Singularities and rigidity in geometric evolution equations

This project focuses on geometric flows, where a geometric object - such as a function, a surface, or a Riemannian metric - evolves over time with the evolution determined by a differential equation modeled on the classical heat equation. The classical heat equation describes the evolution of the temperature as heat spreads out over time. The equations that the PI and collaborators consider were first discovered in materials science, engineering and applied mathematics and are extensively studied in pure mathematics. These geometric flows are nonlinear generalizations of the heat equation and the nonlinear effects lead to new phenomena, including the development of singularities even when starting from a smooth initial configuration. Understanding and modeling these singularities is a fundamental problem, both theoretically and in applied science. The broader impact of the project includes graduate advising, undergraduate mentoring, curriculum reform, writing graduate textbooks, dissemination, seminar and conference organization, and other service to the community including multiple editorial boards. The project studies geometric flows focusing on singularities and rigidity in Ricci and mean curvature flow (MCF). Mean curvature flow is a nonlinear parabolic evolution equation that originated in materials science and has been intensely studied in pure and applied mathematics. A closed surface evolves to decrease its area as efficiently as possible, pulling itself tight. As the surface gets smaller, the flow contracts even faster and, thus, singularities always occur. The key is to understand the singularities. Function theory plays a role, both continuous and discrete, and unique continuation. A second main direction is to understand certain properties of singularities in Ricci flow, including when blowups are unique, which blowups are rigid, and the asymptotic structure of gradient shrinking Ricci solitons. 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.