1st talk:
I give an introduction to approximation properties for nuclear C*-algebras, discuss how they are related to amenability, and describe how finite dimensional approximations lead to intriguing rigidity questions. I will use C*-algebras associated with topological dynamical systems as guiding examples.

2nd talk:
The notion of comparison traces its roots back to the Hausdorff and Banach-Tarski paradoxes and has long layed a fundamental role in both ergodic theory and operator algebras. Its importance is often overshadowed however by related but more refined structural properties such as tileability and finite-dimensional approximation. In this talk I will highlight two instances in dynamics where the teasing out of finer structure hinges around comparison, one of a quantitative nature with connections to entropy theory and the other of a topological nature with connections to C*-algebras.