Optimal Geometry: Theory and Computation

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Sullivan0071520 The investigator, with his collaborators, studies geometric optimization problems like finding minimum-energy shapes for surfaces and knots in space. They extend their recent classification of embedded constant-mean-curvature surfaces with three ends to the more general case of surfaces with any number of coplanar ends, and also investigate in detail surfaces with truncated ends. In addition, the investigator computes these surfaces numerically, in order, for instance, to create interactive computer graphics. This project uses Willmore's elastic bending energy, and its gradient flow, to discover new minimal surfaces in euclidean and spherical space. The Willmore flow has been recently shown to have short-time solutions, but the investigator considers whether it can fail to have long-time solutions. This project also studies configurations for knots which minimize ropelength, giving new lower bounds for the ropelength of small knots, and new asymptotic bounds on the growth of ropelength with crossing number. Finally, the investigator uses his experience with numerical modeling of curves and surfaces to give new understanding of geometrically natural discretizations for quantities related to curvature. Many real-world problems can be cast in the form of optimizing some feature of a shape; mathematically, these become variational problems for geometric energies. For instance, thin films, like those in foams, usually minimize their area and thus are constant-mean curvature surfaces. Cell membranes are more complicated bilayer surfaces which minimize an elastic bending energy known mathematically as the Willmore energy. Knotted curves achieve an optimal shape when a rope is pulled tight, or if a charged knotted wire repels itself electrostatically; understanding such configurations helps explain the behavior of biological molecules like DNA. This project explores such phsically natural problems, which remain challenging from both theoretical and computational standpoints.