Abstract

Expander graphs in general, and Ramanujan graphs , in particular,  have played a major role in combinatorics and computer science in the last 4 decades  and more recently also in pure math.

Approximately 10 years ago a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others. 

In recent years a high dimensional theory of expanders is emerging.  The notions of geomrtric and topological expanders were defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. 

Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013,  but it was left open  if they are also topological expanders. 

By developing new isoperemetric  methods for "locally minimal small"  F_2- co-chains, it was shown recently by 

Kaufman- Kazdhan- Lubotzky for small dimensions and  Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders. 

We will describe these developments and the general area of high dimensional expanders and some of its open problems.