Mathematical Sciences: Spectral Invariants of Non-Simply-Connected Manifolds

该奖项将支持三个相关领域的研究：覆盖空间上由热核导出的非单连通流形的拓扑不变量，谱不变量的高签名扩张，以及非交换空间上的黎曼几何。最后一个领域与粒子理论和数学物理的数学方面有关。在紧致曲面或流形上，局部信息和全局拓扑信息之间的关系是相当清楚的。非紧凑或无界情况下的情况要困难得多。这项研究试图在非紧致空间与紧致空间以精确的方式相关的情况下理解这种关系。

This award will support research in three related areas: topological invariants of non-simply-connected manifolds derived from heat kernels on covering spaces, higher signature extensions of spectral invariants, and Riemannian geometry on noncommutative spaces. The last area is connected with mathematical aspects of particle theory and mathematical physics. The relationship between local information and global topological information is fairly well understood on compact surfaces or manifolds. The situation in the noncompact or unbounded case is much more difficult. This research involves trying to understand this relationship in cases in which the noncompact space is related to a compact space in a precise manner.