Teichmuller theory and Low-Dimensional Geometric Variational Problems

摘要奖：DMS-1007383主要研究人员：迈克尔·沃尔夫主要研究人员将继续他对二维几何变分问题的研究。特别是，他在经典极小曲面理论的几个领域提出了一些项目，这些项目都与使用TeichMuller理论来证明欧几里德三维空间中极小曲面的存在唯一性或性质有关。他建议在泰希穆勒理论的两个领域进行研究：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式的视角。此外，他还建议研究曲面上凸射影结构中出现的问题，以及双曲平面上退化调和拟等距的工作。这些信息来自于他早期在高能调和地图上的工作。近年来，国际教育局对各级教育给予了相当的关注。他指导研究生；他协调莱斯大学的Vigre项目，该项目由大约12个数学科学领域的小研究小组组成，这些小组的垂直组织范围从本科生阶层到研究生和博士后水平，再到永久教员；他自己也参加其中一个小组，他给教师讲课，并担任增进K-12教师数学理解的项目的科学顾问。他服务于更广泛的数学界，在三家期刊的编辑委员会任职，通过组织大大小小的会议和向普通听众发表演讲。他参与了本科生的咨询工作。PI将继续所有这些活动。科学的指导原则之一是自然是有效的：当我们遇到自然现象时，我们希望我们找到的形状将使用最少的材料来建造，或者是可以用材料制成的最薄的形状，或者将跨越尽可能大的区域，或者其他某种优化。与此同时，许多自然系统的一个共同的几何成分是一个二维表面，通常以某种曲面的方式进行配置。这些表面存在于所有尺度上，从蛋白质的边界到大脑表面，再到磁层的边界。在这个项目中，我们研究了曲面形状的各个方面与它们可能最优化的数量之间的关系，以及曲面形状的一个特征变形如何影响该曲面的其他性质。

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.