Anderson and LeBrun plan to study a cluster of related geometric structures on low-dimensional manifolds. Their research program focuses on the discovery and development of fundamental connections between Riemannian geometry, differential topology, and issues in theoretical physics. Anderson will investigate problems concerning conformally compact Einstein metrics and the AdS/CFT correspondence, the structure of Einstein metrics on bounded domains and mathematical aspects of general relativity. Meanwhile, LeBrun will study the existence and moduli of canonical metrics on 4-manifolds, with an emphasis on extremal Kahler metrics, Seiberg-Witten theory, and the twistor geometry of holomorphic disks. This research program aims both to explore fundamental issues in the mathematical field of differential geometry, and simultaneously to discover new links between mathematics and theoretical physics. Much of the planned research activity takes its inspiration from current attempts to bridge the gulf separating Einstein's theory of gravitation from the quantum field theories that describe the forces of nature on a microscopic scale. Some of the research concerns the problem of describing all possible geometries of 4-dimensional universes governed by Einstein's gravitational field equations. Other aspects of the research program are intimately linked to recent developments in string theory. By also training a group of graduate students to pursue research in this area, the project will additionally help foster interactions between mathematics and physics on an immediate, human scale, through its long-term educational impact.
This award supported the work of two mathematicians, Claude LeBrun and Michael Anderson, on problems in global differential geometry. This area of mathematics provides the conceptual framework for Einstein's general theory of relativity, and many of the problems investigated by the supported research are directly or indirectly tied to questions about classical or quantum gravity. More generally, global differential geometry attempts to understand the way that small-scale information about the way a space curves is linked to the large-scale topological structure ("shape") of the space. LeBrun's work primarily concerned Euclidean-signature solutions of the Einstein equations on closed 4-dimensional spaces. He was able to prove new theorems characterizing which compact 4-manifolds admit Einstein metrics, and also proved theorems concerning how many solutions there are when one is known to exist. Many of LeBrun's techniques and results depend on the use of complex variables to help understand the relevant geometrical structures. Anderson's work included work on mathematical problems related to general relativity (Einstein's theory of gravity), and led to new geometrical insights arising from the problem of assigning a total mass to a given region of space-time. Anderson also proved new theorems regarding the relationship between the geometry of a solution of Einstein's equations on an open space and the residual geometry of angles it induces on the boundary surface at infinity; physicists refer to this relationship as "holography," and Anderson's ideas have a direct bearing on the AdS/CFT (Anti-deSitter/Conformal Field theory) correspondence of string theory.
