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

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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: