Function Theory and Topology of Complete Manifolds

本课题主要研究完全流形的对称性与流形上各种分析对象的行为之间的相互作用。更具体地说，他提出将覆盖群的结构和大小与多项式生长调和函数和覆盖空间上的谱数据联系起来。该项目的主要目标包括通过解析方法推导（非紧）流形的拓扑信息，并进一步理解空间的对称性如何影响各种几何微分方程的解的行为，这是几何分析的主要主题之一。本课题旨在研究具有部分对称性的微分方程解的行为与底层空间的整体结构和形状之间的关系。预计这将增强我们对几何分析结构的理解，并对包括工业应用在内的各个领域产生直接影响。

DMS-0072181Jiaping WangIn this project, the principal investigator proposes to study the interactions between the symmetries of a complete manifold and the behaviors of various analytical objects on it. More specifically, he proposes to link the structure and the size of the covering group with the polynomial growth harmonic functions and the spectral data on the covering space. The main objectives of this project include deriving topological information about a (noncompact) manifold via analytical approaches and furthering the understanding of how the symmetries of the space affect the behavior of solutions to various geometric differential equations, which is one of the main themes of the geometric analysis. This project is intended to study how the behavior of the solutions to differential equations is related to the global structure and shape of the underlying space with partial symmetries. It is expected to both enhance our understanding of the geometric-analytic structure and havedirect implications to various fields including possible applications in industry.