International Conference on Mathematical Relativity

University Of Miami, Coral Gables FL

Investigators

Abstract

This award supports participation in the conference "A Celebration of Mathematical Relativity in Miami" held in Coral Gables, Florida, during December 14 - 16, 2018. The Einstein equations, which are the field equations of general relativity, describe how spacetime universe curves in the presence of matter; it is this curvature that is responsible for the effects of gravity. General relativity is a remarkably accurate theory, which describes the formation of black holes, predicts the existence of gravitational waves (now famously detected by LIGO), and governs the large-scale behavior of the entire cosmos. The aim of this conference is to gather together leading experts at the intersection of mathematical general relativity and geometric analysis to discuss recent advances and new directions of research in this area, and to expose graduate students and young mathematicians to these developments. Mathematical general relativity is a very rapidly developing area of research. Geometric analysis has played a remarkable role in this development, beginning with the singularity theorems of Hawking and Penrose. Many advances have ensued in both the elliptic and hyperbolic aspects of the theory, including developments in the theory of black holes, in understanding of Penrose's cosmic censorship conjecture, and in the study of manifolds of nonnegative scalar curvature. Geometric inequalities, such as the positive mass theorem and the Riemannian Penrose inequality, play a deep and fundamental role in general relativity. In recent years new geometric inequalities, involving physical quantities such as mass, angular momentum, charge, etc., have given rise to many interesting open questions. Studies of initial data sets for gravitational fields have proven to be fruitful areas of research. The initial data for the Cauchy problem associated to the Einstein field equations are required to satisfy the so-called constraint equations, a system of nonlinear partial differential equations, the geometric origin of which are the Gauss-Codazzi equations. Great progress has been made in developing techniques for solving the constraint equations, both by the conformal method and by gluing methods. The development of the theory of marginally outer trapped surfaces, which are general initial data versions of minimal surfaces, has led to results concerning the topology of black holes (and the region exterior to black holes), and has made possible a direct proof of the spacetime positive mass theorem in dimensions less than eight. The recent proof of the (Riemannian) positive mass theorem (in all dimensions and without spin assumption) has greatly advanced knowledge about geometric singularities and their roles in initial data sets. Efforts to obtain a localized measure of mass, which includes a contribution from the gravitational field, have led to various notions of quasi-local mass, and continues to be a very active area of research. Geometric flows have played a fundamental role in many of these developments. The conference will address all these topics. Details of the program are available at the program webpage: http://www.math.miami.edu/gg70 This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.