Analytic Torsion, Conical Singularity and Geometric Applications

本项目涉及Atiyah-Singer指标理论中的问题以及几何中与狄拉克算子和/或奇异空间有关的几个问题。Atiyah-Singer指标理论是数学中具有里程碑意义的成果之一，它统一了数学中几个重要的发展和结果，并在数学和物理学中得到了许多显著的应用。这些问题包括奇异空间的几何不变量的研究，狄拉克算子在研究爱因斯坦度量稳定性问题中的应用，热核和伯格曼核的研究以及它们与“最佳”度量的关系。该项目的一部分涉及研究生。解析扭转在Seiberg-Witten理论、双曲几何和镜像对称中发现了许多有趣的联系和重要的应用。契格-穆勒定理在所有这些中都扮演了重要的角色。PI，以及他的合作者和研究生，想要证明具有圆锥奇点的流形的Ray-Singer猜想/Cheeger-Muller定理，这将有助于我们理解更复杂的奇点。正则度量是几何中最重要和被广泛研究的领域之一，理解与变分问题相关的稳定性问题是很重要的。PI和他的学生将通过探索与锥和锥奇点的联系来更好地理解具有正标量曲率的爱因斯坦度规的稳定性。伯格曼核和圆锥奇点已经成为了最近关于you - tian - donaldson猜想的精彩解答的重要组成部分。本文旨在解决具有圆锥奇点的流形的Ray-Singer猜想，更好地理解非紧化流形上的全标量曲率泛函、Bergman核和Ricci流的变分结构。它还探讨了与正质量定理的联系。

This project concerns problems in Atiyah-Singer index theory and several problems in geometry that are related to Dirac operators and/or singular spaces. Atiyah-Singer index theory is one of the landmark results of mathematics that unifies several important developments and results in mathematics and has found many remarkable applications in mathematics and physics. The problems include the study of geometric invariants for singular spaces, the use of Dirac operators in the study of stability problems for Einstein metrics, and the study of the heat kernel and the Bergman kernel as well as their relation with the "best" metrics. Part of the project involves graduate students.Analytic torsion has found many interesting connections and significant applications, in Seiberg-Witten theory, hyperbolic geometry, and mirror symmetry. The Cheeger-Muller theorem has played an important role in all of these. The PI, along with his collaborator and graduate students, would like to prove the Ray-Singer conjecture/Cheeger-Muller theorem for manifolds with conical singularities which should help us understand more complicated singularities. One of the most important and extensively studied areas of geometry is the study of canonical metrics and it is important to understand the stability issue associated with variational problems. The PI and his student will seek better understanding of the stability of Einstein metrics with positive scalar curvature by exploring the connection with cones and conical singularity. Bergman kernels and conical singularity have been essential ingredients in the recent spectacular solutions of the Yau-Tian-Donaldson conjecture. This proposal aims for the solution of the Ray-Singer conjecture for manifolds with conical singularity, better understanding of the variational structure of the total scalar curvature functional, the Bergman kernel, and the Ricci flow on noncompact manifolds. It also explores the connection with positive mass theorems.