Geometry, Analysis and General Relativity

摘要奖：DMS-0707306主要研究人员：拉斯·安德森主要研究与洛伦兹几何、广义相对论和黎曼几何有关的问题。他研究的主要焦点，爱因斯坦方程的柯西问题，与这些领域都有深入的互动。在这个提议中考虑的主题包括宇宙模型的全局稳定性，边缘囚禁表面和动力学视界理论，以及宇宙学奇点的结构。安德森将试图证明，在时空维度至少为11的情况下，存在全局稳定的具有静态奇异性的真空宇宙模型族。在3+1维的情况下，类似的结果预计也适用于爱因斯坦标量场方程。安德森和他的合作者最近已经能够将极小表面理论中开发的技术应用于稳定的边缘陷阱表面的情况，这在广义相对论中扮演着表观视界的角色。安德森计划在他未来的研究中探索边缘表面和最小表面之间的联系，并将这些技术应用于动态视界。除了上述主题，安德森还提出了一项使用解析和数值技术的研究计划，旨在研究宇宙中一般非齐次宇宙的渐近行为。爱因斯坦方程描述了时空的几何形状，这反过来又决定了物体(行星、恒星、星系)在空间中的运动，以及通过各自的场论来确定场和粒子的动力学。爱因斯坦方程提供的对宇宙的几何观点导致了我们的宇宙图景中一些最基本和矛盾的特征，包括最初的时空单一性或大爆炸和黑洞。对极端条件下的时空的数学研究，如在大爆炸奇点附近或在黑洞的奇点附近，已经导致了遥远的猜想，称为“宇宙审查”，这意味着违反直观因果概念的现象必须对观察者隐藏(审查)。在这个项目中，我们使用基于计算机的技术和分析技术的结合来研究与宇宙审查有关的问题，特别关注几何从初始条件的演变。

AbstractAward: DMS-0707306Principal Investigator: Lars AnderssonThe principal investigator is proposing to do research onproblems relevant to Lorentzian Geometry, General Relativity andRiemannian Geometry. The main focus of his research, the Cauchyproblem for the Einstein equations, interacts deeply with each ofthese areas. Among the topics considered in this proposal areglobal stability of cosmological models, the theory of marginallytrapped surfaces and dynamical horizons, and the structure ofcosmological singularities. Andersson will attempt to prove thatin the case of spacetime dimension at least eleven, there arefamilies of vacuum cosmological models which are globally stable,with quiescent singularity. In the 3+1 dimensional case, similarresults are expected to hold for the Einstein-scalar fieldequations. Andersson and collaborators have recently been ableto apply techniques developed in the theory of minimal surfacesto the case of stable marginally trapped surfaces, which play therole of apparent horizons in general relativity. Andersson plansto pursue the connection between marginal surfaces and minimalsurfaces in his future research, and to apply these techniquesalso to dynamical horizons. In addition to the above topics,Andersson proposes a program of research using analytic andnumerical techniques, directed towards investigating theasymptotic behavior of general inhomogenous cosmologies near thesingularity.The Einstein equations describe the geometry of spacetime, whichin turn determines the motion of bodies (planets, stars,galaxies) in space, as well as the dynamics of fields andparticles via their respective field theories. The geometric viewof the universe provided by the Einstein equations has led tosome of the most fundamental and paradoxical features of ourpicture of the universe, including the initial spacetimesingularity or Big Bang and black holes. The mathematical studyof spacetimes in extreme conditions such as near the Big Bangsingularity or near the singularity in black holes has led to afar reaching conjecture called ``Cosmic Censorship'', whichimplies that phenomena which violate the intuitive notions ofcausality must be hidden (censored) from observers. In thisproject, problems related to cosmic censorship are studied usinga combination of computer based and analytical techniques, with aparticular focus on the evolution of geometry from initialconditions.