Problems in Global Analysis

这项研究的目的是通过对流形的整体分析的发展，促进分析和几何的交叉发展，同时探索数学各个领域之间有趣和潜在的富有成效的相互关系。该项目涉及偏微分方程、动力系统、谐波分析和其他不同的分析主题。在这个研究项目期间所获得的结果可以总结如下。偏微分方程及其在几何中的应用：(1)非经典正势Schrodinger算子特征值的渐近分布研究；(2)自旋流形上Dirac算子的等变指标研究；(3)与Ricci曲率有关的全纯向量束孤立奇点可移性的证明。动力系统与泛函微分方程：(1)奇异点附近可积哈密顿系统的正规化；(2)发现了无穷时滞泛函微分方程存在性和稳定性的各种判据。非线性分析：(1)研究反应扩散方程解的几何结构，如模式的形成和奇点的出现；(2)与Yang-Mills函数相关的梯度流渐近稳定性的证明。调和分析与算符理论：(1)关于严格伪凸域上调和函数边界行为的一个fatou型定理的证明；(2)算子代数的序结构与正则补全的相互关系的研究。复流形的分析：C^2上严格伪凸Reinhardt区域上Bergman核与边界的chen - moser不变多项式的关系。

The research had the purpose of contributing to the cross-fertilisation of analysis and geometry, through the development of global analysis on manifolds, and exploring at the same time interesting and potentially fruitful interrelations between various fields of mathematics.The project involved work in partial differential equations, dynamical systems, harmonic analysis and other diverse topics in analysis.The results obtained during the period of this research project can be summarised as follows.1. Partial differential equations and their applications to geometry : (1) study of asymptotic distribution of eigenvalues for Schrodinger operators with non-classical positive potentials ; (2) study of equivariant index of Dirac operators on spin manifolds ; (3) proof of the removability of isolated singularities for holomorphic vector bundles in connection with their Ricci curvature.2. Dynamical system and functional differential equations : (1) reduction of integrable hamiltonian system to the normal form near singular points ; (2) discovery of various criteria for the existence and stability of functional differential equations with infinite delay.3. Nonlinear analysis : (1) study on geometric structures of solutions, such as pattern formation and the appearance of singularities for reaction-diffusion equations ; (2) proof of asymptotic stability of gradient flow, associated with the Yang-Mills functional.4. Harmonic analysis and operator theory : (1) proof of a Fatou-type theorem concerning the boundary behavior of harmonic functions on strictly pseudo-convex domains ; (2) study on the interrelation between the order structure and the regular completion of operator algebras.5. Analysis on complex manifolds : study on the Bergman kernels on strictly pseudo-convex Reinhardt domains in C^2 in connection with the Chern-Moser invariant polynomials of the boundaries.