Research at Undergraduate Institutions Collaborative Research, Energy Functions for Knots

本科院校合作研究，结的能量函数

• 批准号: 9420088
• 负责人: Gregory Buck
• 金额: $ 5.75万
• 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: 1994
• 资助国家: 美国
• 起止时间: 1994-08-15 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9420088&HistoricalAwards=false
• 关键词: Research; Undergraduate; Institutions; Collaborative; Energy

9420088 Buck The investigator and his colleague undertake an extensive computational and analytical study of the energy surfaces given by the energy functions on finite-dimensional conformation spaces of knots. The focus is on two conformation spaces: polygonal knots, and harmonic knots, where harmonic knots are those given by finite Fourier series parametrizations. Recent theorems relate the invariants given by these energy functions to classical invariants such as crossing number. So this project should lead to a better understanding of knots realizable with relatively low degrees of freedom. Knot theory is the mathematical study of the everyday objects called knots. Once thought the purview of theoretical mathematicians, knot theory has recently found a wide range of applications, in fields as diverse as molecular biology and cosmology. For example, DNA molecules are usually knotted, and in the replication process must somehow become untangled. In this project the investigators apply sophisticated computing techniques to the study of knots. In particular, they continue the development of a new approach to knots: physical knot theory. The mathematical knot is given physical properties, and knot types are distinguished by computations involving these properties. For example, knot conformations can be assigned an energy, and then knot types can be distinguished by considering minimum energy conformations. The conceptual basis of the approach is familiar to researchers in biology, chemistry, and physics as well as mathematics. The project produces both theoretical results and software.

小行星9420088 调查员和他的同事进行了广泛的计算和分析研究的能量表面的能量函数有限维构象空间的结。 重点是两个构造空间：多边形结和调和结，其中调和结是由有限傅里叶级数参数化给出的。 最近的定理与这些能量函数的经典不变量，如交叉数给出的不变量。 因此，这个项目应该导致一个更好的了解结实现相对较低的自由度。 纽结理论是对称为结的日常物体的数学研究。 纽结理论曾一度被认为是理论数学家的研究领域，但最近却在分子生物学和宇宙学等领域得到了广泛的应用。 例如，DNA分子通常是打结的，在复制过程中必须以某种方式解开。 在这个项目中，研究人员将复杂的计算技术应用于结的研究。 特别是，他们继续发展一种新的方法来结：物理结理论。 数学结被赋予物理属性，并且结类型通过涉及这些属性的计算来区分。 例如，可以为结构象分配能量，然后可以通过考虑最小能量构象来区分结类型。 该方法的概念基础对于生物学、化学、物理学以及数学领域的研究人员来说是熟悉的。 该项目产生了理论结果和软件。