Spectral gap and definability

Originating in the theory of unitary group representations, the notion of spectral gap has played a huge role in many of the deep results in the theory of von Neumann algebras in the last couple of decades. Recently, with my collaborators, we are slowly understanding the model-theoretic significance of spectral gap, in particular its connection with definability. In this talk, I will discuss a few of our recent observations in this direction and speculate on some further possible developments. I will assume no knowledge of von Neumann algebras nor continuous logic. Various parts of this work are joint with Bradd Hart, Thomas Sinclair, and Henry Towsner.