Lecture 3, PhD course "Applying ideas of mathematical logic to group theory and operator algebras"

The course will cover applications of logic to various mathematical areas. They will be discussed in four largely independent blocks, each consisting of 2 hours of lecturing and one hour of discussions and exercises. 1. First-order logic and group theory: We introduce the notion of a f.g. group to be QFA (quasi-finitely axiomatizable): the group can be uniquely described by a first-order sentence, among the finitely generated groups. We provide examples, and discuss recent work of the analogous notion for the class of profinite groups 2. Descriptive set theory and the isomorphism problem for Polish groups: We show that isomorphism of profinite groups is as complex as possible w.r.t. Borel reducibility, while isomorphism of oligomorphic groups is essentially countable. 3. Computabity and totally disconnected, locally compact (tdlc)groups: We initiate an algorithmic theory of tdlc groups, and discuss an example of a computably tdlc group with noncomputable scale function. 4. Continuous logic and the Connes embedding problem: We discuss an alternative method due to Goldbring and Hart, based on continuous logic, how a recent breakthrough result in quantum information theory can be used to refute a long-open conjecture on von Neumann algebras. This avoids the various equivalences to the CEP that were applied in the original refutation.