Principal Investigator: Dan F. Knopf
The broad goals of the investigator's research program are to find and classify optimal geometries using curvature flows. This subject is currently enjoying a period of intense interest and rapid progress, inspired by Perelman's landmark results in Hamilton's program to resolve the Poincare and Geometrization Conjectures by Ricci flow. In applications of Ricci flow, one evolves a Riemannian metric on a manifold to simplify and improve its geometry. In many cases, these improvements involve changes in topology that are triggered by singularity formation. Such improvements are possible because solutions of Ricci flow (like many other nonlinear PDE) are expected to exhibit very special, highly symmetric profiles in a space-time neighborhood of a developing singularity. A thorough and deep understanding of singularity formation is therefore of critical importance for extracting topological and geometric information from Ricci flow behavior. The investigator's immediate objectives focus on singularity formation for curvature flows, particularly (1) analytic aspects (asymptotics) of singularity formation, (2) the structure of singularities in dimension four, (3) singularities of Kaehler-Ricci flow, (4) the structure of Perelman's entropy and reduced distance, and (5) applications of geometric flows to problems in physics. Progress toward these objectives will lead to new applications of Ricci flow in the geometry and topology of manifolds, productive insights into parallels between various geometric flows, and new applications of curvature flows to problems motivated by materials science and physics. These objectives are well suited to collaborations with other researchers in geometric evolution equations and in related fields like comparison geometry, low-dimensional topology, and nonlinear analysis.
The partial differential equations that arise in curvature flows are remarkably similar to equations used to model heat propagation, the movement of oil in shale and thin films, combustion in porous media, and certain effects in plasma physics. By increasing our understanding of singularity formation for such equations, the research elements of this project may contribute to progress in these applications. The education elements of the project are very tightly integrated with its research focus. The cornerstone of this integration is a series of annual workshops: each will contain a graduate education component, a research component, and an undergraduate outreach component. The education component includes an organized seminar to prepare graduate students for participation in the research component. The outreach component includes invitations to local undergraduate students to interact with the visiting researchers with the goal of attracting those students to further studies in mathematics. The project will benefit undergraduate education in two other ways: the investigator will develop a new course in optimal geometry and integrate it into the liberal-arts honors curriculum at the University of Texas, and the investigator will organize and direct an undergraduate learning-by-discovery project in which students investigate combinatorial curvature flows.
