Dirac Operator, Eta Invariant and Applications

狄拉克算子、Eta 不变量及其应用

• 批准号: 0405890
• 负责人: Xianzhe Dai
• 金额: --
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405890&HistoricalAwards=false
• 关键词: Dirac; Operator; Eta; Invariant; Applications

DMS-0405890Title: Dirac operator, eta invariant and applicationsPI: Xianzhe Dai (University of California, Santa Barbara)ABSTRACTIn this proposal the principal investigator studies various questions related to Dirac operator. This includes the use of Dirac operator in the study of stability of Riemannian manifolds and positive scalar curvature metrics and scalar flat metrics on Calabi-Yau manifolds and other special holonomy manifolds,the study of the significance of eta invariant in conformal geometry and locally symmetric spaces, the behavior of analytic torsion under conical degeneration and the use of conical singularity in the study of geometric quantization in the singular case.Dirac operator and related geometric invariants are playing more and more fundamental and important role in diverse fields of mathematics and physics.They reveal much about the structures of the underlying spaces. This proposal aims for better understanding of conformal structures and special metrics through the use of Dirac operator and explores the connection with the positive mass theorems in the general relativity. As is well known, Einstein's general relativity uses geometry to describe gravity, one of the four fundamental forces in nature, and the most important one in determining the large scale structure of our universe. The understanding of mass and momentum is thus of crucial importance in our ultimate understanding of the universe. Calabi-Yau spaces and other special holonomy spaces are now playing fundamental role in the string theory, generally considered the best candidate for the ``theory of everything''.

DMS-0405890题目：狄拉克算子、eta不变量及其应用研究者：戴贤哲（加州大学，圣巴巴拉）摘要在这个提案中，主要研究者研究了与狄拉克算子有关的各种问题。这包括Dirac算子在黎曼流形稳定性研究中的应用，以及Calabi-Yau流形和其他特殊完整流形上的正数量曲率度量和数量平坦度量，共形几何和局部对称空间中eta不变量的意义的研究，解析挠率在圆锥退化下的行为以及圆锥奇异性在奇异几何量子化研究中的应用Dirac算子及其相关的几何不变量在数学和物理学的各个领域中起着越来越重要的作用，它们揭示了许多关于底层空间结构的信息。这个建议的目的是更好地理解共形结构和特殊的度规，通过使用狄拉克算子，并探讨与广义相对论的正质量定理的联系。众所周知，爱因斯坦的广义相对论使用几何来描述引力，引力是自然界中四种基本力之一，也是决定我们宇宙大尺度结构的最重要的力。因此，对质量和动量的理解对于我们最终理解宇宙至关重要。卡-丘空间和其他特殊的完整空间现在在弦理论中扮演着重要的角色，通常被认为是“万物理论”的最佳候选者。