Homotopy Groups of Automorphism Groups of Classifiable C*-algebras

In the early '90s, Elliott conjectured that separable simple nuclear C*-algebras are classified up to isomorphism by their K-theory groups and traces, in analogy with the Connes--Haagerup classification of separably acting injective factors by their type and flow of weights.  In the last few years, Elliott's classification program has been completed under the two additional hypotheses: Z-stability and the UCT.  In the von Neumann algebraic setting, the classification of injective factors was followed by a delicate analysis of the symmetries of such factors.  For example, Oceanu proved every amenable group admits a outer action on the hyperfinite II_1 factor R, which is unique up to cocycle conjugacy, and Popa and Takesaki proved the automorphism group of R is contractable.  In the spirit of the latter result, I will discuss recent joint work with Jamie Gabe on the homotopy type of automorphism groups of "classifiable" C*-algebras.