In this lecture\, we begin with a brief overview o f the mathematical formulation of Einstein (evolut ion and constraint) equations\, and then focus on some fundamental mathematics research questions in volving the Einstein constraint equations. We beg in with a look at the most useful mathematical for mulation of the constraint equations\, and then su mmarize the known existence\, uniqueness\, and mul tiplicity 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 approac hes for developing "provably good" numerical metho ds for solving these types of geometric PDE system s on 2- and 3-manifolds. We examine how one prove s rigorous error estimates for particular classes of numerical methods\, including both classical fi nite element methods and newer methods from the fi nite element exterior calculus.

This lect ure will touch on several joint projects that span more than a decade\, involving a number of collab orators. The lecture is intended both for mathema ticians 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.

LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR