并通过这种方式给出一些具体的三维流形的Heegaard分解的例子。进一步地，通过这种表示寻求跟曲线复形等经典理论的联系，并探索与其相关的流形几何与动力系统问题。具体包括曲面上简单闭曲线（组）的表示、曲线复形一维骨架的组合结构、基于Heegaard 图的流形识别、特殊Heegaard 图提供的流形动力系统性质等方面的研究。

Heegaard splitting is a classical subject in low dimentional topology, and research on related topics is always an active direction in geometric topology. This project is aimed to give a specific description on Heegaard splitting by the research on Heegaard diagram. Furthermore, this project will try to find the connections with the curve complex theory, and with other problems in geometry and dynamics. Specifically, this study includes the representation of simple closed curves on surfaces, the structure of the 1-skeleton in curve compexes, recognition for special 3-manifolds and dynamic properties on 3-manifolds.