A Look at Some Research Problems in
Mathematical and Numerical General Relativity
Michael Holst
UCSD
Abstract:
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.