15:00 Arick Shao (QMUL): Uniqueness Theorems on Asymptotically Anti-de Sitter Spacetimes
In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (AdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate a correspondence statement as a unique continuation problem for PDEs: Is an asymptotically AdS solution of the Einstein equations uniquely determined by its data on its conformal boundary at infinity? In this presentation, we establish a key step in toward a positive result; we prove an analogous unique continuation result for linear and nonlinear wave equations on fixed asymptotically AdS spacetimes satisfying a positivity condition at infinity. We show, roughly, that if a wave phi on this spacetime vanishes on a sufficiently large but finite portion of its conformal boundary, then phi must also vanish in a neighbourhood of the boundary. In particular, we highlight the analytic and geometric features of AdS spacetimes which enable this uniqueness result, as well as obstacles preventing such a result from holding in other cases. This is joint work with Gustav Holzegel.
16:30 Tadahiro Oh (Edinburgh): On the transport property of Gaussian measures under Hamiltonian PDE dynamics.
Transport properties of Gaussian measures under different transformations have been studied in probability theory. In this talk, we discuss transport properties of Gaussian measures on periodic functions under nonlinear Hamiltonian PDEs such as the nonlinear Schrodinger equations and nonlinear wave equations. Lebowitz-Rose-Speer ’88, Bourgain ’94, and McKean ’95 initiated the study of invariant Gibbs measures for dispersive Hamiltonian PDEs. In the first part of the talk, we give a review on the construction of invariant Gibbs measures and discuss how it lead to a recent development of probabilistic construction of solutions in late 2000’s. In the second part, we discuss the quasi-invariance property of Gaussian measures on Sobolev spaces under certain dispersive Hamiltonian PDEs. We also discuss the importance of dispersion in this quasi-invariance result by showing that the transported measure and the original Gaussian measure are mutually singular when we turn off dispersion. The second part of the talk is based on a joint work with Nikolay Tzvetkov (Universite Cergy-Pontoise) and Philippe Sosoe (Harvard University).