Research in Lorentzian Geometry and Mathematical Relativity

摘要建议：DMS-9803566首席研究员：格雷戈里·加洛韦将在洛伦兹几何和数学相对论领域进行研究。这项拟议研究的中心目标之一是开发研究时空刚性现象的新技术。提出的研究试图建立零超曲面的最大值原理，并利用这些原理得到包含零线的时空的分裂定理/刚性结果。黑洞理论中的问题也有待研究。这些研究零超曲面的新技术将使严格建立非自然正则性假设的问题变得容易，霍金的黑洞面积定理。研究人员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关系。更广泛地说，这项研究项目涉及在宇宙尺度和坍塌恒星的尺度上更好地理解时空宇宙的结构。时空几何学的目的是研究时空宇宙的三个基本方面之间的关系：曲率(根据爱因斯坦方程是由物质的存在产生的)、拓扑(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都有应用。

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.