CAREER: Geometric flows and four-dimensional geometry

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This grant will support a multifaceted and interconnected program of education and research aiming at significant results in both areas. The research goals center around a deeper understanding of the behavior of natural evolution processes arising in physics and geometry. Central examples are the equations governing the evolution of soap bubbles and thin films. A thorough understanding of these equations requires incorporating techniques from physics, analysis, and geometry, and results in a deeper understanding of all of those areas. Aiming at the long term fulfillment of these research goals, we will enact a number of educational activities. These include outreach activities to recruit local high school and community college students into the UC system, opportunities for undergraduates to gain research experience in this area of mathematics, and programs aimed at graduate-level education and training in both research and teaching activities. This project will directly support the NSF goals of maintaining a position of leadership in mathematics research as well as supporting excellence in mathematics education.One of the central themes in geometry is to understand the interplay between the topology and geometry of manifolds, and geometric flows have proven to be powerful tools in understanding this relationship. Despite striking results in three dimensions, Ricci flow alone may not be enough to understand the geometry of four-manifolds. This project centers around a deeper understanding of the long term behavior, singularity formation, and geometric content of certain generalizations of Ricci flow, with a focus on understanding aspects of complex geometry and four-dimensional geometry. We will use a variety of techniques from PDE and geometry to understand these flows, in conjunction with the goal of training students at all levels. These equations appear quite naturally in physical contexts, so progress on the questions outlined in this proposal would yield insights and have impact beyond geometry, specifically to understanding PDE, complex geometry, and mathematical physics. Moreover, these equations, while already worthy of study in their own right, in principle have the potential to address long standing conjectures in topology and geometry. Aiming at the long term fulfillment of these research goals, we will enact a number of educational activities. We will pursue a robust program of graduate student training involving the development of novel courses, learning seminars, and the development of research projects. We also will create a "Future Faculty Program" aimed at training graduate students as future teachers. Building on the success of the Southern California Geometric Analysis Seminar, we will initiate a one-week minicourse for graduate students preceding the conference as well as a poster session for graduate students and postdocs accompanying the conference. Moreover, we will enact a series of recruitment days designed to foster interaction between UC Irvine undergraduates and graduate students and local high school and community college students, with the goal of aiding transfer into the UC system and other four-year institutions.