We propose to further developments related to Perelman's work on the Geometrization Conjecture, using Ricci flow through the space-time formulation, new gradient estimates and monotone quantities and their geometric applications, and to apply new techniques and applications based on the ideas of these methods and estimates with the aim of furthering the understanding of the following interconnected topics: 1. Uniformization of compact and noncompact Kaehler manifolds, combining Kaehler-Ricci flow and the study of holomorphic functions/sections. 2. Analysis of singularities, formulation of weak solutions, and flow past singularities for geometric evolution equations and the duality between the Ricci flow and the mean curvature flow. 3. Study of harmonic/holomorphic functions, function theory, spectrum and the geometry of complete Riemannian/Kaehler manifolds. 4. Existence of Einstein and other canonical Riemannian metrics on manifolds.
Geometric evolution equations are powerful and central tools in the study of the global geometry and topology of manifolds. Recent work of Perelman on Hamilton's program for Ricci flow and its applications towards a possible solution to the Poincare and geometrization conjectures provides a timely and promising opportunity for a group effort on significant advancements in geometric analysis and related areas. The results from the project should lead to new advances in and connections between string/duality theory and renormalization group flow, Ricci flow, mean curvature flow and other geometric evolution equations, and may enhance the understanding of the homogeneity of the universe at large scales, as well as other areas in science. The project will enhance the understanding of geometric analysis, linear and nonlinear partial differential equations, algebraic geometry and mathematical physics.
