Variational Problems in Low Dimensional Geometry and Topology

摘要奖：DMS-0076085首席研究员：罗伯特·库斯纳这个项目继续研究极值曲面和相关的几何变分问题，并将其应用于低维拓扑学和自然科学。这位主要的研究者将致力于(1)最小化Willmore弯曲能的曲面的存在唯一性，(2)确定具有有限拓扑的完备恒定平均温度曲面(CMC)的模空间，(3)具有无限拓扑的适当嵌入的极小曲面上的布朗运动的几何分析和位势理论，以及(4)能量最小化纽结和链环的存在和几何。此外，首席研究员将与资深科学家N·施密特合作，研究如何在黎曼曲面上使用平坦(环)SU(2)丛的单列来构造CMC曲面；这将包括一个实验方面(实例族的计算和可视化)，以及一个理论方面(可积系统与函数/几何分析方法之间的关系)。虽然这项工作的大部分动机主要是美学的，但应该注意的是，极小曲面、CMC曲面和Willmo曲面作为流体之间的界面出现在物理环境中，因此它们的几何形状在预测某些天然和合成材料的行为方面可能有一些价值。例如，当电子元件的自动焊接导致短路或断路时，大量的资源被浪费：首席研究员在CMC表面的一些工作直接应用于这个问题；他(在NIST和其他地方)与试图解决这个问题的人免费分享了他的藏身之处。首席研究员最近关于结和链接的长度的工作(一些发表在普通科学期刊《自然》上)代表了第一次仔细的数学研究--在某些情况下，正确的--文献中关于长聚合物链(如DNA)几何的主张；他在这个主题上积极地与自然科学家合作。施密特在新的CMC表面上的可视化工作被记录在最近的帮派电影《表面，流动和全息》中，其中的场景可以在www.ang.umass.edu.上获得。该奖项由计算数学项目共同资助。

AbstractAward: DMS-0076085Principal Investigator: Robert KusnerThis project continues research on extremal surfaces and relatedgeometric variational problems, with applications tolow-dimensional topology and the natural sciences. The principalinvestigator will work on (1) existence and uniqueness ofsurfaces minimizing the Willmore bending energy, (2)determination of the moduli spaces of complete constant meancurvature surfaces (CMC) with finite topology, (3) geometricanalysis of brownian motion and potential theory on properlyembedded minimal surfaces with infinite topology, and (4) theexistence and geometry of energy-minimizing knots and links. Inaddition, the principal investigator will work at GANG withsenior scientist N. Schmitt on the approach to constructing CMCsurfaces using monodromy of flat (loop) SU(2)-bundles over aRiemann surface; this will include an experimental aspect(computation and visualization of families of examples), as wellas a theoretical aspect (relationship between integrable systemsand the functional/ geometric analysis methods).While the motivation for most of this work is primarilyaesthetic, it should be noted that minimal, CMC and Willmoresurfaces arise in physical situations as interfaces betweenfluids, and thus their geometry may have some value in predictingthe behaviors of certain natural and synthetic materials. Forexample, vast resources are wasted when automatic soldering ofelectronic microcomponents results in short- or open-circuits:some of the principal investigator's work on CMC surfaces hasdirect application to this problem; he has freely shared hisideas (at NIST and elsewhere) with people trying to solve it.The principal investigator's recent work on ropelength of knotsand links (some published in the general science journal, Nature)represents the first careful effort to mathematically investigate-- and, in certain instances, correct -- claims in the literatureabout the geometry of long polymeric chains (such as DNA); he isactively collaborating with natural scientists around the worldon this topic. Schmitt's visualization work on new CMC surfaceshas been documented in the recent GANG film "Surfaces, Flows &Holonomy," scenes of which are available at www.gang.umass.edu.This award is cofunded by the program in Computational Mathematics.