Combinatorics: Amitay Kamber (HU) " Lp Expander Complexes"

Sun, 23/04/201711:00-13:00
Rothberg B221 (CS building)
Speaker: Amitay Kamber, HU
Title: Lp Expander Complexes.
Abstract: In recent years, several different notions of high dimensional expanders have been proposed (which in general are not equivalent), each with its own goal and motivation. The goal of this talk is to propose another generalization, based on ideas from the representation theory of p-adic groups.
By comparing a complex to its universal cover, we show how to define Lp-expanders and in particular L2-expanders, which are Ramanujan complexes, generalizing the notions of expander graphs and Ramanujan graphs.
We discuss two applications of the approach:
1. A well known theorem of Hashimoto says that a graph is Ramanujan if and only if its zeta function, associated with a non-backtracking random walk, satisfies the Riemann hypothesis. Our results show that a similar theorem holds for complexes of dimension n, if we consider n special non-backtracking operators.
2. Recent results of Lubetzky and Peres show that in a q+1 regular Ramanujan graph with |V| vertices, the average distance between vertices is log_q(|V|)+ O(log(log(|V|)), and that the simple random walk has a cutoff phenomena. We show similar results for Ramanujan complexes.