Metric Differential Geometry and Mathematical Gravity

AbstractAward：DMS-0405906首席研究员：Gregory J. Galloway引力理论的最新发展，无论是理论上的还是实验上的，都引起了人们对渐近反德西特时空和渐近德西特时空研究的兴趣。在这个项目中，洛伦兹几何和黎曼几何的技术将被用来解决与这种时空有关的某些问题，以及与黑洞理论的新发展有关的一些问题。本项目的具体目标之一是继续发展一种方法来建立渐近双曲黎曼流形（具有球面共形无穷大）的正质量，而不需要自旋假设。这种方法利用了维滕和姚在AdS/CFT对应的工作中引入的膜作用量，也可以用于解决无穷远处具有环形拓扑的渐近反德西特时空的一些正质量问题。PI还计划继续调查独特性的奇异性现象，以及奇异性在渐近德西特时空中的发展。 黑洞的经典定义，虽然在数学上是优雅的，但在实践中（特别是在数值研究中）是有问题的，因为它需要知道时空的全部未来演化，直到无穷大。 作为一种替代，Ashtekar和Krishnan引入了一种新的，完全非线性和动力学的方法来研究黑洞，基于准局部的概念的adjactic视界。本项目的另一个目的是调查有关发生的一些开放性问题，即，动力学视界的存在性、唯一性和位置。现代引力理论本质上是几何的。引力场和其他场，黑洞和相关的天体，都可以用几何方法来描述和分析。更一般地说，这个项目涉及从这个几何观点，利用空间的数学理论黎曼几何和时空的数学理论洛伦兹几何的工具，研究当前科学感兴趣的引力的某些特征。这些理论提供了一种研究时空宇宙的三个基本方面之间关系的方法：曲率（即，空间或时空的弯曲），拓扑（即，空间或时空的整体形状和复杂性）和混沌结构（即，光线和光锥的大尺度行为）。

AbstractAward: DMS-0405906Principal Investigator: Gregory J. GallowayRecent developments in gravitational theory, both theoretical andempirical, have lead to an increased interest in the study ofasymptotically anti-de Sitter and asymptotically de Sitterspacetimes. In this project, techniques of Lorentzian andRiemannian geometry will be used to attack certain problemspertaining to such spacetimes, as well as some problemsconcerning new developments in the theory of black holes. One ofthe specific aims of this project is to continue development ofan approach to establishing positivity of mass for asymptoticallyhyperbolic Riemannian manifolds (with spherical conformalinfinity) which does not require spin assumption. This approach,which makes use of the brane action introduced by Witten and Yauin their work on the AdS/CFT correspondence, may also be usefulin settling some positive mass conjectures for asymptoticallyanti-de Sitter spacetimes with toroidal topology at infinity. ThePI also plans to continue investigations into uniqueness andrigidity phenomena, and the development of singularities inasymptotically de Sitter spacetimes. The classical definition ofa black hole, though mathematically elegant, is problematic inpractice (especially for numerical studies) in that it requiresknowing the full future evolution of spacetime all the way toinfinity. As an alternative, Ashtekar and Krishnan haveintroduced a new, fully nonlinear and dynamical, approach to thestudy of black holes, based on the quasi-local notion of adynamical horizon. Another aim of this project is to investigatesome open problems concerning the occurence, i.e., existence,uniqueness and position of dynamical horizons.Modern theories of gravity are geometrical in nature. Thegravitational field and other fields, black holes and relatedobjects, may be described and analyzed using geometric methods.In more general terms, this project is concerned with the studyof certain features of gravity of current scientific interestfrom this geometric point of view, utilizing the tools ofRiemannian geometry, a mathematical theory of space, andLorentzian geometry, a mathematical theory of spacetime. Thesetheories provide a method for studying the relationship amongthree fundamental aspects of the spacetime universe: curvature(i.e., the bending of space or spacetime), topology (i.e., theglobal shape and complexity of space or spacetime) and causalstructure (i.e., the large scale behavior of light rays andlight cones).