Geometric Analysis on Complete Manifolds

摘要：“Geometric Analysis on Complete Manifolds”作者Peter Li本项目的目标是进一步了解无界（无限）几何对象（流形）的基本几何和拓扑结构。 所使用的技术将涉及理解的曲率和其他几何不变量方面的流形上的一些偏微分方程的解决方案。 另一方面，这些偏微分方程的解的知识将产生关于流形的几何和拓扑结构的附加信息。 这个总的主题是这个建议的本质，在这个项目中所涉及的各种技术的发展将是非常有用的，在未来的研究完全，非紧流形。 这条线的调查将产生直接影响的理论偏微分方程的行为，许多物理模型和生物模型。 它还涉及到许多工程问题，如液晶，传热和成像。从更广泛的角度来看，所提出的项目将产生对几何对象的进一步理解。 几何对象的形状和结构与许多科学领域都有着重要的关系。 这些例子是物理学中黑洞和虫洞的结构，生物学中由双螺旋结构给出的DNA结构，化学中分子的结构，以及工程中液晶的行为。 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，为培养更多数学家的国家需求做出贡献。 他们的参与形式可能是直接合作、提供咨询和通过研讨会学习新材料。 这一过程将是培养新一代数学家的重要组成部分。

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.