Mathematical Sciences: Polynomial Invariants in the Theory of Knots

小行星9504471 本计画探讨由状态总和模型、量子群与拓扑量子场论所衍生出的纽结、三维流形与高维流形的不变量。 这包括工作的结构琼斯多项式，瓦西里耶夫不变量，并与量子场论和分子生物学的连接，以及技术和progratures有关使用功能集成拓扑结构。 在过去的十年里，纽结理论通过注入数学物理的一系列技术以及与分子生物学、化学和理论物理的一系列相互关系而获得了觉醒。 主要研究者是一个关键人物，在开始这种关系与物理（统计力学）通过他的介绍使用状态求和模型的多项式不变的结和链接。 在状态求和模型中，纽结被视为一个微型物理系统，纽结的拓扑性质被视为物理相互作用的平均值。 这些模型已经发展到包含来自量子场论的技术，并且在拓扑学及其在生物学和化学中分子相互作用研究的应用中非常有用。 在拓扑学中，这些技巧适用于三维和四维空间的结构以及结。 这意味着这些技术对于三维结构的架构至关重要-从分子到通信网络，从原子到星系。 所有科学领域都是以空间结构为基础的。 这个项目的拓扑研究的特别之处在于，它阐明了大型网络中的拓扑组件，其中节点是主要的例子。 这种网络是RNA和DNA等重要结构的基础，拓扑结构是相互作用的关键。 事实上，从这位首席研究员的角度来看，拓扑学在这些学科中的引入只是一个更大的相互作用的开始，拓扑学被视为研究复杂系统的递归和循环体系结构的基础。 一个星系、一个社会、一个有机体和一个物理系统都有一个共同点，那就是通过循环相互作用的封闭系统而产生的稳定性。 这种相互作用传统上是通过控制论和计算机建模来处理的。 这些拓扑研究为控制论和复杂系统增添了新的维度，提出了新的问题和许多新的难题。 所有这些问题都在新书"结和应用程序"中处理，由主要研究者编辑和贡献。 ***

9504471 Kauffman This project investigates invariants of knots, 3-manifolds and higher dimensional manifolds that arise from state summation models, quantum groups and topological quantum field theory. This includes work on the structure of the Jones polynomial, on Vassiliev invariants, and on connections with quantum field theory and molecular biology, as well as techniques and conjectures about the use of functional integration in topology. In the last decade, knot theory has encountered an awakening through the infusion of a remarkable collection of techniques from mathematical physics and a series of interrelationships with molecular biology, chemistry, and theoretical physics. The principal investigator is one of the key figures in the inception of this relationship with physics (statistical mechanics) via his introduction of the use of state summation models for polynomial invariants of knots and links. In a state summation model the knot is regarded as a miniature physical system, and topological properties of the knot are seen as averages of physical interactions. These models have grown to encompass techniques from quantum field theory and have been useful in topology and its applications to the study of molecular interactions in biology and chemistry. In topology these techniques apply to the structure of three- and four-dimensional spaces as well as to knots. This means that these techniques are essential to the architecture of three- dimensional structures - from molecules to networks of communication, from atoms to galaxies. All fields of science are based on the structure of space. What is particular to this project's topological study is that it articulates the topological components in large networks of which knots are the principal example. Such networks are the basis of significant structures such as RNA and DNA, where topology is the key to interaction. In fact, the introduction of topology in these subjects is, from the viewpoint of this principal investigator, just the beginning of a larger interplay, where topology is seen as the basis for studying the recursive and circular architectures of complex systems. A galaxy, a society, an organism and a physical system all have in common the stability that arises through closed systems of circular interactions. Such interactions have traditionally been treated via cybernetics and computer modeling. These topological investigations add a new dimension to cybernetics and complex systems, providing new questions and a host of new problems. All these issues are treated in the new book "Knots and Applications," edited and contributed to by the principal investigator. ***