Principal Investigator: Michael T. Anderson, Claude R. LeBrun
Working both jointly and independently, Anderson and LeBrun plan to study a cluster of related geometric structures on low-dimensional manifolds. Their research program focuses on relationships linking geometry to topological structures, algebraic geometry, and theoretical physics. Jointly, Anderson and LeBrun plan to study the existence and moduli of canonical metrics on 4-manifolds, with an emphasis on the Kahler case. Anderson will also investigate problems concerning asympotically hyperbolic Einstein metrics, the AdS/CFT correspondence, and mathematical aspects of general relativity. LeBrun will also study the curvature of 4-manifolds, Seiberg-Witten theory, extremal Kahler metrics, and the twistor geometry of holomorphic disks. The proposed work is expected to have a significant broader impact on two different fronts. On one hand, it will promote further interactions between researchers in differential geometry and those in topology, in algebraic geometry and, especially, in theoretical physics. The project also has a significant educational impact at the graduate level, in that Anderson and LeBrun oversee a large number of graduate students who are working on problems closely tied to the research proposal.
The relations between mathematics and theoretical physics are at the forefront of research in both disciplines, leading on both sides to remarkable new insights and developments. Much of the planned research activity takes its inspiration from current attempts to bridge the gulf between Einstein's theory of space-time geometry and gravitation (General Relativity), and the quantum field theories that are currently understood to govern the other fundamental forces of nature. Some of the research has to do with the problem of describing all possible geometries of 4-dimensional universes governed by Einstein's gravitational equations. One of the toolsto be used involves use of the Seiberg-Witten equations, which originated in the theory of short-range nuclear interactions, but nonetheless turn out to play a major role in our understanding of the large-scalestructure of many 4-dimensional universes. Other aspects of the planned research are closely linked to developments in string theory, which is arguably the most promising and exciting avenue of research in theoretical physics, and which has had a profound and multi-faceted impact on recent progress in mathematics.
