The first part of the project involves two nonlinear partial differential equations that arise in a number of different disciplines in science and engineering, as well as from within mathematics. These equations describe the motion of a curved surface or curved object which evolves so as to simplify its shape as efficiently as possible over time. An important feature is the formation of singularities. Singularities are what enable the solutions to model situations where topology changes, for example when a soap bubble elongates and splits into two bubbles. On the one hand, this flexibility has led to numerous profound applications; on the other, it creates great intellectual challenges which have occupied mathematicians for more than 40 years. The proposed research aims to build on successes in the last few years, to address some of the main open problems. The second part of the project applies ideas from geometry and analysis to study the structure of rough objects, including fractals. This area has been developing very rapidly in the last 20 years, due to new connections between different parts of mathematics, and applications to problems from computer science. The project also involves substantial training of PhD students.
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 solution of the Poincare conjecture. The primary objective of the proposed research on these equations is to study issues related to the formation of singularities in their solutions, including their structure and stability. The resolution of fundamental conjectures in geometry and topology is an outgrowth of this work. 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 15-20 years. Another application of similar ideas is to embedding problems in theoretical computer science: Cheeger, Naor, and the PI were able to substantially improve the previous best known results on the embedding of spaces of negative type, in connection with the quantitative version of the Goemans-Linial conjecture. This project specifically addresses geometric evolution equations, embedding problems, analysis on metric spaces, and geometric group theory. The evolution equations in the project 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.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
