光滑函数的Morse 理论是现代数学中最漂亮，最深刻的理论之一。它的思想、技巧在不同方向的推广已对现代数学的许多分支产生了巨大影响。Floer同调是它在辛几何拓扑中一种新形式推广，是研究辛流形上哈密顿动力系统与拉格朗日子流形的几何拓扑重要工具，并还激发出开/闭弦Gromov-Witten不变量理论、Fukaya范畴与辛场论等当今重要研究领域。我们将进一步探讨这种同调的新特征，解决旧问题，建立新理论。在非线性分析方面，我们力争发展新形式的Gromoll-Meyer 的裂开引理以推进计算临界群的方法及证明高阶椭圆方程解的存在性与多重性，解决一些几何变分问题。

The Morse theory of smooth functions is one of the most smart.and the most profound theories in modern mathematics. The generalizations of the.ideas and techniques of it in different directions have yielded the enormous effects.on different subjects of modern mathematics. Floer homology is a new generalization version of it in symplectic geometry topology, and an important tool to study .Hamiltonian dynamics on symplectic manifolds and geometry topology of Lagrangian.submanifolds; it also motivate三 current important research fields such as open/closed string Gromov-Witten invariants theory,Fukaya category and symplectic field theory and so on. We shall further explore new characteristics of this homology, solve old questions and establish new theories. In the nonlinear analysis direction we devote every effort to developing of new splitting lemmas so as to promote computation methods of critical groups, to prove existence and multipicity of.solutions of elliptic equations of higher order and to solve some geometric .variational problems.