Variational Problems in Low Dimensional Geometry and Topology

项目编号：dms -0076085项目负责人：Robert kusner该项目继续研究极端曲面和相关的几何变分问题，并将其应用于低维拓扑和自然科学。主要研究者将研究(1)最小化Willmore弯曲能量的曲面的存在性和唯一性，(2)确定有限拓扑的完全恒定平均曲率曲面（CMC）的模空间，(3)具有无限拓扑的适当嵌入最小曲面上的布朗运动和势理论的几何分析，以及(4)最小化能量节和链路的存在性和几何。此外，首席研究员将在GANG与资深科学家N. Schmitt合作，研究在aRiemann表面上使用扁平（环）SU(2)束的单峰构造cmc表面的方法；这将包括实验方面（计算和实例族的可视化），以及理论方面（可积系统和功能/几何分析方法之间的关系）。虽然大多数工作的动机主要是美学，但应该注意的是，最小、CMC和willmore表面作为流体之间的界面出现在物理情况下，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，大量的资源被浪费了：一些主要研究者在CMC表面的工作直接应用于这个问题；他（在NIST和其他地方）自由地与试图解决这个问题的人分享他的想法。首席研究员最近对结和链长度的研究（其中一些发表在普通科学杂志《自然》上）是第一次认真地用数学方法研究——在某些情况下，还纠正了文献中关于长聚合链（如DNA）几何结构的说法；他积极与世界各地的自然科学家合作研究这个课题。Schmitt在新的CMC表面上的可视化工作被记录在最近的GANG电影“表面，流动和全息”中，该电影的场景可在www.gang.umass.edu.This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.