Geometric Applications of Dirac Operator and Atiyah-Singer Index Theory

狄拉克算子和Atiyah-Singer指数理论的几何应用

• 批准号: 1007041
• 负责人: Xianzhe Dai
• 金额: $ 13.7万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007041&HistoricalAwards=false
• 关键词: Geometric; Applications; Dirac; Operator; Atiyah

AbstractAward: DMS-1007041Principal Investigator: Xianzhe DaiThis proposal concerns several problems in geometry that are related to Dirac operators and Atiyah-Singer index theory. It includes the use of Dirac operator and spinors in the study of stability problem for Einstein metrics. The PI will seek better understanding of the stability of Einstein metrics with positive scalar curvature by exploring the connection with Killing spinors and Sasakian-Einstein metrics. Another problem involves the study of heat kernel and Bergman kernel using local index theory technique and the study of their relation with canonical metrics. The PI would like to study the spectral gap of Dirac operators and the asymptotic expansion of Bergman kernel in the semipostive case. The PI will also study Ricci flows on a class of noncompact manifolds, the ALE spaces. The question of long time convergence will be the main focus. Finally, the PI will investigate the behavior of geometric invariants under metric degeneration, including adiabatic limit and conical degeneration. In particular, one of the applications will be the Ray-Singer conjecture for manifolds with conical singularities.Einstein's General relativity geometrizes gravity, one of the four fundamental forces in nature and the dominating one in shaping our universe. Einstein manifolds play essential role in mathematics and physics. It is important to understand the stability of Einstein manifolds. Stability issue is also important in the study of geometric evolution equations such as the Ricci flow. Recent development shows the extraordinary power of the Ricci flow. Dirac operators and related geometric invariants are playing significant and important role in diverse fields of mathematics and physics. This proposal aims for better understanding of the stability of Einstein manifolds, of the Ricci flow on noncompact manifolds, and of geometric invariants.

摘要奖项：DMS-1007041 首席研究员：戴贤哲该提案涉及与狄拉克算子和 Atiyah-Singer 指数理论相关的几个几何问题。它包括在研究爱因斯坦度量的稳定性问题时使用狄拉克算子和旋量。 PI 将通过探索与 Killing 旋量和 Sasakian-Einstein 度量的联系来更好地理解具有正标量曲率的爱因斯坦度量的稳定性。 另一个问题涉及使用局部指数理论技术研究热核和伯格曼核以及它们与规范度量的关系的研究。 PI 希望研究狄拉克算子的谱间隙和半正定情况下伯格曼核的渐近展开。 PI 还将研究一类非紧流形（ALE 空间）上的 Ricci 流。长时间收敛问题将是主要焦点。最后，PI 将研究度量退化下几何不变量的行为，包括绝热极限和圆锥退化。特别是，应用之一将是具有圆锥奇点的流形的雷-辛格猜想。爱因斯坦的广义相对论将引力几何化，引力是自然界的四种基本力之一，也是塑造我们宇宙的主导力。 爱因斯坦流形在数学和物理学中起着至关重要的作用。 了解爱因斯坦流形的稳定性很重要。稳定性问题在几何演化方程（如里奇流）的研究中也很重要。最近的发展显示了利玛窦流的非凡力量。狄拉克算子和相关的几何不变量在数学和物理的各个领域中发挥着重要作用。 该提案旨在更好地理解爱因斯坦流形、非紧流形上的里奇流以及几何不变量的稳定性。