Michael Gage will continue his research on curve shortening problems. These questions ask for information on the asymptotic behavior of a curve, in the plane or in a surface, which is evolving according to its curvature. For example this situation would occur if an elastic curve were allowed to slide in a surface. The techniques to be applied are a mixture of analysis and geometry. Many of the problems will find applications in the evolution of vortex filaments in fluid dynamics and wave fronts in excitable media. Gage's immediate goals are the study of the curve shortening problem on complete surfaces with bounded Gauss curvature; the analysis of the curve shortening flow for locally convex immersed curves; and the analysis of the size of the blow up set for the curvature. He will also study related flows. In particular these include area preserving curve shortening processes and higher dimensional analogues in which curves are replaced by hypersurfaces.
