Mathematical Sciences: Partial Differential Equations

将继续开展涉及数学分析、几何和代数相互作用的研究项目。在保形几何和生物全纯几何以及Fuchsian偏微分方程中出现的问题将是研究的重点。第一个项目是一个持续的合作，旨在构建一个共形或柯西-黎曼流形的所有标量不变量。在共形情况下，奇维的完全解似乎是可以得到的。这个问题在不变量理论中被简化为一个纯粹的代数问题。偶维的部分解也得到了。本文的最终目的是给出复n维空间中光滑有界严格伪凸区域的Bergman核渐近展开的显式不变描述。一个相关的项目涉及到表示理论方法在抛物不变量理论中的应用。目的与上述相同，除了另一种方法是使用Verma模块进行分析，该模块与保形结构的射流空间具有相同的结构。用Verma模块表述问题带来了更多的结构和工具，并提供了一大类测试问题，可以根据这些问题开发方法。第三个项目是研究共形紧流形上由爱因斯坦度量和调和映射描述的Fuchsian微分方程的存在性、唯一性和正则性。这些方程是由在流形边界处退化的椭圆型非线性系统给出的。作为第一个目标，我们将寻找具有恒定负里奇曲率的共形紧度量，并预先指定共形类。这将扩展在球上定义的恒定负曲率双曲度规的模型情况。这里，共形类是球体的标准度规

Work will continue on research projects involving the interplay of mathematical analysis, geometry and algebra. Questions arising in conformal and biholomorphic geometry and Fuchsian partial differential equations will be the focus of the research. The first project is a continuing collaboration aimed at the construction of all scalar invariants of a conformal or Cauchy- Riemann manifold. A complete solution for odd dimensions in the conformal case appears to be accessible at this time. The problem has been reduced to a purely algebraic problem in invariant theory. A partial solution in even dimensions has also been found. The ultimate goal of this work is to give an explicit invariant description of the asymptotic expansion of the Bergman kernel of a smooth bounded strictly pseudoconvex domain in complex n-dimensional space. A related project involves the application of representation-theoretic methods to parabolic invariant theory. The objective is the same as above except that an alternate approach is to be analyzed using Verma modules which have the same structure as the space of jets of a conformal structure. A formulation of the problem in terms of Verma modules brings more structure and tools to bear as well as providing a large class of test problems on which to develop methods. The third project concerns the study of existence, uniqueness and regularity of Fuchsian differential equations described by Einstein metrics and harmonic maps on conformally compact manifolds. These equations are given by nonlinear systems of elliptic type which degenerate at the manifold boundary. As a first goal, work will be done in finding conformally compact metrics having constant negative Ricci curvature, with the conformal class specified in advance. This would extend the model case of the hyperbolic metric of constant negative curvature defined on a ball. Here, the conformal class is the standard metric of the sphere.//