Low-dimensional topology and topological methods in condensed matter physics

所提出的研究涉及低维拓扑，特别是拓扑4流形的几何分类和量子拓扑的某些方面。Krushkal提出了一种在同伦A-B切片问题的背景下解决拓扑4维外科猜想的方法。拓扑仲裁者为这种方法提供了一个方便的框架，这是最近与M.弗里德曼合作提出的一个概念。该项目包括量子拓扑学中的一些问题，涉及自旋网络的具体问题，以及作为特例的平面图的色多项式与Temperley-Lieb代数之间的范畴关系。该项目还旨在利用拓扑方法解决自由和特定类别的相互作用费米子的分类问题，这是凝聚态物理中的一类重要系统。拓扑不变量在Kitaev等人最近对非相互作用费米子有隙相的分类工作中发挥了重要作用。事实证明，K理论和Bott周期性是对具有各种对称性的系统进行分类的关键因素。Krushkal计划致力于一些相关的问题，包括自由费米子族的不变量和某些相互作用系统的扩展。该项目的一个目标是对局部看起来像欧几里德空间的物体的可能大规模形状进行分类。三维和四维形状的分类是一个特别重要和具有挑战性的问题。这个项目的目的是为了更好地理解具有较大基本群的四维对象，这些对象包含许多无法收缩的回路。该项目的另一部分涉及量子拓扑学，这是一个受物理和数学思想影响的学科，与统计力学、表象理论、拓扑学和组合学有关。特别是，目标是在最近开发的分类数学框架中，获得关于自旋网络的新概念视角，这一概念起源于量子物理。该项目的第三部分探索拓扑学和凝聚态物理之间的跨学科联系，重点是拓扑绝缘体。与物理学家合作，这项研究将建立在对这类非凡的物理系统进行分类的最新进展的基础上。

The proposed research concerns low-dimensional topology, specifically geometric classification of topological 4-manifolds and certain aspects of quantum topology. Krushkal proposes an approach to solving the topological 4-dimensional surgery conjecture in the context of the homotopy A-B slice problem. A convenient framework for this approach is provided by topological arbiters, a notion recently introduced in collaboration with M. Freedman. The project includes a number of problems in quantum topology, specific questions concern spin networks, and as a special case a categorified relation between the chromatic polynomial of planar graphs and the Temperley-Lieb algebra. The project also aims to use topological methods to address classification problems for free and certain classes of interacting fermions, an important class of systems in condensed matter physics.Topological invariants have played an important role in recent work on classification of gapped phases of non-interacting fermions by Kitaev and others. K-theory and Bott periodicity turned out to be crucial ingredients in classifying systems with various symmetries. Krushkal plans to work on a number of related problems, including invariants of families of free fermions and extensions to certain classes of interacting systems.One goal of the project concerns classification of possible large scale shapes of objects that locally look like the Euclidean space. The classification of three- and four-dimensional shapes is a particularly important and challenging problem. This project is aimed at a better understanding of 4-dimensional objects with large fundamental groups, which contain many loops that cannot be contracted. Another part of the project concerns quantum topology, a subject influenced by ideas from both physics and mathematics and connected with statistical mechanics, representation theory, topology, and combinatorics. In particular, the goal is to gain a new conceptual perspective on spin networks, a notion whose origins are in quantum physics, in the recently developed mathematical framework of categorification. The third part of the project explores interdisciplinary connections between Topology and Condensed Matter Physics, focusing on topological insulators. This research, in collabor ation with physicists, will build on recent advances in classification of this remarkable class of physical systems.