Research in Lorentzian Geometry and Mathematical Relativity

摘要建议：DMS-9803566首席研究员：Gregory Galloway该奖项下的研究将在洛伦兹几何和数学相对论领域进行。 这项研究的中心目标之一是开发研究时空刚性现象的新技术。 拟议的研究旨在建立零超曲面的最大值原理，并使用这些来获得分裂定理/刚性结果的时空包含零线。 黑洞理论中的问题也将被研究。 这些研究零超曲面的新技术将使严格建立霍金黑洞面积定理的非自然规律性假设的问题变得容易解决。 研究者还将继续研究拓扑审查及其与黑洞拓扑的关系，重点是渐近结构和黑洞拓扑之间的联系。更一般地说，这个研究项目涉及在宇宙学尺度和恒星尺度上更好地理解时空宇宙的结构。 时空几何学的目的，在这个项目中被利用和发展，是研究时空宇宙的三个基本方面之间的关系：曲率（由爱因斯坦方程产生的物质），拓扑学（即，时空的整体形状）和因果结构（即，光锥的行为）。 这项研究在广义相对论和引力理论中有应用。

AbstractProposal: DMS-9803566Principal Investigator: Gregory GallowayResearch under this award will be conducted in the fields ofLorentzian geometry and mathematical relativity. One of the centralaims of the proposed research is to develop new techniques forstudying spacetime rigidity phenomena. The proposed research seeks toestablish maximum principles for null hypersurfaces and use these toobtain splitting theorems/rigidity results for spacetimes whichcontain null lines. Problems in the theory of black holes are also tobe investigated. These new techniques for studying null hypersurfaceswill make amenable the problem of rigorously establishing undernatural regularity assumptions Hawking's black hole area theorem. Theinvestigator will also continue research on topological censorship andits relationship to the topology of black holes, focusing on theconnection between asymptotic structure and black hole topology.In more general terms, this research project is concerned withachieving a better understanding of the structure of the spacetimeuniverse at both the cosmological scale and the scale of collapsingstars. The aim of spacetime geometry, as to be utilized and developedin this project, is to study the relationship between threefundamental aspects of the spacetime universe: curvature (which by theEinstein equations is generated by the presence of matter), topology(i.e., the global shape of spacetime) and causal structure (i.e., thebehavior of light cones). This research has applications to generalrelativity and gravitation theory.