Geometry of Curves and Surfaces

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

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. Yet despite centuries of study, and an abundance of potential applications, there are still many fundamental open problems in this area which are strikingly intuitive and elementary to state. Studying these problems may stimulate useful developments in mathematics, or lead to wider applications in science and technology. For instance, the Principal Investigator's work on rigidity problem for surfaces may have applications for stability of complicated domes in modern architecture, or various physical frameworks. Further, various techniques which the Principal Investigator is proposing to develop could be useful in computer aided design, mathematical physics, and the emerging field of discrete differential geometry. Finally, these problems are ideal for introducing the general public to the exciting world of modern day mathematics, and arousing the interest of beginning students in Geometry. The PI's research on curves and surfaces, and more generally Riemannian submanifolds, spans a wide range of topics including isometric embeddings, h-principle theory, isoperimetric problems, geometric knot theory, polyhedral approximations, and connections with real algebraic geometry. Some recurring themes throughout these investigations are various notions of convexity or optimization, and the interaction between geometric and topological concepts, or local versus global properties of submanifolds. More specifically, a typical problem is how restrictions on curvature, intrinsic metric, or various boundary conditions, influence the global shape of a curve or a hypersurface, or even allow an embedding of that object in a Euclidean space of low codimension. A fundamental problem in this area is that of isometric rigidity of surfaces: can one continuously deform a smooth closed surface in Euclidean space without changing its intrinsic metric? The PI also considers a number of related problems involving the self-linking number or vertices of closed curves, spherical images of closed surfaces, and various deformations of submanifolds which preserve the sign or magnitude of the curvature. Other projects include unfoldings of convex polytopes, and the study of the cut locus or medial axis of contractible regions in Euclidean space.