Geometric Invariants and Metric Degenerations

9704296戴这个项目解决了光谱几何和几何不变量领域的各种问题。研究具有锥奇性的流形，以期在更一般的环境中定义和理解某些几何不变量，如解析挠率；研究退化度量在各种过程下的谱几何，包括绝热极限和黎曼塌陷；研究几何不变量及其在外科手术下的行为。光谱几何是一个经常使用解析技术来研究几何对象的领域。这个项目特别涉及通过热方程技术定义某些几何不变量-热方程是描述热流现象的较著名的偏微分方程组之一，并且对该方程有很多了解。

9704296 Dai This project deals with a variety of problems in the area of spectral geometry and geometric invariants. The investigator is to study manifolds with conical singularity with a view towards defining and understanding certain geometric invariants such as analytic torsion in a more general setting; to study the spectral geometry of degenerating metrics under various processes including the adiabatic limit, and Riemannian collapsing; to study geometric invariants and their behavior under surgery. Spectral geometry is an area where often analytic techniques are used to study geometric objects. This project in particular deals with defining certain geometric invariants via heat equation techniques - the heat equation is among the more well-known partial differential equations, describing the heat flow phenomenon, and much is known about the equation.