Geometric flows and analysis on metric spaces

New York University, New York NY

Investigators

Abstract

The project focusses on 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 simply its shape as efficiently as possible over time. An important feature of the motion is the formation of singularities that 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 leads to numerous profound applications; on the other, it creates great intellectual challenges. The PI will build on recent progress to address some of the main open problems in this area. 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 25 years, due to new connections between different parts of mathematics, and applications to problems from computer science. Graduate students will be trained in this project. The proposed research studies geometric evolution equations and analysis on metric spaces. The evolution equations in the proposal are mean curvature flow and Ricci flow, and the projects focus on different aspects of regularity. The projects in analysis on metric spaces are concerned with geometric mappings such as bilipschitz, quasi-conformal, Sobolev mappings in the sub-Riemannian setting. The project will make progress in solving longstanding problems at the interface of geometric group theory and geometric mapping theory, and analysis of PDE, especially the regularity of weak solutions to the contact system in Carnot group groups. 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.