Geometry, Analysis and General Relativity

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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.