1st talk:
K-theoretic classification works surprisingly well for C*-algebras which - are separable and simple - have approximations (in terms of finite-dimensional algebras) which also are topologically finite-dimensional in a suitable sense - have some underlying (amenable) dynamical structure. A long-term project within the CRC is it to understand and perhaps even to classify such underlying structures for a given classifiable C*-algebra. I will explain how the principles above can be used to write the well-known CAR algebra (an infinite tensor product of 2x2 matrices) in new ways, with substantially different underlying dynamical structures, and how the latter can be distinguished in terms of dimension-type invariants. The examples are based on the so-called paper-folding subshift from symbolic dynamics. (This is joint work with Grigoris Kopsacheilis.)

2nd talk:
I will talk about stable rank one, that is, the density of invertible elements in a C*-algebra. For commutative C*--algebras, this property occurs precisely in topological dimension zero or one, and can therefore be interpreted as a notion of noncommutative low dimensionality. Stable rank one guarantees more accessible computations of K-theory and other C*-algebraic invariants. The space Act(G,X) of actions of a countable group G on the Cantor set X by homeomorphisms carries a natural Polish topology, making it amenable to methods from Baire category theory. By restricting to suitable Polish subspaces of Act(G,X), we investigate situations in which stable rank one occurs generically, that is, on a dense G_\delta subset, for the associated crossed product C*-algebras. This is joint work with Jamie Bell and David Kerr.