Nonlinear Geometric Flows: Singularity analysis, ancient solutions 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 norm that is used to measure such quantities breaks up. It may be related to the understanding of turbulence, of black holes, the accumulation of cancer cells in human bodies, or the behavior of neurons in the brain. These phenomena are often described mathematically via a differential equation which involves time and space. Studying the qualitative behavior of the solutions of such equations becomes crucial for understanding the related physical problem and is also essential for computing. The main goal of this project is to study the singular behavior of partial differential equations that is related to physical problems, as discussed above, and also to see how the fundamental shapes of spheres, cones, and cylinders, appear in the singularity formation of these equations. The goal of the studies is to enhance our knowledge of the behavior of solutions near singularities. The project provides research training opportunities for graduate students and postdoctoral scholars. The Ricci flow is a geometric equation that describes the intrinsic change in shape according to its Ricci curvature, a notion of curved space that played a fundamental role in the theory of relativity. G. Perelman, in his seminal 2002 work on the Ricci flow and the resolution of the 100 years old Poincare Conjecture, showed that high curvature regions are modeled on ancient solutions to the flow, that is solutions that have existed for a very long time. He also formed a conjecture regarding the classification of such solutions in three dimensions, under certain natural geometric conditions. During the past several years, the Principal Investigator (PI) and her collaborators resolved this conjecture. However, the higher dimensional problem has remained open since it requires the advancement of new methods. The PI investigates this problem, and work on the related problem of Ricci flow singularities as part of this project. In addition, the PI intends to work on the classification of asymptotically conical ancient solutions and study the analytical behavior of solutions near cylindrical and conical singularities. The interplay between analytical and geometric techniques will be crucial for the resolution of these problems. The project lies at the intersection of several active areas of mathematics, in particular nonlinear partial differential equations, geometry, and classical analysis. Applications to topology, quantum field theory, and the theory of relativity are also explored. 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.