3. Postdoctoral Program - MSRI

comprehensive lectures explaining the philosophy, technical constructions and conclusions of his result.
As his work is a revolutionary tour de force both relying on and altering the understanding of the
evolution problem in General Relativity, it was further aided by two introductory lectures addressing
that precise subject.
The workshop started on Monday with the introductory remarks by the MSRI Director Robert Bryant
and the first introductory lecture by Mihalis Dafermos on the evolution problem in General Relativity,
followed by the first lecture of Christodoulou. The day ended with the second introductory lecture
by Alan Rendall on the characteristic initial value problem in General Relativity. The framework of
characteristic value problem is a crucial element of the Christodoulou’s result.
The second day of the conference began with the second lecture by Christodoulou, followed by a lecture
by Greg Galloway on topology of the marginally trapped surfaces in 2+1 and higher dimensional gravity.
The day ended with an excellent lecture by Jim Isenberg in which he surveyed a very active field of the
constraint equations in General Relativity. Both lectures provided an invaluable complementary point
of view to the lectures of Christodoulou concerned with the evolution problem in General Relativity.
The third day continued with another (two 1 hour lectures) installment from the series of lectures by
Christodoulou, followed by a talk by Spyros Alexakis on the recent progress on the black hole rigidity
problem. This is a fascinating long standing question concerning uniqueness of the Kerr family of
black hole solutions in the class of stationary asymptotically flat spacetimes. The initial breakthrough
was achieved in the 70’s in the work of Carter, Israel, Robinson and Hawking culminating in the
rigidity statement for the analytic spacetimes. Alexakis explained recent results aimed at removing the
analyticity assumption. The last talk of the day was by Robert Bartnik on the topic of the Hamiltonian
formulation of General Relativity.
On Thursday, Christodoulou delivered the last of his lectures. The rest of the day focused on the
subjects related to another outstanding open problem in General Relativity – stability of black holes.
While the work of Christodoulou established the sufficient conditions under which a trapped surface,
and thus potentially a black hole, forms in evolution, the problem of nonlinear stability of basic black
hole solutions – Schwarzschild and Kerr families – has been open since their respective discoveries in
1916 and 1963. The last 5 years have seen a significant progress towards the solution of the problem
as punctuated both by the results on the linear stability and development of the methods relevant
to the nonlinear problem. Mihalis Dafermos gave a sweeping overview of the problem and the recent
breakthrough results in the subject. His lecture was followed by 4 shorter (half-hour) talks of the junior
participants on their results. Pieter Blue and Mihai Tohaneanu spoke about the decay and Strichartz
estimates for the wave equation in Schwarzshild and Kerr spacetimes, Gustav Holzegel explained his
new results on the boundedness for solutions of the Klein-Gordon equation in Kerr-anti de Sitter spacetime,
and Dean Baskin gave a talk on the parametrix construction for the Klein-Gordon equation in an
asymptotically de Sitter spacetime.
The last day of the workshop started with the lecture by Igor Rodnianski who described the almost
immediate impact of the discussed work of Christodoulou given in the extension and reinterpretation of
the new short-pulse method. The last two talks were by Maciej Zworski and Mu-Tao Wang. Zworski’s
lecture addressed the subject of quasi-normal modes of solutions of the wave equation on stationary
black hole spacetimes and their importance to numerical and observational relativity and high energy
physics. Wang’s lecture focused on another important subject of General Relativity – quasi-local mass
and described recent important results in the area.
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