Geometric Analysis on Complete Manifolds

完整流形的几何分析

• 批准号: 0801988
• 负责人: Peter Li
• 金额: $ 37.08万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801988&HistoricalAwards=false
• 关键词: Geometric; Analysis; Complete; Manifolds

Abstract: "Geometric Analysis on Complete Manifolds" by Peter Li The goal of this project is to obtain further understanding of the underlying geometric and topological structure of an unbounded (infinite) geometric object (manifold). The techniques being utilized will involve understanding solutions of some partial differential equations on the manifold in terms of the curvature and other geometric invariants. On the other hand, knowledge of the solutions of these partial differential equations will yield additional information on the geometric and topological structure of the manifold. This general theme is the essence of this proposal and the development of various techniques involved in this project will be very useful in future studies of complete, noncompact manifolds. This line of investigation will yield direct implications to the theory of partial differential equations governing the behavior of many physical models and biological models. It is also related to many engineering problems, such as, liquid crystals, heat transfer, and imaging. In a broader point of view, the proposed project will yield further understand of a geometric object. The shape of and structures of geometric objects are important related to many scientific fields. Examples of these are the structure of black holes and worm holes in physics, the structure of DNA given by the double helix in biology, the structure of molecules in chemistry, and behavior of liquid crystal in engineering. This Principal Investigator will also contribute to the national needs of producing more mathematicians by involving graduate students and postdoctoral scholars in his research. Their involvement may be in the form of direct collaborations, advising, and learning new material through seminars. This process will be an important component in training the new generation of mathematicians.

摘要：“完整流形的几何分析”作者：Peter Li 该项目的目标是进一步了解无界（无限）几何对象（流形）的基础几何和拓扑结构。 所使用的技术将涉及理解流形上一些偏微分方程在曲率和其他几何不变量方面的解。 另一方面，了解这些偏微分方程的解将产生有关流形的几何和拓扑结构的附加信息。 这个总体主题是本提案的精髓，该项目中涉及的各种技术的开发将在未来完整、非紧流形的研究中非常有用。 这一研究方向将对控制许多物理模型和生物模型行为的偏微分方程理论产生直接影响。 它还涉及许多工程问题，例如液晶、传热和成像。从更广泛的角度来看，拟议的项目将产生对几何对象的进一步理解。 几何物体的形状和结构与许多科学领域都有重要关系。 这些例子包括物理学中的黑洞和虫洞的结构、生物学中双螺旋给出的DNA结构、化学中的分子结构以及工程中的液晶行为。 这位首席研究员还将通过让研究生和博士后学者参与他的研究，为国家培养更多数学家的需求做出贡献。 他们的参与可能是直接合作、提供建议以及通过研讨会学习新材料的形式。 这一过程将成为培养新一代数学家的重要组成部分。