Analytic Torsion, Conical Singularity and Geometric Applications

解析扭转、圆锥奇异性和几何应用

• 批准号: 1611915
• 负责人: Xianzhe Dai
• 金额: $ 15.32万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611915&HistoricalAwards=false
• 关键词: Analytic; Torsion; Conical; Singularity; Geometric

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.

这个项目涉及Atiyah-Singer指数理论中的问题和与Dirac算子和/或奇异空间有关的几个几何问题。Atiyah-Singer指数理论是数学的里程碑式的成果之一，它统一了数学中的几个重要发展和结果，在数学和物理学中有许多引人注目的应用。这些问题包括奇异空间的几何不变量的研究，在爱因斯坦度量的稳定性问题的研究中使用狄拉克算子，以及热核和伯格曼核的研究以及它们与“最佳”度量的关系。解析挠率在塞伯格-威滕理论、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。Cheeger-Muller定理在所有这些方面都发挥了重要作用。PI，沿着他的合作者和研究生，想证明Ray-Singer猜想/Cheeger-Muller定理的流形与圆锥奇点，这应该有助于我们理解更复杂的奇点。几何学中最重要和最广泛研究的领域之一是正则度量的研究，理解与变分问题相关的稳定性问题是很重要的。PI和他的学生将寻求更好地理解爱因斯坦度量的稳定性与正标量曲率探索与锥和锥奇点。Bergman核和圆锥奇点是最近Yau-Tian-唐纳森猜想的壮观解中的重要组成部分。该方案的目的是解决Ray-Singer猜想的锥奇点流形，更好地理解的变分结构的全标量曲率泛函，Bergman核，和Ricci流的非紧流形。它还探讨了与正质量定理的联系。