Mathematical Sciences: Some Problems in Geometric Analysis

保罗·杨将致力于几何学中的三个核心问题 分析.其中第一个涉及调查哪些 球面上的函数是高斯曲率函数， 与标准度量一致的度量。第二 是找出在多大程度上， 负曲率的表面类似于球。在决赛中 问题的目的是研究分析和几何 用作抽样时提供可靠数据的区域的性质 地区 刻画高斯分布的尼伦伯格问题 曲率函数是一个微妙的鉴于必要的 卡兹丹和华纳观察到的情况。杨将集中精力 对只允许两个人的职能进行调查的努力 临界点。在统一化问题的背景下， 杨将调查的猜想，普遍覆盖的 负截面曲率的Kahler流形是 复域中有界域的双全纯等价 n空间庞皮尤问题将根据 两个对偶的无穷本征函数列的存在性 超定边界条件系统。

Paul Yang will work on three central problems in geometric analysis. The first of these involves investigating which functions on the sphere are Gaussian curvature functions of metrics conformally equivalent to the standard metric. The second is to find out to what extent the universal cover of certain surfaces of negative curvature resembles the ball. In the final question the objective is to study the analytic and geometric nature of regions which give reliable data when used as sampling regions. The Nirenberg problem concerning characterizing Gaussian curvature functions is a delicate one in view of the necessary condition observed by Kazdan and Warner. Yang will focus his efforts on an investigation of functions admitting only two critical points. In the context of the uniformization problem Yang will investigate the conjecture that the universal cover of a Kahler manifold of negative sectional curvature is biholomorphically equivalent to a bounded domain in complex n-space. The Pompieu problem will be reformulated in terms of the existence of an infinite sequence of eigenfunctions to two dual systems of overdetermined boundary conditions.