Analytic aspects of long-time dynamics in general relativity

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The Einstein equation of general relativity is a geometric, nonlinear PDE with a fundamentally hyperboliccharacter. The last few decades, and particularly recent years, have seen a lot of activity in applying themethods of mathematical analysis to understanding the existence, uniqueness, continuous dependence, andlong-term behaviour of solutions to this equation. This has included proof of the stability of empty space andsome cosmological models, while the stability of black holes and singularities remain major open problems.Key methods include proving decay for linear equations via methods that are sufficiently robust to extend tononlinear problems and understanding the structure of nonlinearities. This project will apply these techniques toinvestigate problems in long-term dynamics in general relativity using these techniques.