Dirac operator, Atiyah-Singer index theory, and applications

本文讨论了几何中与狄拉克算子和阿蒂亚-辛格指数理论有关的几个问题。几何中最重要和被广泛研究的领域之一是规范度量的研究。几何变分问题和几何流是研究规范度量的两种最有成果的方法。理解与变分问题相关的稳定性问题是很重要的。稳定性问题在几何流动的研究中也很重要。我们的目标之一是利用狄拉克算子研究爱因斯坦度规的稳定性问题以及Calabi-Yau流形上的正标量曲率和标量平坦度规并阐明所谓的Ricci平坦流形的正质量问题。在热核和伯格曼核及其与规范度量的关系的研究中，发展了局部指标理论技术。特别是，研究者（与合作者）想要寻找一个类似的，对于轨道，Donaldson的定理，与常数标量曲率的Kahler度量的存在性和代数几何的稳定性概念有关。本提案的另一个重点是研究一类非紧流形（ALE空间）上的Ricci流，并探讨其与著名的正质量定理的联系。最后研究几何不变量的意义和退化问题，如不变量和解析扭转。最近的发展显示了几何流（如利玛窦流）的非凡力量。狄拉克算子及相关几何不变量是受物理学启发，由Atiyah-Singer指标理论衍生而来，在数学和物理的各个领域发挥着重要的作用。它们揭示了底层空间的拓扑结构、几何结构和解析结构。本文旨在更好地理解重要几何泛函和几何流的变分结构，利用几何不变量研究几何结构的模空间和Calabi-Yau流形上的特殊度量。它还探讨了与正质量定理的联系。根据爱因斯坦的广义相对论，引力是空间曲率的表现。引力是自然界四种基本力之一，也是塑造我们宇宙的主导力量。对质量和动量的理解对于我们对宇宙的最终理解至关重要。Calabi-Yau流形和特殊度量在物理学中起着重要的作用。拟议的活动将对所有这些方向产生影响。该提案还将对教育产生影响，因为它涉及研究生和博士后。

This proposal concerns several problems in geometry that are related to Dirac operators and Atiyah-Singer index theory. One of the most important and extensively studied areas of geometry is the study of canonical metrics. Geometric variational problems and geometric flows have been two of the most fruitful approaches to the study of canonical metrics. It is important to understand the stability issue associated to variational problems. Stability issue is also important in the study of geometric flows. One of the goals here is to use the Dirac operator in the study of stability problems for Einstein metrics and positive scalar curvature and scalar flat metrics on Calabi-Yau manifolds and shed light on the so called positive mass problem for Ricci flat manifolds. It will also develop local index theory technique in the study of heat kernel and Bergman kernel and their connection with canonical metrics. In particular, the investigator (with collaborators) would like to search for an analogue, for orbifolds, of Donaldson's theorem relating the existence of Kahler metrics with constant scalar curvature with algebro-geometric notions of stability. Another focus of the proposal is the study of Ricci flow on a class of noncompact manifolds, the ALE spaces, and explores the connection with the famous Positive Mass Theorem. Finally the investigator will study the significance and degeneration problems of geometric invariants such as eta invariant and analytic torsion. Recent development shows the extraordinary power of the geometric flows such as the Ricci flow. Dirac operators and related geometric invariants, inspired by Physics and coming out of the Atiyah-Singer index theory, are playing significant and important role in diverse fields of mathematics and physics. They reveal much about the topological, geometric and analytic structures of the underlying space. This proposal aims for better understanding of the variational structure of important geometric functionals and geometric flows, the use of geometric invariants in studying moduli spaces of geometric structures and special metrics on Calabi-Yau manifolds. It also explores the connection with positive mass theorems. According to Einstein's general relativity, gravity is the manifestation of curvature of the space. Gravity is one of the four fundamental forces in nature and the dominating one in shaping our universe. The understanding of mass and momentum is of crucial importance in our ultimate understanding of the universe. Calabi-Yau manifolds and special metrics play essential role in physics. The proposed activities would have impact on all these directions. The proposal will also have educational impact as it involves graduate students and post-doctors.