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

拓扑研究几何对象在变形下保持不变的特性。它提供了一些问题的答案，比如：物体有多少个孔？上面有没有不可收缩的环？它有内部和外部吗？阿提亚和西格尔在数学物理学中使用了拓扑方法：他们定义了拓扑量子场论（TQFT），它在描述分数量子霍尔效应、量子引力和拓扑量子计算中有应用。 TQFT的对称性，编码在它们的表示理论中，具有由模张量范畴给出的丰富的数学结构。Freed、霍普金斯和Teleman的一个深入结果用一个称为扭曲等变K理论的拓扑不变量来表达一类有趣的TQFT族的对称性。它概括了拓扑K-理论，这是由阿蒂亚，赫泽布鲁赫和格罗滕迪克，赢得他们菲尔茨奖。Dadarlat和Pennig证明了扭曲K-理论在其最一般的形式下可以用算子K-理论来表达，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，在那里它们描述了可观测量。在这个项目中，你将深入研究C *-代数，它概括了Freed-Hopkins-Teleman定理的核心结构，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，并回答有关它们与TQFT关系的开放性问题。因此，这个项目提供了一个切入点，进入几个蓬勃发展的纯数学领域：

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: