RUI: Algebraic topology of knot and link spaces

奖项：DMS 1205786，首席研究员：Ismar VolicInsert摘要此处为奖项推荐。该项目的主要目标是更好地理解结和链接空间的拓扑结构以及更一般的嵌入空间。主要研究者提出利用函子演算、协简空间、操作数、位形空间积分等技术，证明不同维欧几里德空间中结、链、同伦链、辫的同调与同伦结果。特别地，这些建议的项目包括一个程序来描述这些空间的有理同伦类型；将组态空间积分与Milnor不变量理论相结合；为了更好地理解，如果不是完全解决，用有限型不变量分离结点和链接的问题。此外，首席研究员计划统一操作数在结理论中函子演算应用中的各种方式，并进一步理解和使用组态空间积分。他的一些长期项目将尝试将嵌入理论中的函子演算观点与Khovanov同调以及对曲面嵌入和映射类群的研究联系起来。结和链接空间代表了拓扑学中一些最有趣的研究对象，因为它们易于定义和可视化，因为它们是物理学家和化学家感兴趣的。一些关于结的基本问题，比如它们的分类，或者有效区分结的方法的构建，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这些问题的答案。此外，他计划使用的技术相当通用，并指出拓扑、几何、组合学和物理学之间的新联系。这些连接可能有助于回答关于结和连接空间结构的几个重要猜想，并在代数拓扑中引入新的观点。

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.