Geometry and Analysis of Einstein Metrics

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

This project concerns investigations on several aspects of the Einstein equations. These equations arose from Einstein's fundamental breakthroughs in understanding space, time, and gravity in his theory of general relativity. In physics, the equations govern the large scale structure of the universe, but are also crucial in understanding fine scale structures. For example, the GPS system would be unworkable without a full understanding of the Einstein equations. They also play an important role in the main efforts to quantize gravity, namely string theory. Mathematically, the Einstein equations are the most interesting equations relating geometry and analysis on curved spaces. The investigations to be undertaken involve a cross-fertilization of ideas and methods from several areas. A main focus will be global issues (existence and uniqueness) for boundary value problems for Einstein metrics, including applications to classical differential geometry such as the isometric embedding problem. Research in general relativity will include studies of quasi-local mass and the initial boundary value problem for the Einstein equations. Studies in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence from string theory and the existence and structure of conformally compact Einstein metrics will also be undertaken.