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。本研究计画主要研究连结同调理论与拓扑量子场论。链同调理论是代数定义的链不变量，它推广和增强了经典的不变量，如亚历山大，琼斯和量子sl（n）-多项式，它们通常被认为是限制在3维空间中的链和链配边的TQFT。有秩2 Frobenius扩展，发挥了重要作用，在这样的理论，和主要研究者计划使用的技术开发在过去几年中在该地区的分类，以找到新的纠缠和链接同源理论是相关的秩n-Frobenius扩展任意n 2。该项目还旨在加深对现有链接同源性理论的理解，包括Khovanov-Rozansky同源性，并改进目前已知的有色琼斯多项式的简化。这项研究的新奇在于通过网络和泡沫链接同源性的新方法，以及它们对结和链接的协边性的扩展。这种新的方法也激发了该项目的另一个目标，即构建扩展的TQFT定义在某些cobordisms与接缝，也称为foams.The拟议的研究项目涉及的结和链接的理论。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简化，产生了链接同源性，并取得了重大发展。该项目的重点是更好地理解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论。 主要研究者期望更好地理解量子sl（n）不变量以及纽结理论和表象理论之间的相互作用。该研究的结果应该打开新的视角，从该领域的同调不变量的结和链接在数学和理论物理的各个分支，包括表示论，范畴理论和拓扑量子场论的方法的应用。

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.