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.

We will present Ozawa's remarkable sum of squares certificate for property (T), which subsumes Żuk's criterion/trickle down 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 (Alon)

Exercises 

Lecture 1 recording (Sidhanth) Lecture 2 recording (Alon)

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