Problems in Global Analysis

这项研究的目的是通过发展流形上的整体分析，促进分析和几何的交叉融合，同时探索数学各个领域之间有趣和可能富有成果的相互关系。该项目涉及偏微分方程、动力系统、谐波分析和其他分析中的不同主题。在本研究项目期间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）研究具有非经典正位势的Schrodinger算子特征值的渐近分布;（2）研究自旋流形上Dirac算子的等变指数;（3）证明全纯向量丛孤立奇点的可移性与Ricci曲率有关.动力系统与泛函微分方程：（1）将可积Hamilton系统约化为奇点附近的规范形;（2）发现了无穷时滞泛函微分方程存在性与稳定性的各种判据.非线性分析：（1）研究反应扩散方程解的几何结构，如斑图的形成和奇点的出现;（2）证明梯度流的渐近稳定性，与Yang-Mills泛函有关.调和分析与算子理论：（1）证明了关于严格伪凸域上调和函数的边界性态的Fatou型定理;（2）研究了算子代数的序结构与正则完备化之间的相互关系.复流形分析：研究C^2中严格伪凸Reinhardt域上的Bergman核与边界的Chern-Moser不变多项式。

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.