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)提供资金的。本研究项目主要涉及链环同调理论和拓扑量子场理论。环同调理论是环的代数定义的不变量，推广和加强了经典的不变量，如Alexander，Jones和量子sl(N)-多项式，它们通常可以被认为是局限于三维空间中的环和链余边线的TQFT。在这类理论中，有一些等级2的Frobenius扩张扮演着重要的角色，首席调查者计划使用最近几年在范畴化领域发展起来的技术来寻找与任意n-2的秩n-Frobenius扩张相关的新的缠结和链同调理论。该项目还旨在加深对现有的链同调理论的理解，包括Khovanov-Rozansky同调，并改进目前已知的有色Jones多项式的范畴分类。这项研究的新奇之处在于提出了一种新的方法，通过网络和泡沫将同源联系起来，并将它们扩展到结和环的密码学。这一新的方法还激发了该项目的另一个目标，即构造定义在某些带接缝的肋骨上的扩展的TQFT，也称为泡沫。这一领域为DNA理论、分子构型和物理学提供了模型和应用，并在最近通过对量子不变量的分类取得了重大发展，产生了链接同调。该项目的重点是更好地感知现有的链接同调理论，改进它们的一些特性，以及寻找新的同调和拓扑量子场理论。首席研究员期望对量子sl(N)不变量以及纽结理论和表示理论之间的相互作用有更好的理解。这项研究的结果将为纽结和链环的同调不变量领域的方法在数学和理论物理的各个分支中的应用开辟新的视角，包括表象理论、范畴理论和拓扑量子场论。

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.