Expanders, Property (T) & Sum of Squares

Expanding graphs are ubiquitous throughout mathematics and computer science and have cemented themselves a fundamental objects of interest. This minicourse joint with Sidhanth Mohanty will give a powerful framework from group theory (Kazhdan's property (T)) and operator algebras to construct explicit expanders in a wide variety of settings.

This talk will present Ozawa's remarkable sum of squares certificate for property (T), which subsumes Żuk's criterion presented previously. We then prove Ozawa's theorem showing that property (T) is in fact equivalent to possessing a sum of squares certificate. This gives a semidefinite programming approach towards establishing property (T), yielding computer based proofs of expansion. Time permitting, we will discuss (T) for automophism groups of free groups.

Lecture notes

Exercises

This is part of ICM 2026 Satellite Conference: Spectral Theory, High-Dimensional Expansion, and Pseudorandomness, Simons Institute for the Theory of Computing, August 2026.