K-theory of C*-algebras with Applications in Topological Quantum Field Theory

拓扑学研究几何物体在变形时保持不变的性质。它提供了以下问题的答案：这个物体有多少个洞？上面有不可收缩的环吗？它有内外之分吗？Atiyah和Segal在数学物理中使用了拓扑方法：他们定义了拓扑量子场论（TQFTs），该理论在描述分数量子霍尔效应、量子引力和拓扑量子计算方面具有应用。tqft的对称性编码在其表示理论中，具有模张量范畴给出的丰富数学结构。Freed、Hopkins和Teleman的一个深入研究结果用一种叫做扭曲等变k理论的拓扑不变量表达了一个有趣的tqft族的对称性。它推广了由Atiyah， Hirzebruch和Grothendieck提出的拓扑k理论，并为他们赢得了菲尔兹奖。Dadarlat和penning已经证明了扭曲k理论的最一般形式可以用k -算子表示，这是另一个可以应用于C*-代数的推广。例如，这些代数出现在多体量子物理学中，它们描述的是可观测物。在这个项目中，你将深入研究C*代数，它推广了fred - hopkins - teleman定理中的核心结构，并使用谱序列和其他拓扑工具研究它们的k群。我们将一起使用这些计算并回答关于它们与tqft关系的开放性问题。因此，这个项目提供了一个进入几个蓬勃发展的纯数学领域的切入点：

Topology studies properties of geometric objects that remain unchanged under deformations. It provides answers to questions like: How many holes does the object have? Are there any non-contractible loops on it? Does it have an inside and an outside? Atiyah and Segal used topological methods in mathematical physics: They defined topological quantum field theories (TQFTs), which have applications in the description of the fractional quantum Hall effect, quantum gravity and topological quantum computing. Symmetries of TQFTs, encoded in their representation theory, have a rich mathematical structure given by modular tensor categories. A deep result of Freed, Hopkins and Teleman expresses the symmetries of an interesting family of TQFTs in terms of a topological invariant called twisted equivariant K-theory. It generalises topological K-theory, which was developed by Atiyah, Hirzebruch and Grothendieck, earning them the Fields Medal. Dadarlat and Pennig have shown that twisted K-theory in its most general form can be expressed by operator K-theory, another generalisation that can be applied to C*-algebras. These algebras appear for example in many-body quantum physics, where they describe observables. In the project you will have an in-depth look at C*-algebras that generalise the core construction in the Freed-Hopkins-Teleman theorem and study their K-groups using spectral sequences and other topological tools. Together we will put these computations to use and answer open questions about their relation to TQFTs. This project therefore provides an entry point into several flourishing areas of pure mathematics: