Award: DMS 1308777, Principal Investigator: Mohammad Ghomi
The principal investigator proposes to continue his work on the theory of curves and surfaces in Euclidean space, and more generally on Riemannian submanifolds of low dimension or codimension. He specializes in applying contemporary methods such as curvature flows and h-principle theory to solve classical problems which often have simple intuitive statements, while their solutions may require sophisticated techniques. The PI's research in this area spans a wide range of topics including isometric embeddings, 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? We also consider 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 inequalities for mean chord lengths of submanifolds, regularity of real algebraic hypersurfaces, and unfoldings of convex polyhedra.
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 pure 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 pure mathematics, or lead to wider applications in science and technology. For instance, the PI's work on rigidity problem for surfaces may have applications for stability of complicated domes in modern architecture, or various physical frameworks. The polyhedral approximation techniques which the PI is proposing could be useful in computer aided design, and the emerging field of discrete differential geometry. The related studies of the Gauss maps of surfaces could be useful in computer vision and optics, while studying isoperimetric problems has been a significant source of enrichment in calculus of variations and mathematical physics. Further, folding-unfolding problems have numerous applications ranging from deployment of satellite dishes in space to implantation of stents in human arteries. Another impact of the proposed activity would be development of connections between various fields, as in the PI's work on tangent cones, which combines concepts from geometric measure theory, algebraic geometry, and convex analysis. 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.
