Research in Geometric Analysis

几何分析研究

• 批准号: 9803399
• 负责人: Jie Qing
• 金额: $ 7.02万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803399&HistoricalAwards=false
• 关键词: Research; Geometric; Analysis

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.

摘要提案：DMS 9803399 项目负责人：杰青该项目由两个相关的研究课题组成。 第一个重点是开发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，这些问题主要来自微分几何背景下的理论物理，包括调和映射、杨-米尔斯-希格斯理论、调和映射流和曲率驱动流。 主要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压缩空间（调和图）的某种更精细的结构。 这是理解奇点的形成和性质的一个重要问题。 第二部分是几何与分析相互作用的研究。 中心问题是理解等周常数的尖锐索博列夫型不等式中的最佳常数，并理解局部解析量如何与全局几何甚至拓扑相关。 其中一种途径是研究与几何相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。自然界奇点的形成和性质引起了人类的巨大兴趣。 拟议项目中将研究的奇点类型对材料科学和气象学尤其感兴趣。 例如，金兹堡-朗道问题中奇点的形成和性质作为杨-米尔斯-希格斯理论的一个特例，被提出来研究超导和其他超流体现象。 事实证明，用来描述系统的规模的效果非常重要。 这说明了解任何在尺度变化下生存的事物是非常重要的。 因此，我们的主要方法是基于缩放方法。 拟议项目中的第二个研究主题基本上旨在开发和理解一些数学工具，这些工具预计对第一个主题至关重要。 拟议的项目还与 UCSC 数学系的研究生项目密切相关，计划开展研究活动，以造福于对该领域感兴趣的研究生。