辛几何拓扑与非线性分析中 Morse 理论方法

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.

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。

对Finsler度量的能量泛函在自然的H^1曲线的Hilbert流形上建立了广义Morse引理,作为应用将黎曼流形上关于多个测地线存在性的Bangert-Klingenberg 定理及Grove-Tanaka 定理推广Finsler流形. 构造了Hamiltonian微分同胚群上一族双不变度量并得到了对应的 Hofer不等式与Sikorav 不等式. 推广Seidel关于标准复投映空间的辛微分同胚群的工作到多个复投映空间的乘积. 研究了紧旋流形上非线性增长的狄拉克方程的解并构造了藕合非线性增长的狄拉克方程组的Rabinowitz-Floer同调. 还研究了凯勒流形上狄拉克算子特征值的下界估计, 以及有非线性奇性的非局部椭圆方程与方程组的广义解的存在性.