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

9625641摩根这项建议适用于极小曲面和更一般的常值平均曲率曲面。研究人员计划使用几何测量理论和校准以及计算机模拟的方法。更具体地说，提出的主题包括：空间的最小周长划分成相等的体积；最小化平衡配置的性质。还有一个涉及双泡沫和最小集群的本科生研究项目--这是在本科生机构研究计划下完成的。关于用最小面积方法封闭和分开两个等体积区域的所谓双气泡猜想，最近给出了一个计算机证明，这将在这个项目中得到进一步的探索。最小曲面和最小化簇在各种环境中作为物理曲面出现，因为它们具有某些极端性质，例如，表面积最小化和能量最小化。SMAP薄膜和肥皂泡表面提供了现成的极小表面和恒定平均曲率表面的例子。此外，最小团簇和周期最小曲面也可用于模拟复合聚合物体系和一般凝聚态物质。

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.