Proposal: DMS-9802646 Principal Investigator: Eugenio Calabi
Eugenio Calabi proposes to continue his research on variational problems and partial differential equations arising from various aspects of the geometry of manifolds. Herman Gluck proposes to continue his research on spectral geometries for the writhing of knots and the helicity of vector fields, on embedding and knotting of positive curvature surfaces in 3-space, and on the existence and uniqueness of volume-minimizing cycles in Grassmann manifolds. Julius Shaneson plans to apply higher dimensional Euler-MacLaurin formulae with remainder to some problems in number theory related to counting of lattice points, and to continue his research on singular spaces, characteristic classes, and related classification problems. Wolfgang Ziller proposes to continue his research on positively curved cohomogeneity-one manifolds, minimal isometric immersions into spheres, weakly symmetric spaces, and primitive subgroups of Lie groups.
One aspect of Calabi's work is an application to computerized image enhancing by a method that does not interfere with the optical distortion due to the projection into a photographic plate. One aspect of Gluck's work applies to the writhing and coiling of DNA and to the writhing of magnetic field lines in the Crab Nebula. Shaneson plans to use Euler-MacLaurin formulae with remainder to study the numer of lattice points in curved regions of the plane and higher-dimensional spaces, with a view towards progress on some basic questions in number theory and also some practical applications to problems involving complex networks. Ziller plans to continue, in joint work with Karsten Grove of the University of Maryland, the study of cohomogeneity-one metrics with positive or nonnegative sectional curvature; finding new examples with this property is important for geometry and also very difficult, as one can see by the fact that new ones are discovered only about every 15 years.
