Research in Geometric Analysis

摘要建议：DMS 9803399主要研究人员：杰青该项目由两个相关的研究课题组成。第一个重点是发展和使用偏微分方程的非线性分析来研究各种问题中奇点的形成，这些奇点主要来自于理论物理中的微分几何环境，包括调和映射、Yang-Mills-Higgs理论、调和映射流和曲率驱动流。主要的方法是基于缩放方法。我现在正在研究的一个问题是理解解的紧致空间(调和映射)的更精细的结构。这是一个理解奇点的形成和性质的基本问题。第二部分是几何与分析相互作用的考察。中心问题是用等周常数来理解尖锐的Sobolev型不等式中的最佳常数，以及了解局部解析量如何与整体几何甚至拓扑相联系。一种方法是研究与几何有关的各种自然微分算子的谱理论，特别是共形协变几何微分算子。自然界中奇点的形成和性质是人类非常感兴趣的问题。将在拟议的项目中研究的那种奇点对材料科学和气象学特别感兴趣。例如，Ginzburg-Landau问题中奇点的形成和性质是杨-Mills-Higgs理论的一个特例，它被用来研究超导和其他超流体现象。一个人用来描述系统的尺度的作用被证明是非常重要的。这解释了，理解任何在尺度变化中存活下来的东西是非常关键的。因此，我们的主要方法是基于尺度方法。拟议项目中的第二个研究主题基本上是为了开发和理解一些数学工具，这些工具预计对第一个主题是必不可少的。拟议的项目还与加州大学伯克利分校数学系的研究生项目紧密联系在一起，计划产生有利于对这一领域感兴趣的研究生的研究活动。

AbstractProposal: DMS 9803399Principal Investigator: Jie QingThe proposed project consists of two related research topics. Thefirst is focused in the investigations in developing and usingnonlinear analysis of partial differential equations to studyformations of singularities in various problems, arising fromtheoretical physics mostly in differential geometrical settings,including harmonic maps, Yang-Mills-Higgs theory, harmonic map flows,and curvature driven flows. The main approach is based on scalingmethods. One problem that I am working on now is to understand somefiner structure of the compactified spaces of solutions (harmonicmaps). This is an essential problem for one to understand theformation and nature of singularities. The second part is theinvestigation of the interplay of geometry and analysis. The centralproblems are to understand the best constants in sharp Sobolev typeinequalities in terms of the isoperimetric constants, and tounderstand how the local analytic quantities are related to the globalgeometry even the topology. One of the approach is to study thespectral theory of various natural differential operators associatedwith the geometry, especially the conformally covariant geometricdifferential operators.The formation and nature of singularities in nature are of enormous interests to human beings. The kind of singularities that will be studied in the proposed project are particularly of great interests to the material sciences and meteorology. For instance, the formation and nature of singularities in Ginzburg-Landau problems as a particular case of the Yang-Mills-Higgs theory was proposed to study super-conductivity and other super-fluids phenomena. The effect of the scale that one uses to describe the system turns out to be very important. This explains that to understand anything that survives under changes of scales is very crucial. Therefore our main approach is based on the scaling methods. The second research topics in the proposed project is basically aiming at developing and understanding some mathematical tools that are expected to be essential for the first topics. The proposed project is also closely tied with the graduate program in the department of Mathematics at UCSC by a plan to generate research activities to the benefit of graduate student interested in this area.