在1940年代为了研究3-维流形上可积向量场的存在性问题Ehresmann和Reeb提出了叶状结构的概念。随后，叶状结构出现在可积系统，黎曼几何，辛几何以及非交换几何等不同的数学分支中并且发展成为许多数学分支的一个交叉点。在过去几十年，许多数学家从黎曼几何的角度研究叶状空间的几何与拓扑并且取得的许多经典的重要结果；相比而言，从辛几何的角度研究叶状空间的工作还比较少。在本项目中我们将从辛几何的角度研究横截辛叶状空间的几何与拓扑，我们期望从叶状空间理论的角度能够为奇异辛空间特别是一些non-Hausdorff的奇异辛空间的研究提供一个统一的几何框架。结合辛几何与叶状空间两方面的理论，在本项目中我们将重点探索以下两个基本问题：横截辛叶状空间上Hamiltonian群作用的几何以及等变basic上同调的局部化定理及其在toric几何中的应用。

In 1940s, to study the existence problem of integrable vector fields on 3-manifolds Ehresman and Reeb introduced the foliation into mathematics. Since then foliations appeared in many different fields of mathematics such as integrable systems, Riemannian geometry, symplectic geometry and non-commutative geometry and so on; particularly, it has become a crossing point of different fields of mathematics. In the past decades, people achieved many important results in studying geometry and topology of foliations from the viewpoint of Riemannian geometry, namely, Riemannian foliations. However, comparing with the extensive study of the Riemannian foliations, limited works have been done in the study of foliations from the symplectic geometry point of view. In this project we will study the geometry and topology of transversely symplectic foliations from the viewpoint of symplectic geometry and we expect to construct a unified geometric framework for the study of some singular symplectic spaces, especially some non-Hausdorff singular symplectic spaces, from the foliation point of view. Combining the theories of symplectic geometry and foliation, we will focus on the following two basic problems: geometry of the Hamiltonian group action on a transversely symplectic foliation and the localization theorem of equivariant basic cohomology and its applications in toric geometry.