Logic Seminar

In the theory of unitary group representations, the following theorem of Elmar Thoma from the early 1960s is fundamental: A countable discrete group is "type I" if and only if it has an abelian finite index subgroup. By way of a celebrated theorem of Glimm from the same period, a group being "type I" is equivalent to saying that the irreducible unitary representations of the group admits a smooth classification in the familiar sense of Borel reducibility, and in fact they are all finite-dimensional in this case. Glimm's theorem, and later work by Hjorth, Farah and Thomas, implies that if a group is not type I, then it is quite hard to classify the irreducible unitary representations.

In this talk I will give an overview of the descriptive set-theoretic perspective on the classification of irreducible representations, and I will discuss a new proof of Thoma's theorem due to F.E. Tonti and the speaker.