Principal Investigator: Dan F. Knopf
The project will advance the search for canonical geometries by means of geometric heat flows. In light of the landmark progress made recently by Perelman in Hamilton's program to resolve the Geometrization and Poincare' Conjectures, this research area is undergoing a rapid and productive expansion. The powerful innovations and profound insights in Perelman's work contribute to the extraordinary power of Ricci flow as a tool for investigating the geometry and topology of Riemannian and complex manifolds. In virtually all known applications of Ricci flow, it is critical to have a deep understanding of the mechanisms of singularity formation. Therefore, the project will investigate four aspects of singularity formation. These four objectives are chosen to build upon the prior results and current research program of the PI and to be highly relevant to promising new applications of Ricci flow. The objectives are to study (1) asymptotics of Ricci flow singularity formation, (2) analysis of Ricci flow singularities in dimension four, (3) analysis of singularity models for Kaehler-Ricci flow, and (4) the structure of reduced geometry.
A manifold is an object that - like our universe - looks like Euclidean space locally, but whose global topology and geometry may be much different. The broad goals of this project are to find optimal geometric structures with which to categorize manifolds. The methods used are certain partial differential equations called geometric heat flows. The idea is to let a geometric object evolve in time in such a way that its geometry improves and simplifies, possibly after a change in topology. A geometric heat flow called the Ricci flow has just yielded major breakthroughs in two of the most difficult open problems in mathematics. These successes provide great incentives to apply it to other challenging open problems and make it a very active and competitive field of research. The types of partial differential equations studied in Ricci flow have much in common with those used to model the movement of oil in shale and in thin films, combustion in porous media, heat propagation, avalanches, population dispersal, the spreading of microscopic droplets, and certain effects in plasma physics. For this reason, methods developed in this project may have important applications to those areas of applied mathematics. The project will focus on the delicate analysis needed to understand such equations as they become singular. This analysis should have important broad applications, among which are the following. (i) The methods developed, especially asymptotic analysis, should extend to the practical applications mentioned above. (ii) The project will promote interdisciplinary interactions with physics, where there are many applications for geometric classification and flow techniques. For example, theorists in general relativity want to classify possible topologies of four-dimensional space-times. Researchers in string theory and mirror symmetry are interested in understanding certain six-dimensional manifolds. The Ricci flow itself is an approximation to the renormalization flow for an important model in quantum field theory. (iii) The project will benefit graduate education, because the PI will invest time helping students develop expertise in relevant areas of geometry, analysis, and topology.
