The Erwin Schrödinger Lectures are directed towards a general audience of mathematicians and physicists. In particular it is an intention of theses lectures to inform non-specalists and graduate students about recent developments and results in some area of mathematics or physics.

Abstract:

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in computer science in the last 4 decades and more recently also in pure mathematics. In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov 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. This question was answered recently (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by T. Kaufman and S. Evra for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders.

Alex Lubotzy holds the Maurice and Clara Weil chair in Mathematics at the Einstein Institute ofMathematics of the Hebrew University Jerusalem.

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