Selberg's 3/16 theorem, spectral gap and expander graphs
Ecole Normale Superieure, France.
A famous theorem of Selberg says that every positive eigenvalue of the Laplace operator on the hyperbolic surface associated to a congruence subgroup is at least 3/16. The original proof utilized the theory of modular forms. Expanders are graphs on which the first positive eigenvalue of the discrete Laplace operator is bounded away from 0. They are important objects which have many applications in computer science and in pure mathematics. These two phenomenons of spectral gap turned out to be equivalent. Through this equivalence, the recent work of Bourgain, Gamburd, Varju etc, gives a new proof of (a weak form of) the Selberg's theorem.
This talk will be an introduction to this fascinating connection between number theory, geometry, group theory and discrete mathematics. Only undergraduate mathematics will be required to understand and I hope both mathematics and computer science students will find something interesting in this talk.