This project is concerned with questions in differential geometry, which is the study of higher-dimensional shapes and their curvature. These objects play a central role in general relativity where gravitation is reflected by the curvature of space-time. Geometric objects can often be improved if we evolve them by a differential equation, so that the curvature dissipates in a way analogous to heat dissipation. However, one important difference is that, while heat dissipation is a linear phenomenon, all the geometrically interesting equations are nonlinear. This project is aimed at understanding these nonlinear phenomena. Of particular interest is the process of singularity formation: That is to say, what happens when a surface is about to break apart and the curvature becomes very large? Understanding these questions is an important problem within mathematics. In addition, discrete versions of these equations are used in engineering and computer science.
The most important example of a geometric evolution equation is Hamilton's Ricci flow, which is a key tool in Perelman's proof of the Poincare and Geometrization conjectures, as well as in the proof of the Sphere Theorem. Another important example is the mean curvature flow for surfaces in Euclidean space. One of the main concept is that of an ancient solutions.Â Ancient solutions are solutions which can be extended infinitely far backward in time.Â They oftenÂ arise as models for a solution to a geometric flow right before a singularity forms. As such, they play an important role in understanding singularity formation.Â One of the goals here is to classify all ancient solutions in low dimensions, subject to a noncollapsing assumption. This is analogous to the problem of classifying entire solutions to elliptic equations. The PI is also planning to study singularity formation in higher dimensions, under suitable curvature restrictions.Â In another direction, the PIÂ is interested in studying problems related to minimal surfaces and free boundary value problems, as well as applications of partial differential equations to general relativity.
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.