Dirac operator, Atiyah-Singer index theory, and applications

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。几何学最重要和最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富有成果的方法。了解与变分问题相关的稳定性问题非常重要。稳定性问题在几何流的研究中也很重要。这里的目标之一是使用狄拉克算子研究爱因斯坦度量和卡拉比-丘流形上的正标量曲率和标量平坦度量的稳定性问题，并阐明所谓的里奇平坦流形的正质量问题。它还将开发局部指数理论技术来研究热核和伯格曼核及其与规范度量的联系。特别是，研究人员（与合作者）希望寻找唐纳森定理的类比，即轨道折叠，该定理将具有恒定标量曲率的卡勒度量的存在性与代数几何稳定性概念联系起来。该提案的另一个重点是研究一类非紧流形（ALE 空间）上的 Ricci 流，并探索与著名的正质量定理的联系。最后研究者将研究几何不变量如eta不变量和解析挠率的意义和退化问题。最近的发展显示了几何流（例如里奇流）的非凡力量。狄拉克算子和相关的几何不变量受物理学的启发并源自 Atiyah-Singer 指数理论，在数学和物理的各个领域中发挥着重要的作用。它们揭示了许多有关底层空间的拓扑、几何和分析结构的信息。该提案旨在更好地理解重要几何泛函和几何流的变分结构，在研究几何结构的模空间和卡拉比-丘流形上的特殊度量时使用几何不变量。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，也是塑造宇宙的主导力。对质量和动量的理解对于我们最终理解宇宙至关重要。卡拉比-丘流形和特殊度量在物理学中发挥着重要作用。拟议的活动将对所有这些方向产生影响。该提案还将产生教育影响，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.