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: Topological spaces - such as classifying spaces of small categories and spacetimes - often admit extra temporal structure. Such "directed spaces" often arise as geometric realizations of simplicial sets and cubical sets; the temporal structure encodes orientations of simplices and 1-cubes. Directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Nevertheless, we present simplicial and cubical approximation theorems for a homotopy theory of directed spaces. In our directed setting, ordinal subdivision plays the role of barycentric subdivision and cubical sets equipped with coherent compositions of higher cubes serve as analogues of Kan complexes. We consequently show that geometric realization induces an equivalence between certain weak homotopy diagram categories of cubical sets and directed spaces. As applications, we show that directed analogues of homotopy groups of spheres are uninteresting, sketch constructions of a (more interesting) cubical singular cohomology theory for directed spaces, and calculate such "directed cohomology" monoids for various directed spaces of interest.