Evolution equations in geometry and related fields

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Evolution equations are fundamental objects in the sciences, describing how natural phenomena change over time. The two main parts of this project concern evolution equations or results that should lead to applications to evolution equations. The first part is about mean curvature flow which has been used and studied in materials science for almost a century to model things like cell, grain, and bubble growth. The second is related to the new methods for dealing with the collections of transformations of (the gauge group) for non-compact spaces with applications to Ricci flow. The mean curvature flow’s computational and theoretical study and its applications as well as that of other similar flows have had enormous impact in diverse areas of pure and applied mathematics, theoretical physics,material science, and engineering. For broader impacts the PI will continue mentoring and advising graduate students and post-doctoral scholars, write a second graduate level book on “Heat equation in Analysis, Geometry and Probability", and disseminate their work via lectures, workshops and conferences. In mean curvature flow (MCF) the project will study questions about stable structures; both for hypersurfaces and submanifolds of higher codimension. In the second project a new way of dealing with the diffeomorphism group will be investigated that should be useful in many settings. Indeed, in many problems across a wide swath of areas singularities occur. Understanding them and the set where they form becomes a question of great importance. When the objects sit inside some canonical space, one can use ``extrinsic'' reference points to make progress on important questions like these. However, many key problems are defined intrinsically and there are no canonical reference points. In those problems, the infinite dimensional diffeomorphism group (the gauge group) becomes a major issue and dealing with it, and understanding it, becomes a major obstacle. Ricci flow is such an example. The PI expects that the projects on MCF, Ricci flow and gauge groups will have implications to a number of other fields of mathematics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.