Geometry of Low Dimensional Manifolds

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

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.