Mathematical Sciences: Partial Differential Equations

将继续开展研究项目， 数学分析、几何和代数的相互作用。 共形和双全纯几何中出现的问题， Fuchsian偏微分方程将是重点的 research.第一个项目是持续的合作， 在构造共形或 柯西-黎曼流形。 奇数维的完全解 在适形情况下，此时似乎可以接近。 的 问题已被简化为一个纯粹的代数问题， 不变量理论 在偶数维中的部分解也有 这项工作的最终目标是给一个 的渐近展开的显式不变描述 光滑有界严格伪凸域的Bergman核 在复杂的n维空间中。 一个相关的项目涉及应用 表示理论方法到抛物不变理论。 目标与上述相同，除了替代的 方法是使用Verma模块进行分析， 与共形结构的射流空间相同的结构。 一 在Verma模块方面的问题公式化带来了更多 结构和工具，以及提供一个大类， 测试问题，以开发方法。 第三个项目是关于存在的研究， Fuchsian微分方程的唯一性和正则性 共形上的爱因斯坦度量和调和映射所描述的 紧流形 这些方程由非线性方程组给出。 在流形上退化的椭圆型方程组 边界 作为第一个目标，工作将在寻找 具有常负Ricci的共形紧度量 曲率，具有预先指定的保形类。 这 将扩展模型的情况下，双曲度量的常数 球上定义的负曲率。这里，保形类 是球面的标准度量。//

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.//