Geometric Invariants and Metric Degenerations

[9704296] Dai本项目涉及光谱几何和几何不变量领域的各种问题。研究者将研究具有圆锥奇点的流形，以期在更一般的情况下定义和理解某些几何不变量，如解析扭转；研究了退化度规在绝热极限和黎曼坍缩等不同过程下的谱几何；研究几何不变量及其在手术条件下的行为。光谱几何是一个经常使用解析技术来研究几何物体的领域。这个项目特别涉及通过热方程技术来定义某些几何不变量——热方程是比较著名的偏微分方程之一，描述了热流现象，关于这个方程我们知道的很多。

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.