CAREER: Classical Problems in Differential Geometry, Topology, and Convexity

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Abstract Award: DMS-0332333 Principal Investigator: Mohammad Ghomi The principal investigator is interested primarily in the interplay between the geometry and topology of submanifolds, including low dimensional problems in Euclidean space (curves and surfaces). These investigations often involve some notion of convexity, and include the following categories: (i) Shadows (or shades) on illuminated hypersurfaces, and their application to geometric variational problems; (ii) Embeddings of manifolds in Euclidean space without creating parallel or intersecting tangent lines (totally skew embeddings), and their relation to quadric hypersurfaces and nonsingular bilinear maps; (iii) Global properties of locally convex hypersurfaces with boundary, including connections with Monge-Ampere equations, and a new convex hull property which is dual to that of negatively curved surfaces; (iv) Certain deformations of space curves (unfoldings), and their application to study of extremals of knot energies and distortion. Curves and surfaces are to geometry what numbers are to algebra. They form the basic ingredients of our visual perception, and inspire the development of far reaching mathematical tools. For instance, those aspects of the PI's work dealing with shadows on illuminated surfaces are motivated in part by a study of soap films, and have connections to computer vision (the ``shape from shading" problems). Further, the investigations on knot energies may be of interest in studying DNA. Yet, despite an abundance of potential applications and centuries of pure study, there are still numerous open problems in submanifold geometry and topology which are strikingly intuitive and elementary to state. The PI believes that advertising these problems at an early stage is an excellent tool for sparking the interest of students in mathematical research. With the aid of computer workshops, courses, seminars, and the lecture series proposed in this project, the PI plans to communicate the beauty and excitement of geometric problems to as wide an audience as possible.