Mathematical Sciences: RUI: Minimal Surfaces, Clusters, and Singular Geometry

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9625641 Morgan This proposal lies in the area of minimal surfaces and more generally constant mean curvature surfaces. The investigator plans to use methods from geometric measure theory and calibrations along with computer simulation. More specifically, the proposed topics include: least perimeter partitions of space into equal volumes; minimizing properties of equilibrium configurations. There is also an undergraduate research project involving double bubbles and minimal clusters - this is done under the Research at Undergraduate Institutions Program. The so called Double Bubble Conjecture on the least-area way to enclose and separate two regions of equal volume was recently given a computer proof, and this is to be further pursued in this project. Minimal surfaces and minimizing clusters arise as physical surfaces in a variety of setting as they have certain extremal properties, e.g., surface area minimizing and energy minimizing. Smap film and soap bubble surfaces provide readily available examples of minimal surfaces and constant mean curvature surfaces. Also, minimal clusters and periodic minimal surfaces can be used to model compound polymer systems and condensed matter in general.