Algebraic Dynamics of Knots and Links

DMS-0071004 Daniel S. Silver和Susan G. WilliamsKnot的理论应用，从一个新兴的动力学领域，代数起源的动力系统的研究，将得到发展。将链接群的派生群的紧群表示的集合作为一个动力系统来研究。期望在结对称和连杆分支覆盖方面得到新的结果。最近发现的符号动力系统和拓扑量子场论之间的关系将被探讨。该研究将解决结理论和动力学中的开放问题，并扩展这些领域的先前结果。它将继续一个新的研究项目，结合符号动力系统的思想，研究结理论。结和链的研究不仅在数学界，而且在更大的科学界都引起了极大的关注。例如，分子生物学和等离子体物理学中发现了打结和连接现象，而理论物理学中经常出现计算方法的惊人相似之处。所提出的研究将通过使用符号动力系统（信息论的一个数学分支）的技术，提供对结点和链接的新理解。该研究还将加强拓扑学和动力学领域研究人员之间的交流。

DMS-0071004 Daniel S. Silver and Susan G. WilliamsKnot theory applications from a newly emerging area of dynamics, the study of dynamical systems of algebraic origin, will be developed. The set of representations into a compact group of the derived group of a link group will be studied as a dynamical system. New results about knot symmetries and branched covers of links are expected. A recently discovered relationship between symbolic dynamical systems and topological quantum field theory will be explored. The research will address open questions in knot theory and dynamics and expand on previous results in these fields. It will continue a new program of research in knot theory that incorporates ideas from symbolic dynamical systems. The investigation of knots and links has attracted great attention not only in the mathematical world but also in the larger scientific community. Knotting and linking phenomena are found in molecular biology and plasma physics, for example, while striking similarites in computational methods arise frequently in theoretical physics. The proposed research will provide new understanding of knots and links by using techniques from symbolic dynamical systems, a mathematical branch of information theory. The research will also strengthen interaction between researchers in the fields of topology and dynamics.