Voiculescu used free entropy theory (which roughly measures the amount of matrix approximations for a tuple of operators in a tracial von Neumann algebra) to give the first proof that free group von Neumann algebras do not have Cartan subalgebras, and his method has had many generalizations since then. Jung implicitly and Hayes explicitly defined a variant of free entropy, known as 1-bounded entropy, which distills the ideas in Voiculescu's proof and behaves well with respect to von Neumann algebraic operations such as joins, unions of increasing chains, and various weak normalizers. This lecture will introduce 1-bounded entropy and describe its relationship with free products, property (T), and vanishing cohomology, as well as consequences of the recent resolution of the Peterson-Thom conjecture. I will also describe the generalization of the Jung-Hayes entropy to model-theoretic types rather than non-commutative laws.

This is partly based on joint work with Hayes and Kunnawalkam Elayavalli.