Metric Differential Geometry and Mathematical Gravity

本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。这个项目的具体目标之一是研究渐近局部反de Sitter时空的全局结构，特别是与ADS/CFT通信有关的结构。我们继续研究渐近局域的反de Sitter时空的拓扑与其无穷大的构象边界的关系，并研究高维黑洞的某些拓扑和几何性质，以及弦理论中出现的相关物体，包括某些被猜想为最低能量组态的孤子。这个项目的另一个目标是继续研究与类时线和零线的出现有关的时空刚性现象。将考虑零点分裂定理的应用，并提出一种新的方法来研究关于奇点定理刚性的著名猜想。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。更广泛地说，这个项目从这个几何角度，利用黎曼几何和洛伦兹几何的工具，研究当前科学兴趣的某些引力特征。黎曼几何是一种空间数学理论，洛伦兹几何是一种时空数学理论。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).