Calculus of Variations

Fundamental laws in physics often can be derived from variational principles. Similarly, in differential geometry, Riemannian manifolds enjoying special properties can be characterized variationally. The Calculus of Variations has proven to be both a means of understanding the laws of nature as well as a universal tool for finding solutions and for analyzing their properties.

Partial Differential Equations

With the aim of understanding qualitative properties of solutions to various nonlinear partial differential equations that arise in geometric analysis and mathematical physics one current focus is the class of geometric flows, including scalar curvature flow and the harmonic map heat flow, or wave maps; another current focus is on problems involving metrics of prescribed curvature.