My research centers on geometric evolution equations, notably the Ricci flow and related curvature flows. I plan to study seven areas in which I have obtained prior results, and where continued work is likely to yield new and useful mathematics. [1] When a flow converges, it is valuable to study the stability of its limit, in order to improve our global understanding of the dynamics of flows. [2] If a flow fails to converge but behaves in a nonsingular way, one can still study the dynamics of this collapse by classifying the asymptotic behavior of nearby solutions. [3] In most cases, a flow does become singular; so it is of paramount importance (particularly in regard to Hamilton's program to resolve Thurston's Geometrization Conjecture) to develop a better classification of singularities. [4] The basic method of studying singularities is the construction of a sequence of parabolic dilations (blow-ups). To take limits of these solutions, one must obtain (partial) injectivity radius estimates by various means. [5] The most powerful (but perhaps most difficult) way to obtain such injectivity radius estimates would be to study and extend existing Harnack estimates of the type pioneered by Li and Yau and further developed by Hamilton. [6] It is also useful to study the asymptotic behavior and stability of parabolic dilations at certain model singularities (a method which has been very fruitful in studying the mean curvature flow). [7] Further information about singularities can be obtained by constructing and studying solitons: self-similar solutions that often arise as limits of blow-ups. Moreover, Kaehler Ricci solitons have interesting connections with complex geometry and algebraic geometry.
Geometric evolution studies the way an object's shape changes. In some cases, such as the mean curvature flow and porous media flow, the motivation is to model certain physical phenomena such as the motion of an interface in forming metallic alloys, the shape of a thin film of highly viscous oil, or the flow of oil in shale. In other cases, the goal is to improve the shape of an object, either to find optimal (most efficient) shapes, or else to help mathematicians recognize and classify geometric objects. My own research is part of a large program to resolve one of the most compelling open questions in mathematics: the desire to understand and classify all possible 3-dimensional shapes. But regardless of whether their motivation comes from material science or pure mathematics, all geometric evolution problems have much in common; so that the field benefits from rich cross-fertilization. In particular, ideas and techniques that are developed for any of these highly nonlinear problems are usually quickly adaptable to related applications.
