Expander Graphs in Number Theory -- Ioan Filip
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In this talk, we introduce expander graphs and illustrate some
connections with number theory. We start from first principles in
graph theory, Cayley graphs and the basic theory of expanders. Next,
we describe the spectral properties of the discrete Laplacian from a
geometric perspective and outline a basic construction of families
of expander graphs by Lubotzky, Phillips and Sarnak. Time
permitting, we include at the end an overview of the recent results
on expansion in linear groups over finite fields and some of their
applications to finiteness statements in arithmetic geometry. The
major part of the talk is mostly self-contained, assuming only basic
linear algebra.