Dirac Operator, Eta Invariant and Applications

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DMS-0405890Title: Dirac operator, eta invariant and applicationsPI: Xianzhe Dai (University of California, Santa Barbara)ABSTRACTIn this proposal the principal investigator studies various questions related to Dirac operator. This includes the use of Dirac operator in the study of stability of Riemannian manifolds and positive scalar curvature metrics and scalar flat metrics on Calabi-Yau manifolds and other special holonomy manifolds,the study of the significance of eta invariant in conformal geometry and locally symmetric spaces, the behavior of analytic torsion under conical degeneration and the use of conical singularity in the study of geometric quantization in the singular case.Dirac operator and related geometric invariants are playing more and more fundamental and important role in diverse fields of mathematics and physics.They reveal much about the structures of the underlying spaces. This proposal aims for better understanding of conformal structures and special metrics through the use of Dirac operator and explores the connection with the positive mass theorems in the general relativity. As is well known, Einstein's general relativity uses geometry to describe gravity, one of the four fundamental forces in nature, and the most important one in determining the large scale structure of our universe. The understanding of mass and momentum is thus of crucial importance in our ultimate understanding of the universe. Calabi-Yau spaces and other special holonomy spaces are now playing fundamental role in the string theory, generally considered the best candidate for the ``theory of everything''.