Title: Non-classification of free Araki-Woods factors and $τ$-invariants,
Abstract: We define the standard Borel space of free Araki-Woods factors and prove that
their isomorphism relation is not classifiable by countable structures. We also
prove that equality of $\tau$-topologies, arising as invariants of type III
factors, as well as coycle and outer conjugacy of actions of abelian groups on
free product factors are not classifiable by countable structures.