Nonlinear Geometric Flows: Ancient Solutions, Non-Compact Surfaces, and Regularity

Some of the most important problems in mathematics and physics are related to the understanding of singularities. These are anomalies in the behavior of a physical quantity where the mathematical expressions that are used to measure such quantities break down. It may be related to the understanding of black holes or turbulence to the accumulation of cancer cells. These physical phenomena are often described via a differential equation which involves time and space. Studying the qualitative behavior of the solutions of such equations often results in a better understanding of the related physical problem. To capture a singularity one uses a blow up procedure which allows one to focus near the potential singularity and use the scaling properties of the differential equation involved. Because of the change in the scaling of space and time, the new solution after the blow up is defined for all space and time -- in other words it is a 'global solution'. The classification of such solutions, when possible, sheds new insight into the singularity analysis and the related physical problem.

This project addresses the questions of existence, uniqueness and qualitative behavior of global solutions to nonlinear geometric elliptic and parabolic partial differential equations. Emphasis is given to the classification of ancient solutions and the study of singularities. The interplay between analytical and geometric techniques will be crucial for the resolution of the proposed research activities. The project links a wide range of active fields of mathematics, in particular nonlinear partial differential equations, geometry and classical analysis. The PI intends to study the applications of the mathematical problems to other disciplines such as quantum field theory and image processing. Results will be disseminated to the research community at various meetings and by publication of research articles. Special emphasis will be given to the training of PhD students and the encouragement of minorities and women to pursue a successful career in mathematics.

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.