The proposed research is in geometric evolution equations, analysis on metric spaces, and geometric group theory. The evolution equations in the proposal are mean curvature flow and Ricci flow, and the problems pertain to the structure of the singular set, and issues such as rectifiability, uniqueness of tangents, and uniqueness of model flows. The proposed research in analysis on metric spaces has two focal points: bilipschitz embedding problems (and related issues) and the structure of boundaries of Gromov hyperbolic spaces. The projects in geometric group theory are a continuation of my earlier work, which is influenced by Gromov's papers "Hyperbolic groups" and "Asymptotic invariants of infinite groups", rigidity theory, 3-manifolds, and geometric mapping/function theory; their principal aim is to address rigidity and uniformization/geometrization problems for groups by analyzing their asymptotic structure with a variety of tools from geometry, analysis, topology, dynamics and combinatorics.
The project aims to study two nonlinear analogs of the heat equation: evolution of surfaces by mean curvature, and Hamilton's Ricci flow. Evolution by mean curvature has been studied for decades as a natural model for evolving surface interfaces. Ricci flow describes an evolving geometry, and was used in Perelman's recent solution of the Poincare conjecture. The primary objective of the proposed research on these equations is to study singularities and show that they have a very special form. Another component of the research program is an investigation of spaces which 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, in order to reveal hidden symmetries, and otherwise show that no hidden symmetries exist. This is very useful for understanding the asymptotic shape of infinite groups, and is part of confluence of several research trends over the last 10-15 years. Another application of similar ideas is to embedding problems in theoretical computer science: Cheeger and the PI were able to give a natural, new counterexample to the Goemans-Linial conjecture in computer science.
. These are very active areas of fundamental research in mathematics. Geometric group theory is a part of mathematics that uses many different ideas and tools of a geometric character to study different types of symmetry, especially infinite symmetry. It has strong ties with many other areas of mathematics, such as topology (especially the topology of 3-dimensional spacs), the theory of spaces with special curvature properties such as flat space or hyperbolic space, and the geometry of certain fractal spaces, Analysis on metric spaces is an area of mathematics that derives substantial motivation from the need to generalize classical ideas from geometry or analysis in Euclidean space, such as calclulus, to a much more general setting; part of this motivation comes from computer science and data analysis. The geometric flows considered in the project are mean curvature flow and Ricci flow; these are two types of geometric objects that change with time according to a simple geometric rule. Both types of flows have been central topics in mathematics for many years. Part of the research is connected with another field --- computer science ---- as it is related to the design and estimation of algorithms. Highlights of the PI's achievements during this period include the best known techniques for estimating distortion in certain embedding problems in computer science, a substantial improvement on earlier treatments of mean curvature flow, and an extension of the famous work of Perelman on the Poincare conjecture to the case of orbifolds. During this period the PI mentored 5 PhD students on topics related to the project.
