Abstract: The automorphism groups of infinite dimensional associative algebras are often difficult to describe. Precise answers are only known for very particular algebras and most of the known results are about the existence of wild automorphisms.

In this talk we will state and give a full proof of a rigidity theorem for automorphisms of completed quantum tori. It has a broad range of applications to classifying the automorphism groups of quantum algebras. (The talk will concentrate on the rigidity theorem rather than the applications.)