NSF Award
2405007
Geometric flows are evolution equations that describe motions of surfaces, or their higher dimensional analogues, with speeds determined by their curvatures. Of significant interest in this research are the mean curvature flow and the Ricci flow, which are often likened to the diffusion of heat and the heat equation. These flows leverage the concept of diffusion to canonically deform geometric objects towards equilibrium states, which are significantly easier to characterize or analyze. Geometric flows have thus demonstrated valuable geometric applications, particularly in classification theorems and geometric inequalities. Notable examples include the utilization of Ricci flow to the resolution of the long-standing Poincare conjecture, and the application of inverse mean curvature flow in establishing the Riemannian Penrose inequality, a crucial statement in general relativity. Beyond geometry, geometric flows find extensive use in various physical problems. For instance, mean curvature flow plays a role in describing interface evolution in multiphase physical models. Likewise, it is employed in material science to model the growth of cells, grains, and bubbles. Additionally, a discrete version of Ricci flow is applied in data science, such as in community detection. The study of geometric flows and their diverse applications represents a vibrant and influential area of mathematics. This project aims to enhance the field's impact by involving graduate students in different facets of research and fostering collaborations among researchers across diverse disciplines and institutions.<br/><br/>The forthcoming research will concentrate on ancient solutions to the mean curvature and Ricci flows. Ancient solutions refer to solutions that have existed for all times in the past. They are of particular interest due to their significance in studying singularities, which present challenges to the geometric applications of these flows. Understanding the geometry and behavior of ancient solutions is therefore crucial. Despite significant advances in this area in recent decades, there remain large classes of such solutions about which little is known. The proposed project seeks to offer classification results for ancient solutions of mean curvature, mean curvature type, and Ricci flow, without making noncollapsing or compactness assumptions. Our strategy involves developing a method to construct new collapsed ancient solutions and classify them based on specific symmetry assumptions. Drawing on our established methods for constructing and characterizing ancient convex solutions to mean curvature flow, which have proven successful and adaptable, we aim to explore the potential existence of a dichotomy theorem for Ricci flow akin to the one for mean curvature flow, by introducing a width concept for Ricci flow. Finally, by refining and enhancing these techniques, we aim to furnish a more comprehensive classification result for convex collapsed solutions of mean curvature flow.<br/><br/>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.
