Research at Undergraduate Institutions, Collaborative Research: Physical Knot Theory

本科院校研究、合作研究：物理结理论

• 批准号: 9706865
• 负责人: Gregory Buck
• 金额: $ 6.52万
• 依托单位: Saint Anselm College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706865&HistoricalAwards=false
• 关键词: Research; Undergraduate; Institutions; Collaborative; Physical

Buck 9706865 The investigator collaborates with Prof. J. Simon, University of Iowa, to study knots and their applications. Knots are closed loops in space that may be tangled in complicated and essential ways. While knots usually have been studied as idealized one dimensional filaments, the focus here is on "physical knots," knots made of real physical stuff, from rope to DNA or other large flexible molecules. These are modeled as mathematical knots endowed with physical like properties such as self repelling energy or thickness. The twin goals of physical knot theory are to mathematically model and help understand the real physical systems, and to use the physically inspired measures of knot-complexity to develop novel methods for knot recognition and classification. The specific targets of this project include: determining precise relations between different measures of knot complexity, understanding critical points and the configuration space of polygonal knots, understanding the connections between parametrizations and topological properties of harmonic knots, extending knot energy ideas to other structures such as graphs, modeling gel electrophoresis of DNA loops, and using knot energies to help understand physics phenomena such as self-radiating tubes and knotted vortices. Knotting and tangling happen at every scale studied by science, from microscopic DNA loops to everyday rope to tangled magnetic field loops in the solar corona. The investigators and their students will study problems that are fundamental and arise in all the physical systems: How are knots and tangles created? What mathematical properties of the physical systems lead to knots becoming simplified or completely untangled? How do the mathematical properties of different kinds of knots influence their physical behavior? One focus task is to gain a better understanding of gel electrophoresis of DNA loops, hence of the gel process itself; this is one of the most important b iotechnology techniques, so a solid understanding is important. The project requires powerful computing and visualization, so it will train students, as well as stretch the technology, in these areas.

巴克9706865研究人员与爱荷华大学的J·西蒙教授合作研究结及其应用。结是空间中的闭合环，可能会以复杂和必要的方式缠绕在一起。虽然结通常被研究为理想化的一维细丝，但这里的重点是“物理结”，即由真正的物理材料制成的结，从绳子到DNA或其他大的柔性分子。它们被建模为具有类似物理属性的数学结，例如自排斥能量或厚度。物理结点理论的双重目标是对真实的物理系统进行数学建模和帮助理解，并使用物理启发的结点复杂性度量来开发新的结点识别和分类方法。该项目的具体目标包括：确定不同节点复杂性度量之间的精确关系，了解关键点和多边形节点的配置空间，了解参数化与调和节点的拓扑性质之间的联系，将节点能量的概念扩展到其他结构如图形，对DNA环的凝胶电泳进行建模，以及使用节点能量来帮助理解物理现象，如自辐射管和打结的涡流。打结和纠缠发生在科学研究的每一个尺度上，从微小的DNA环到日常生活中的绳索，再到太阳日冕中纠结的磁场环。研究人员和他们的学生将研究在所有物理系统中出现的基本问题：结和缠结是如何产生的？物理系统的哪些数学特性导致结变得简化或完全解开？不同类型的结的数学特性如何影响它们的物理行为？其中一个重点任务是更好地了解DNA环的凝胶电泳，从而了解凝胶过程本身；这是最重要的生物技术之一，因此，扎实的理解是重要的。该项目需要强大的计算和可视化能力，因此它将培训学生，并在这些领域推广技术。