Colloquium: Tuesday, December 27, 12 pm. Speaker: Uriya First (UBC). Title: “Ramanujan Complexes”.

Ramanujan graphs are regular graphs whose vertex adjacency matrix has “very condensed” spectrum. The latter manifests in many desired combinatorial properties, most notable of which is the fact that Ramanujan graphs are expanders, i.e. they admit high connectivity among their nodes despite having a small number of edges. Only few concrete infinite families of Ramanujan graphs are known (thanks to Lubotzky, Phillips, Sarnak, Margulis, Morgenstern and others), and the existence of such families for any vertex valency was established only recently by Marcus, Spielman and Srivastava.

In the last decade, high dimensional generalizations of Ramanujan graphs, called Ramanujan complexes, began to emerge. Like Ramanujan graphs, these too have many good combinatorial properties, including various types of expansion. I will survey this exciting new field and discuss some new constructions arising through deep results in number theory.