Dirac Operator, Eta Invariant and Applications

DMS-0405890标题：Dirac算子，ETA不变量及其应用PI：戴贤哲(加州大学圣巴巴拉分校)在这项建议中，主要研究人员研究了与Dirac算子有关的各种问题。这包括利用Dirac算子研究黎曼流形的稳定性以及Calabi-Yau流形和其他特殊完整流形上的正标量曲率度量和标量平坦度量，研究ETA不变量在共形几何和局部对称空间中的意义，研究锥形退化下的解析扭转行为，以及利用锥奇性研究奇异情况下的几何量子化。Dirac算子和相关的几何不变量在数学和物理的各个领域发挥着越来越重要的作用，它们揭示了许多关于基础空间结构的信息。这一建议旨在通过使用狄拉克算符来更好地理解保角结构和特殊度规，并探索与广义相对论中的正质量定理的联系。众所周知，爱因斯坦的广义相对论使用几何来描述重力，重力是自然界的四个基本力之一，也是决定我们宇宙大尺度结构的最重要的一个。因此，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空间和其他特殊的完整空间现在正在弦理论中扮演着重要的角色，弦理论通常被认为是“万物理论”的最佳候选者。

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''.