Wouter van Limbeek : A weak version of Selberg's 3/16 theorem

Posted by Wouter Van Limbeek , part of the Graduate Groups and Dynamics Seminar.

At

Nov. 6, 2018, 3 p.m.

In

1227 SEO

Abstract

In 1965, Selberg proved that for every congruence cover of the principal modular curve, the spectrum of the Laplacian is uniformly bounded below by 3/16. This implies that the Cayley graphs for SL(2,F_p) form an expander. We will explain the connection with expanders, and discuss recent arguments by Sarnak-Xue (simplified further by Gamburd and Tao) that establish a weak version of Selberg's theorem with smaller gap, but which are more amenable to generalization.