Collaborative Research: FRG: Minimal Surfaces, Moduli Spaces, and Computation

合作研究：FRG：最小曲面、模空间和计算

• 批准号: 0139887
• 负责人: Michael Wolf
• 金额: $ 42.92万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139887&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Minimal; Surfaces

The global theory of minimal surfaces in space is in a phase ofexplosive growth. Many new methods of constructing completeembedded minimal surfaces have recently been found; in place of adearth of examples just a few years ago, we now have a quitevaried collection of surfaces, including infinite families. Abasic problem is to classify these examples, i.e. collect theminto families with common properties and understoodlimits. Fruitful approaches have recently been developed thatcombine numerical simulation with methods from the theory ofgeometric structures on surfaces and classical complex analysis,notably Teichmuller theory. Some of the problems the team willattack are: Are there embedded minimal surfaces with oneheliciodal end and arbitrary genus? Is the classical Scherksurface the unique desingularization of a pair of planes? Ofwhat families is the Scherk surface the limit point? At the sametime, the group hopes to make progress on simulation of minimalsurfaces. For example, we hope to set up a library of Weierstrassrepresentations of minimal surfaces which is reproducible, fullydocumented, and useful as a research tool.A guiding philosophy in many areas of science, from physics tobiochemistry to ecology, is that nature is maximally efficient;indeed, many explanations of natural phenomena have at theirfoundation the assumption that the phenomenon has optimized someor several of its features in the expression we witness. At itsbase, this philosophical principle is mathematical in nature: wesearch for principles in science that can be formulated asextremal problems. In mathematics, we can make this assumption ofoptimality very rigorous by expressing it as an equation. Thisleaves us with the problem of understanding all of the solutionsof that equation. In this project, we aim to study one very richtype of optimization problem, the minimal surface problem, whichis already known to have a number of quite subtlecharacteristics. (A minimal surface is one for which each smallpiece has less area than any other surface with the sameboundary.) The study of these surfaces has its origins inphysical problems studied first by Euler; then, a century later,the problem also arose in the studies of the behavior of rotatingdroplets and soap films by F. Plateau. Today the applicationsrange from cosmology to the understanding of the structure ofstable periodic structures in compound copolymers. As in manyother optimization problems, for the minimal surface problem, wedo not have much general information about solutions to theequation expressing extremality. At present though, we do have awide variety of examples which help to guide our intuition, andwhich we are beginning to organize. It is thus a good modelproblem, enriching our understanding of all optimizationproblems.

空间中最小表面的全球理论正在爆炸的阶段。 最近发现了许多构建完整贴合的最小表面的新方法。几年前，我们现在有了包括无限家庭在内的表面，我们现在有很多表面。卑鄙的问题是对这些例子进行分类，即收集具有共同属性和可理解限制的家庭。最近已经开发出了富有成果的方法，该方法采用了表面上的几何结构理论和经典复杂分析（尤其是Teichmuller理论）的方法来模拟。 Willattack团队的一些问题是：是否有嵌入式最小表面具有oneheliciodal端和任意属的属性？经典的scherksurface是一对平面的独特降落化吗？ Scherk表面是什么？在Sametime，该小组希望在模拟最小值的模拟上取得进展。例如，我们希望建立一个最小表面的WeierStrassentations库，这些库可再现，充分记录并用作研究工具。在许多科学领域的指导性哲学，从物理化学学院到生态学到生态学，是最大程度地提高了某些人的自然现象，这是最大程度地提出的，这是一定程度的启发。我们目睹的表达。 在其基础上，这种哲学原理本质上是数学的：科学原则的搜索，可以被提述为无性问题。在数学中，我们可以通过将其表达为方程式来使这种假设非常严格。 这使我们解决了所有方程式解决方案的问题。在这个项目中，我们旨在研究一个非常富有优化问题的非常重要的表面问题，该问题已经众所周知，该问题具有许多相当细微的细分。 （最小的表面是每个天花的面积比具有同一孔的任何其他表面的面积少。）对这些表面的研究的起源是Euler首先研究的起源不体问题。然后，一个世纪后，F. Plateau的旋转曲折和肥皂膜的行为的研究也引起了问题。如今，从宇宙学到对复合共聚物中稳定周期结构的结构的理解。与许多其他优化问题一样，对于最小的表面问题，Wedo对表达极端性的方案的解决方案没有太多的一般信息。不过，目前，我们确实有各种各样的示例，可以帮助我们进行直觉，并开始组织什么。因此，这是一个很好的模型问题，丰富了我们对所有优化问题的理解。