Problems in Global Analysis

这项研究的目的是通过发展流形上的全局分析，促进分析和几何的交叉融合，同时探索不同数学领域之间有趣和潜在富有成效的相互关系。该项目涉及偏微分方程组、动力系统、调和分析和其他不同的分析主题。偏微分方程及其在几何中的应用：(1)研究具有非经典正势的薛定谔算子本征值的渐近分布；(2)研究自旋流形上Dirac算子的等变指数；(3)证明全纯向量丛的孤立奇点与其Ricci曲率有关的可去性。动力系统与泛函微分方程：(1)可积哈密顿系统在奇点附近的正规化；(2)无穷时滞泛函微分方程存在性和稳定性的各种判据的发现。非线性分析：(1)研究解的几何结构，如图案的形成和反应扩散方程奇点的出现；(2)梯度流渐近稳定性的证明，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边界性质的Fatou型定理；(2)研究了算子代数的序结构与正则完备性之间的关系。复流形分析：C^2中严格伪凸Reinhardt域上的Bergman核与边界的Chern-Moser不变多项式有关的研究。

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.