Abstract: In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Abstract: In the chromatic take on stable homotopy theory, the homotopy type of a finite $p$-local spectrum $X$ is reassembled from its localizations with respect to the various Morava $K$-theories. In the early 1990s, Hopkins proprosed a brash conjecture for how the reassembly process works. I'll review the conjecture and the state of the art -- including a verfication of the conjecture at $p=3$ and chromatic level $2$, where the question is not simply algebraic and where there has been a proposed counterexample. This is joint work with Hans-Werner Henn.