Computational Methods in Mathematics and the Physical Sciences

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Norman 9626804 The investigator and his colleagues pursue interdisciplinary research and training in an environment supporting modern scientific computation and advanced computer graphics. The areas of investigation involve partial differential equations, integrable systems, and variational problems in geometry and topology. Particular topics include: spin structures on surfaces with applications to the geometry of immersions, particularly to rigidity and period problems; surfaces that minimize bending energy subject to natural constraints; the moduli spaces of constant mean curvature embeddings and immersions, using mechanics, symplectic geometry and integrable systems; construction of embedded minimal surfaces by desingularizing immersed minimal surfaces; elastic curves and their higher order soliton analogues; electrostatic energies associated with knots and links, and the behavior of their gradient flows; curve and surface interfaces between distinct thermodynamic phases in disequilibrium; and statistical equilibrium solutions to the Euler equations for incompressible fluid flow. The Center for Geometry, Analysis, Numerics & Graphics (GANG) pursues interdisciplinary research and training in an environment supporting modern scientific computation and advanced computer graphics. The mathematical work at GANG is motivated by - and can shed light upon - many interesting natural systems that arise in both "applied" and "basic" research. For example, the folding and entanglement of long molecules, like proteins, DNA or synthetic polymers, can be dynamically modeled using elastic and electrostatic curve energies that were originally developed at GANG to study the geometry and topology of knots and links; such models may be useful in predicting the strength of polymer materials or the biochemical effect of pharmaceuticals. And a quite unexpected and fundamental new phenomenon - the "conformal diffusion" of tiny phospholipid vesicles, reported by French physicists in the Augu st 1995 issue of Science - was stimulated by work at GANG on the bending energy of surfaces. The computation and visualization facilities at the GANG laboratory serve an important and interrelated triple duty by permitting the pioneering mathematical experiments to be carried out in the first place, providing a fertile environment for the education and training of students, and aiding communication with the general public (to paraphrase the late physicist, Eugene Wigner) on the remarkable effectiveness of mathematics in the natural world.