Learn about topological tensor products, and approximation properties for [math]C^*[/math]-algebras, such as nuclearity and exactness, and their relation with amenability, and write down a brief account of what you learned.

Read the Murray-von Neumann and Connes papers about amenability and hyperfiniteness in the von Neumann algebra setting, and then the fact that a discrete quantum group [math]\Gamma[/math] is amenable precisely when [math]L(\Gamma)[/math] is hyperfinite.

Clarify the fact that a discrete quantum group [math]\Gamma[/math] is amenable precisely when the associated planar algebra, or subfactor, is amenable.

Draw the Cayley graphs of the duals of the main quantum groups,

with respect to suitably chosen fundamental representations, and compute the growth.

Find examples and counterexamples for the notion of connectedness, for the compact quantum groups.

Find examples and counterexamples for the notion of normality of subgroups, for the compact quantum groups.

Work out explicitely the helf-liberation formulae

appearing as particular cases of the theory developed above.