Dirac operator, Atiyah-Singer index theory, and applications

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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.