Teichmuller theory and Low-Dimensional Geometric Variational Problems

Teichmuller理论和低维几何变分问题

• 批准号: 1007383
• 负责人: Michael Wolf
• 金额: $ 14.2万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007383&HistoricalAwards=false
• 关键词: Teichmuller; theory; Low; Dimensional; Geometric

AbstractAward: DMS-1007383Principal Investigator: Michael WolfThe principal investigator will continue his research in two-dimensional geometric variational problems. In particular, he proposes projects in several areas of classical minimal surface theory, all concerned with using Teichmuller theory to prove existence or uniqueness or properties of minimal surfaces in Euclidean three-space. He proposes research in two areas of Teichmuller theory: Weil-Petersson geometry and grafting. Both bring a Riemannian perspective to problems often studied via other methods. In addition, he proposes studying problems arising in convex projective structures on surfaces as well as work on degenerating harmonic quasi-isometries of the hyperbolic plane. These last are informed by his earlier work on high energy harmonic maps. In recent years, the PI has focused considerable attention upon education at all levels. He supervises graduate students; he coordinates a VIGRE program at Rice University encompassing roughly a dozen small research groups in the mathematical sciences whose vertical organization ranges from the undergraduate stratum through the graduate and postdoctoral level to the permanent faculty; he participates in one of these groups himself, he lectures to teachers and acts as a scientific advisor to programs to enhance the mathematical understanding of K-12 teachers. He serves the broader mathematical community by serving on the editorial board of three journals, through organizing large and small conferences and through lectures to general audiences. He is quite involved with undergraduate advising. The PI will continue with all of these activities.One of the guiding principles of science is that Nature is efficient: when we encounter natural phenomenon, we expect that the shapes we find will use the least material for their construction, or be the thinnest that can be made with a material or will span the largest region possible, or some other sort of optimization. At the same time, a common geometric component of many natural systems is a two-dimensional surface, usually configured in some curved manner. These surfaces occur at all scales, from the boundary of a protein to the surface of the brain to the frontier of the magnetosphere. In this project, we investigate problems that relate aspects of the shapes of surfaces to quantities they might optimize, and also how deforming one feature of the shape of a surface affects other qualities of that surface.

摘要奖项：DMS-1007383 首席研究员：Michael Wolf 首席研究员将继续他在二维几何变分问题方面的研究。特别是，他提出了经典最小曲面理论的几个领域的项目，所有这些项目都涉及使用泰希米勒理论来证明欧几里得三空间中最小曲面的存在性或唯一性或属性。他提出了泰希米勒理论的两个领域的研究：韦尔-彼得森几何和嫁接。 两者都为经常通过其他方法研究的问题带来了黎曼视角。此外，他建议研究表面凸射影结构中出现的问题以及双曲平面的简并调和准等距研究。最后这些是由他早期在高能谐波图方面的工作得出的。近年来，PI高度重视各级教育。他指导研究生；他负责协调莱斯大学的 VIGRE 项目，该项目包括大约十几个数学科学领域的小型研究小组，其垂直组织范围从本科生到研究生和博士后，再到常任教职人员；他亲自参加其中一个小组，为教师讲课，并担任旨在增强 K-12 教师数学理解的项目的科学顾问。 他通过担任三种期刊的编辑委员会成员、组织大大小小的会议以及向普通观众演讲，为更广泛的数学界服务。他积极参与本科生的咨询工作。 PI 将继续开展所有这些活动。科学的指导原则之一是自然是高效的：当我们遇到自然现象时，我们期望我们发现的形状将使用最少的材料来构建，或者是可以用材料制成的最薄的形状，或者跨越尽可能大的区域，或者进行某种其他类型的优化。 同时，许多自然系统的常见几何组件是二维表面，通常以某种弯曲的方式配置。 这些表面出现在各个尺度上，从蛋白质的边界到大脑的表面再到磁层的前沿。 在这个项目中，我们研究了将表面形状的各个方面与它们可能优化的数量联系起来的问题，以及使表面形状的一个特征变形如何影响该表面的其他质量。