CAREER: Investigating Ricci flow singularity formation

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AbstractAward: DMS-0545984Principal Investigator: Dan F. KnopfThe broad goals of the investigator's research program are to find andclassify optimal geometries using curvature flows. This subject iscurrently enjoying a period of intense interest and rapid progress,inspired by Perelman's landmark results in Hamilton's program toresolve the Poincare and Geometrization Conjectures by Ricci flow. Inapplications of Ricci flow, one evolves a Riemannian metric on amanifold to simplify and improve its geometry. In many cases, theseimprovements involve changes in topology that are triggered bysingularity formation. Such improvements are possible becausesolutions of Ricci flow (like many other nonlinear PDE) are expectedto exhibit very special, highly symmetric profiles in a space-timeneighborhood of a developing singularity. A thorough and deepunderstanding of singularity formation is therefore of criticalimportance for extracting topological and geometric information fromRicci flow behavior. The investigator's immediate objectives focus onsingularity formation for curvature flows, particularly (1) analyticaspects (asymptotics) of singularity formation, (2) the structure ofsingularities in dimension four, (3) singularities of Kaehler-Ricciflow, (4) the structure of Perelman's entropy and reduced distance,and (5) applications of geometric flows to problems inphysics. Progress toward these objectives will lead to newapplications 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 bymaterials science and physics. These objectives are well suited tocollaborations with other researchers in geometric evolution equationsand in related fields like comparison geometry, low-dimensionaltopology, and nonlinear analysis.The partial differential equations that arise in curvature flows areremarkably similar to equations used to model heat propagation, themovement of oil in shale and thin films, combustion in porous media,and certain effects in plasma physics. By increasing our understandingof singularity formation for such equations, the research elements ofthis project may contribute to progress in these applications. Theeducation elements of the project are very tightly integrated with itsresearch focus. The cornerstone of this integration is a series ofannual workshops: each will contain a graduate education component, aresearch component, and an undergraduate outreach component. Theeducation component includes an organized seminar to prepare graduatestudents for participation in the research component. The outreachcomponent includes invitations to local undergraduate students tointeract with the visiting researchers with the goal of attractingthose students to further studies in mathematics. The project willbenefit undergraduate education in two other ways: the investigatorwill develop a new course in optimal geometry and integrate it intothe liberal-arts honors curriculum at the University of Texas, and theinvestigator will organize and direct an undergraduatelearning-by-discovery project in which students investigatecombinatorial curvature flows.