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 there are still many fundamental open questions in this area that are strikingly intuitive and elementary to state. Studying these questions may stimulate useful developments in pure mathematics and lead to wider applications in science and technology. For instance, the isoperimetric inequality has numerous applications due to its connections with a host of other important inequalities, including the Sobolev inequality in mathematical analysis and the Faber-Krahn inequality in spectral analysis. Furthermore, the rigidity of surfaces may have applications for stability of complicated domes in modern architecture, while the medial axis, or cut locus of distance functions, is an important tool in shape recognition, which is of interest in computer graphics and mathematical biology. These questions are ideal for introducing the public to the exciting world of modern mathematics and arousing the interest of beginning students in geometry. This project will engage in a range of activities, including accessible public lectures and articles, to promote these topics. This project is concerned with curves and surfaces, and more broadly Riemannian submanifolds, spanning a wide range of topics and tools including isoperimetric problems, isometric embeddings, geometric knot theory, polyhedral approximations, h-principle theory, and curvature flows. Some recurring themes throughout these investigations, which the PI conducts in joint work with his students and collaborators, 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 question is how restrictions on curvature, intrinsic metric, or various boundary conditions influence the global shape of a curve or a hypersurface or allow an isometric embedding of that object in a space of low codimension. For instance, the PI recently established Zalgaller’s conjecture on the length and shape of the shortest closed curve that contains the unit sphere in its convex hull. Other projects include rigidity of isometric embeddings, unfoldability of convex polyhedra or Durer’s conjecture, optimization problems for space curves, and the study of the cut locus of distance functions, or medial axis, of contractible regions in Riemannian manifolds. 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.