Geometric Flows and Analysis on Metric Spaces

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The project aims to study two nonlinear analogs of the heat equation that have been the subject of intense investigation by mathematicians for decades. The primary objective of this research is to study issues related to the formation of singularities in the solutions of these equations, including their structure and stability, to enable mathematicians to make deductions about the ways in which space can be deformed.  The resolution of fundamental conjectures in geometry and topology is an outgrowth of this work.  An additional component of the research program is an investigation of spaces that have a self-similar or fractal character, using analytic tools that have been developed in the last few years.  Here one of the goals is to deform the space into an optimal form, if possible, to reveal hidden symmetries, and otherwise show that no hidden symmetries exist.  This is part of confluence of several research trends over the last fifteen years. Another application of similar ideas is to embedding problems in theoretical computer science. The primary objective of this research is to study singularities of geometric evolution equations, embedding problems, analysis on metric spaces, and geometric group theory. The evolution equations in the proposal are mean curvature flow and Ricci flow.  The research in analysis on metric spaces clusters in three areas: (1) bilipschitz embedding problems and related issues, (2) the structure of spaces satisfying Poincare inequalities, (3) the structure of boundaries of Gromov hyperbolic spaces.  Common themes in all three areas are spaces satisfying Poincare inequalities, and rescaling arguments leading to singular limit spaces. A bilipschitz embedding in a certain Banach space is a topic of interest to theoretical computer scientists; in previous work, the PI and collaborators improved the best-known results on the embedding of spaces of negative type, in connection with the quantitative version of the Goemans-Linial conjecture.