Dirac Operator, Eta Invariant and Applications

题目：Dirac算子、eta不变量及其应用[j]：戴宪哲（美国加州大学圣巴巴拉分校）摘要：本课题主要研究与Dirac算子相关的各种问题。这包括Dirac算子在研究riemann流形、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面度量的稳定性中的应用，在共形几何和局部对称空间中不变量的意义的研究，圆锥退化下解析扭转的行为以及圆锥奇点在奇异情况下几何量化研究中的应用。狄拉克算子及其相关几何不变量在数学和物理的各个领域中发挥着越来越重要的基础作用。它们揭示了很多底层空间的结构。本文旨在通过狄拉克算子更好地理解共形结构和特殊度规，并探讨其与广义相对论中正质量定理的联系。众所周知，爱因斯坦的广义相对论用几何学来描述引力，引力是自然界四种基本力之一，也是决定我们宇宙大尺度结构的最重要的一种力。因此，对质量和动量的理解对于我们对宇宙的最终理解至关重要。Calabi-Yau空间和其他特殊的完整空间在弦理论中起着基础性的作用，弦理论通常被认为是“万有理论”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''.