Variational Problems in Low Dimensional Geometry and Topology

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Abstract Award: DMS-0076085 Principal Investigator: Robert Kusner This project continues research on extremal surfaces and related geometric variational problems, with applications to low-dimensional topology and the natural sciences. The principal investigator will work on (1) existence and uniqueness of surfaces minimizing the Willmore bending energy, (2) determination of the moduli spaces of complete constant mean curvature surfaces (CMC) with finite topology, (3) geometric analysis of brownian motion and potential theory on properly embedded minimal surfaces with infinite topology, and (4) the existence and geometry of energy-minimizing knots and links. In addition, the principal investigator will work at GANG with senior scientist N. Schmitt on the approach to constructing CMC surfaces using monodromy of flat (loop) SU(2)-bundles over a Riemann surface; this will include an experimental aspect (computation and visualization of families of examples), as well as a theoretical aspect (relationship between integrable systems and the functional/ geometric analysis methods). While the motivation for most of this work is primarily aesthetic, it should be noted that minimal, CMC and Willmore surfaces arise in physical situations as interfaces between fluids, and thus their geometry may have some value in predicting the behaviors of certain natural and synthetic materials. For example, vast resources are wasted when automatic soldering of electronic microcomponents results in short- or open-circuits: some of the principal investigator's work on CMC surfaces has direct application to this problem; he has freely shared his ideas (at NIST and elsewhere) with people trying to solve it. The principal investigator's recent work on ropelength of knots and links (some published in the general science journal, Nature) represents the first careful effort to mathematically investigate -- and, in certain instances, correct -- claims in the literature about the geometry of long polymeric chains (such as DNA); he is actively collaborating with natural scientists around the world on this topic. Schmitt's visualization work on new CMC surfaces has been documented in the recent GANG film "Surfaces, Flows & Holonomy," scenes of which are available at www.gang.umass.edu. This award is cofunded by the program in Computational Mathematics.