RUI: Algebraic topology of knot and link spaces

RUI：结和链接空间的代数拓扑

• 批准号: 1205786
• 负责人: Ismar Volic
• 金额: $ 15.1万
• 依托单位: Wellesley College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-15 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205786&HistoricalAwards=false
• 关键词: RUI; Algebraic; topology; knot; link

AbstractAward: DMS 1205786, Principal Investigator: Ismar VolicInsert abstract here for an award recommendation.The main goal of this project is a better understanding of the topology of knot and link spaces as well as more general embedding spaces. The principal investigator proposes to use techniques such as calculus of functors, cosimplicial spaces, operads, and configuration space integrals to prove results about homology and homotopy of knots, links, homotopy links, and braids in Euclidean spaces of various dimensions. In particular, these proposed projects include a program to describe the rational homotopy type of these spaces; to combine configuration space integrals with the theory of Milnor invariants; and to better understand, if not complete resolve, of the issue of the separation of knots and links by finite type invariants. Moreover, the principal investigator plans to unify the various ways in which operads appear in the applications of calculus of functors in knot theory and further the understanding and uses of configuration space integrals. Some of his long-term projects will attempt to connect calculus of functors point of view in embedding theory to Khovanov homology as well as to the study of embeddings of surfaces and the mapping class group.Knot and link spaces represent some of the most interesting objects of study in topology because they are easy to define and visualize and because they are of interest to physicists and chemists. Some fundamental questions about knots, such as their classification, or construction of efficient ways of telling them apart, still generate a wealth of exciting research. The principal investigator's proposed work intends to bring us closer to answering these questions. Furthermore, the techniques he plans to use are quite general and point to new connections between topology, geometry, combinatorics, and physics. These connections will potentially help answer several important conjectures about the structure of knot and link spaces and introduce new points of view in algebraic topology in general.

AbstractAward：DMS 1205786，首席研究员：Ismar VolicInsert abstract here for an award recommendation.这个项目的主要目标是更好地理解结和链接空间的拓扑以及更一般的嵌入空间。 主要研究者建议使用的技术，如演算函子，余单纯空间，运算，和配置空间积分，以证明结果的同源性和同伦的结，链接，同伦链接，辫子在欧氏空间的各种尺寸。 特别是，这些建议的项目包括一个程序来描述这些空间的合理同伦类型;联合收割机配置空间积分与Milnor不变量理论;并更好地理解，如果不是完全解决，问题的分离的结和链接有限型不变量。此外，首席研究员计划统一操作出现在纽结理论中的函子微积分应用中的各种方式，并进一步理解和使用配置空间积分。 他的一些长期项目将试图连接微积分的函子的观点嵌入理论的霍瓦诺夫同源性以及研究嵌入的表面和映射类group.Knot和链接空间代表一些最有趣的对象的研究在拓扑结构，因为他们很容易定义和可视化，因为他们感兴趣的物理学家和化学家。关于结的一些基本问题，如它们的分类，或区分它们的有效方法的构建，仍然产生了大量令人兴奋的研究。 首席研究员提出的工作旨在使我们更接近回答这些问题。 此外，他计划使用的技术是相当普遍的，并指出拓扑学，几何学，组合学和物理学之间的新联系。 这些连接将潜在地帮助回答关于结和链空间的结构的几个重要问题，并在一般代数拓扑学中引入新的观点。