Link Homology, Categorification and extended Topological Quantum Field Theory

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本课题主要研究链路同调理论和拓扑量子场理论。连杆同调理论是由代数定义的连杆不变量，它推广和增强了经典不变量，如Alexander、Jones和量子sl(n)-多项式，它们通常可以被视为局限于三维空间中的连杆和连杆协矩阵的tqft。2阶Frobenius扩展在这类理论中扮演着重要的角色，项目负责人计划利用近几年在分类领域发展起来的技术，寻找与任意n- n-Frobenius扩展相关的新的缠结和链接同源理论。该项目还旨在加深对现有链接同调理论的理解，包括Khovanov-Rozansky同调，并改进目前已知的有色琼斯多项式的分类。提出的研究的新颖之处在于通过网和泡沫连接同源性的新方法，并将其扩展到结和链接的协同。这种新方法也激发了项目的另一个目标，即构建在接缝（也称为泡沫）的特定协同上定义的扩展tqft。拟议的研究项目涉及结和链的理论。这一领域为DNA理论、分子构型和物理学提供了模型和应用，最近通过量子不变量的分类，产生了链接同源性，取得了重大进展。该项目的重点是更好地认识现有的链接同调理论，改进其某些特性，以及寻找新的同调和拓扑量子场理论。首席研究员期望更好地理解量子sl(n)不变量以及结理论和表示理论之间的相互作用。本研究结果将为结点和连杆的同调不变量在数学和理论物理的各个分支，包括表示论、范畴论和拓扑量子场论中的应用开辟新的前景。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This research project deals mainly with link homology theories and topological quantum field theories (TQFTs). Link homology theories are algebraically defined invariants of links that generalize and enhance classical invariants such as the Alexander, Jones and quantum sl(n)-polynomials, and they can often be regarded as TQFTs restricted to links in the 3-dimensional space and link cobordisms. There are rank 2 Frobenius extensions that play an important role in such theories, and the Principal Investigator plans to use the techniques developed in the last few years in the area of categorification to find new tangle and link homology theories that are related to rank n-Frobenius extensions for arbitrary n 2. The project also aims to deepen the understanding of existing link homology theories, including the Khovanov-Rozansky homologies, and to improve the currently known categorifications of the colored Jones polynomial. The novelty of the proposed research lies in a new approach to link homologies via webs and foams and to their extension to cobordisms of knots and links. This new approach also motivates another goal of the project, namely that of constructing extended TQFTs defined on certain cobordisms with seams, also called foams.The proposed research project concerns the theory of knots and links. This area has provided models and applications to DNA theory, molecular configurations and physics, and has gone through a significant development recently through categorifications of quantum invariants, giving rise to link homologies. The focus of the project is to better perceive the existing link homology theories, to improve some of their features, as well as to find new homology and topological quantum field theories. The Principal Investigator anticipates a better understanding of the quantum sl(n) invariants and of the interplay between knot theory and representation theory. The findings of the study should open new perspectives for applications of methods from the field of homological invariants of knots and links in various branches of mathematics and theoretical physics, including representation theory, category theory, and topological quantum field theory.