Geometric Applications of Dirac Operator and Atiyah-Singer Index Theory

AbstractAward：DMS-1007041主要研究者：戴贤哲该提案涉及到与Dirac算子和Atiyah-Singer指标理论有关的几个几何问题。包括狄拉克算子和旋量在爱因斯坦度规稳定性研究中的应用。 PI将通过探索与Killing旋量和Sasakian-Einstein度量的联系，寻求更好地理解具有正标量曲率的爱因斯坦度量的稳定性。 另一个问题是利用局部指标理论研究热核和Bergman核，并研究它们与正则度量的关系。PI希望研究Dirac算子的谱隙和Bergman核在半正情形下的渐近展开。PI还将研究一类非紧流形（ALE空间）上的Ricci流。长期趋同问题将是主要焦点。最后，PI将研究几何不变量在度量退化下的行为，包括绝热极限和圆锥退化。爱因斯坦的广义相对论将引力几何化，引力是自然界中四种基本力之一，也是塑造我们宇宙的主导力。 爱因斯坦流形在数学和物理中起着重要的作用。 理解爱因斯坦流形的稳定性是很重要的。稳定性问题在诸如Ricci流等几何发展方程的研究中也很重要。最近的发展显示了里奇流的非凡力量。狄拉克算子及其相关的几何不变量在数学和物理的各个领域都起着重要的作用。 这个建议的目的是为了更好地理解爱因斯坦流形的稳定性，非紧流形上的里奇流，几何不变量。

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.