Variational Problems in Low Dimensional Geometry and Topology

低维几何和拓扑中的变分问题

• 批准号: 0076085
• 负责人: Robert Kusner
• 金额: $ 12.08万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0076085&HistoricalAwards=false
• 关键词: Variational; Problems; Low; Dimensional; Geometry

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.

AbstractAward：DMS-0076085首席研究员：Robert Kusner该项目继续研究极值曲面和相关的几何变分问题，并应用于低维拓扑和自然科学。 主要研究方向为：（1）极小化Willmore弯曲能的曲面的存在性和唯一性;（2）确定具有有限拓扑的完全常数平均曲率曲面（CMC）的模空间;（3）布朗运动的几何分析和具有无限拓扑的适当嵌入极小曲面上的势理论;（4）极小化能量的纽结和链环的存在性和几何。 此外，首席研究员将在GANG与资深科学家N。施密特关于利用黎曼曲面上的平坦（环）SU（2）-丛的单值性构造CMC曲面的方法这将包括一个实验方面，（计算和可视化的家庭的例子），以及理论方面（可积系统与泛函/几何分析方法之间的关系）。虽然这项工作的动机主要是美学，但应该注意的是，CMC和Willmoresurfaces出现在物理情况下，作为流体之间的界面，因此它们的几何形状可能在预测某些天然和合成材料的行为方面具有一定的价值。 例如，当电子微元件的自动焊接导致短路或开路时，大量的资源被浪费：一些主要研究人员在CMC表面上的工作直接应用于这个问题;他自由地分享他的想法，（在NIST和其他地方）人们试图解决它。主要研究者最近对绳结和链环的长度的研究（其中一些发表在普通科学杂志《自然》上）代表了第一次仔细的数学研究--在某些情况下，正确--在文献中关于长聚合链（如DNA）的几何结构的主张;他正积极地与自然科学家围绕这一主题展开合作。 施密特在新CMC表面上的可视化工作已被记录在最近的GANG电影“表面，流动整体”中，其中的场景可在www.gang.umass.edu上获得。