Nonlinear parabolic equations and related geometric problems

Columbia University, New York NY

Investigators

Abstract

This project addresses questions concerning the regularity of and the analysis of singularities of solutions to nonlinear parabolic equations, primarily in connection with more complex problems in differential geometry. A significant part of this project is devoted to the understanding of global-in-time solutions to geometric flows such as the Ricci flow, the Yamabe flow, and the mean curvature flow, and the related problem of the classification of associated singularities and solitons. Other projects include free boundary problems arising from the mathematics of finance, the qualitative behavior to solutions of fully nonlinear elliptic and parabolic equations, and the study of the Ricci flow on Lorentzian manifolds. The project links a wide range of active fields of mathematics, in particular, nonlinear partial differential equations, geometry, and classical analysis. The principal investigator intends to study the applications of the mathematical problems to other disciplines such as quantum field theory, relativity theory, and mathematical finance. Results will be disseminated to the research community at various meetings and by publication of research articles. New courses linking partial differential equations and geometric analysis for graduate students will be designed and implemented.