Dirac operator, Atiyah-Singer index theory, and applications

这一建议涉及与狄拉克算子和Atiyah-Singer指数理论有关的几个几何问题。几何学中最重要和最广泛研究的领域之一是对规范度规的研究。几何变分问题和几何流动一直是研究正则度量的两种最有成效的方法。理解与变分问题相关的稳定性问题是很重要的。稳定性问题也是几何流研究中的一个重要问题。本文的目的之一是利用Dirac算子研究Calabi-Yau流形上的爱因斯坦度量、正标量曲率度量和标量平坦度量的稳定性问题，并阐明Ricci平坦流形的所谓正质量问题。它还将发展局部指数理论技术来研究热核和Bergman核以及它们与正则度量的关系。具体地说，研究者(与合作者)想要寻找与唐纳森定理的类比，将具有常数量曲率的Kahler度量的存在性与代数几何稳定性概念联系起来。该方案的另一个重点是研究一类非紧流形--ALE空间上的Ricci流，并探索它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的意义和退化问题，如ETA不变量和解析挠率。最近的发展显示了几何流的非凡力量，例如Ricci流。狄拉克算子和相关的几何不变量是受物理学启发，由Atiyah-Singer指数理论发展而来的，在数学和物理的各个领域中发挥着重要的作用。它们揭示了许多关于底层空间的拓扑、几何和解析结构。这一建议旨在更好地理解重要几何泛函和几何流的变分结构，利用几何不变量来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，重力是空间曲率的表现形式。重力是自然界的四种基本力量之一，也是塑造我们宇宙的主导力量。对质量和动量的理解在我们对宇宙的最终理解中至关重要。Calabi-Yau流形和特殊度规在物理学中起着至关重要的作用。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.