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

数学科学：RUI：最小曲面、簇和奇异几何

• 批准号: 9625641
• 负责人: Frank Morgan
• 金额: $ 9.6万
• 依托单位: Williams College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-15 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625641&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Minimal; Surfaces

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.

9625641 Morgan 该提案涉及最小曲面和更一般的恒定平均曲率曲面领域。研究人员计划使用几何测量理论和校准方法以及计算机模拟。更具体地说，提出的主题包括：将空间划分为等体积的最小周长；最小化平衡构型的性质。还有一个涉及双气泡和最小簇的本科研究项目 - 这是在本科机构研究计划下完成的。所谓的双气泡猜想，即以最小面积方式封闭和分离两个等体积区域的猜想，最近得到了计算机证明，这将在本项目中进一步研究。 最小表面和最小簇在各种设置中作为物理表面出现，因为它们具有某些极值属性，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最小表面和恒定平均曲率表面的现成示例。此外，最小簇和周期性最小表面通常可用于模拟复合聚合物系统和凝聚态物质。