Harmonic Forms and Topology of Complete Manifolds

DMS-0404817P.I.：王家平，明尼苏达大学双城分校题目：完备流形的调和形式与拓扑研究本文将完备非紧流形的分析与拓扑学联系起来。特别地，我们建议寻找De Rham-Hodge理论的推广，并将Hodge拉普拉斯的谱与流形的拓扑联系起来。这个项目的主要目的是描述物体的一些大尺度结构，这些结构是无界表面的高维模拟。方法是检查这类物体上自然定义的能量的物理稳态。这些结果和技术有望用于描述来自物理学、生物学和其他科学分支的更复杂的系统。

DMS-0404817P.I.: Jiaping Wang, University of Minnesota-Twin CitiesTitle: Harmonic forms and topology of complete manifoldsABSTRACTThis proposed project is about relating the analysis and topology on complete noncomapct manifolds. In particular, it is proposed to searchfor an extension of the de Rham-Hodge theory, and to relate the spectrum ofthe Hodge Laplacian to the topology of manifolds.The main objective of this project is to describe some large scalestructuresof objects which are high dimensional analogue of the unbounded surfaces.The approach is to inspect the physical steady states of a naturallydefinedenergy on such objects. The results and techniques are expected to beusefulin describing more complicated systems arising from physics, biology andotherbranches of science.