Challenges in geometric partial differential equations

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I propose to study three central questions in geometric analysis. Two come from Einstein's equations of the general theory of relativity, while the third is an inverse problem,  a very active area of applied mathematics. *** Einstein's equations are thought to provide the correct mathematical description of gravity. They form a system of nonlinear wave equations, as is the case for many equations of mathematical physics. Thus, robust mathematical methods developed in this area have the potential for broader applicability. The two questions I wish to study deal with black hole solutions which are dynamical (i.e. they change in time). They both stem from a central prediction in the subject  (coined the ``establishment view'' by Penrose in the 1970s), which deals with the large-time behaviour of the exterior regions of such solutions. The prediction asserts that the emission of gravitational waves should have a stabilizing effect on the underlying metric, enabling it to settle down (asymptotically) to a non-radiating state. Such a state must be stationary (i.e. time-independent), we are told. The question then becomes to understand the possible stationary solutions of the equations. *** ***The first project I propose is to verify the last two assertions: That solutions that do not emit gravitational radiation must indeed be stationary. And that such stationary states must agree with the Kerr solutions, as has been verified (by Hawking, Carter and Robinson) in the very restrictive  setting of real-analyticity. The second challenge I wish to address was again proposed by Penrose and is the celebrated inequality that bears his name. This proposes to bound the space-time's mass content from below by the area of trapped surfaces. The inequality was originally proposed as a test of the ``establishment view'', which (in its entirety) lies completely out of the methods of modern mathematical analysis. *** The final question is an inverse problem: The challenge is to derive information on the interior of an object's interior, purely by making measurements at its boundary, i.e with no intrusion into the interior. This field has very wide applicability ranging from medical imaging to seismology and geophysics. The type of data one collects at the boundary varies according to the application one has in mind. I propose a challenge inspired by tomography, where the measurements correspond to applying a charge on the boundary of an object (a human body) and measuring the boundary electric current. The objective is to detect the conductivity of tissue, as a gauge for the interior structure. This conductivity is modelled by an (unknown) coefficient of an elliptic partial differential operator which describes the distribution of the electrostatic potential. *** All the above projects are meant to be undertaken by a team of researchers, tightly connected with centres of excellence in these subjects. *** *** *** *** **