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Some Research Problems in Mathematical and Numerical General Relativity

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GCSW02 - Structure preservation and general relativity

The 2017 Nobel Prize in Physics was awarded to three of the key scientists involved in the development of LIGO and its eventual successful first detections of gravitational waves. How do LIGO (and other gravitational wave detector) scientists know what they are detecting? The answer is that the signals detected by the devices are shown, after extensive data analysis and numerical simulations of the Einstein equations, to be a very close match to computer simulations of wave emission from very particular types of binary collisions.

In this lecture, we begin with a brief overview of the mathematical formulation of Einstein (evolution and constraint) equations, and then focus on some fundamental mathematics research questions involving the Einstein constraint equations. We begin with a look at the most useful mathematical formulation of the constraint equations, and then summarize the known existence, uniqueness, and multiplicity results through 2009. We then present a number of new existence and multiplicity results developed since 2009 that substantially change the solution theory for the constraint equations. In the second part of the talk, we consider approaches for developing “provably good” numerical methods for solving these types of geometric PDE systems on 2- and 3-manifolds. We examine how one proves rigorous error estimates for particular classes of numerical methods, including both classical finite element methods and newer methods from the finite element exterior calculus.

This lecture will touch on several joint projects that span more than a decade, involving a number of collaborators. The lecture is intended both for mathematicians interested in potential research problems in mathematical and numerical general relativity, as well as physicists interested in relevant new developments in mathematical and numerical methods for nonlinear geometric PDE .

This talk is part of the Isaac Newton Institute Seminar Series series.

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