Metric Differential Geometry and Mathematical Gravity

该奖项下的研究将在洛伦兹和黎曼几何领域进行，并应用于广义相对论和弦理论。该项目的一个具体目标是研究渐近局部反德西特时空的全局结构，特别是与AdS/CFT对应的关系。我们继续研究渐近局部反德西特时空的拓扑结构与无穷远处共形边界的拓扑结构之间的关系，并研究高维黑洞的某些拓扑和几何性质，以及弦理论中产生的相关物体，包括被推测为最小能量构型的某些解。该项目的另一个目的是继续研究与时间线和零线的出现有关的时空刚性现象。将考虑零分裂定理的应用，并提出一种新的方法来研究有关奇异性定理刚性的已知猜想。现代引力理论本质上是几何的。引力场和其他场、黑洞和相关物体可以用几何方法来描述和分析。更一般地说，这个项目是从这个几何的角度，利用黎曼几何（空间的数学理论）和洛伦兹几何（时空的数学理论）的工具，研究当前科学感兴趣的重力的某些特征。这些理论为研究时空宇宙的三个基本方面之间的关系提供了一种方法：曲率(即。例如，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）和因果结构（即光线和光锥的大尺度行为）。

DMS-0104042Gregory G. Galloway Research under this award will be conducted in the areas ofLorentzian and Riemannian geometry, with applications to GeneralRelativity and String Theory. One of the specific aims of thisproject is to investigate the global structure of asymptoticallylocally anti-de Sitter spacetimes, especially in connection withthe AdS/CFT correspondence. We continue investigations into therelationship of the topology of an asymptotically locallyanti-de Sitter spacetime and that of its conformalboundary-at-infinity, and undertake a study of certaintopological and geometrical properties of higher dimensionalblack holes, and related objects arising in string theory,including certain solitions conjectured to be least energyconfigurations. Another aim of this project is to continueinvestigations into spacetime rigidity phenomena associated withthe occurence of timelike lines and null lines. Applications ofthe null splitting theorem will be considered, and a newapproach to well-known conjectures concerning the rigidity ofthe singularity theorems is proposed for study.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).