Proposal: DMS 9802722 Principal Investigators: Michael Anderson and Claude LeBrun
Anderson and LeBrun will explore new relations between the geometries and topologies of low-dimensional manifolds. The main theme of this research is the relation between the scalar curvature functional and manifold topology. In dimension 3, Anderson will continue his program to prove Thurston's geometrization conjectures from this point of view, and study relations between this and Einstein metrics on 4-manifolds. In dimension 4, LeBrun will study relations between the scalar curvature functional, Weyl curvature, Seiberg-Witten theory, and the existence of Einstein metrics. LeBrun will also work on related problems in higher dimensions.
The scalar curvature functional was introduced by Einstein and Hilbert, and plays a fundamental role in Einstein's theory of General Relativity. Anderson's investigations have a direct bearing on questions regarding the physics of black holes and the shape of the universe. In a similar way, LeBrun's work studies fundamental questions arising from Hawking's quantum gravity program, and ties the Seiberg-Witten equations of high energy physics to the study of gravitational phenomena.
