Number Theory Seminar

Wednesday 3/19/03 3:30-4:30 PM in Hylan 1104, and
Wednesday 4/2/03 3:30-4:30 PM in Hylan 1104:
Arnie Pizer- "Ramanujan graphs, Hecke Operators, and Quaternion algebras"

ABSTRACT:

Ramanujan graphs are a special type of expander graphs which are best possible (or extremal) in an asymptotic sense. We will discuss Ramanujan graphs and show how to construct large classes of Ramanujan graphs using Hecke operators associated to quaternion algebras. In a second talk we show how combining ideas used in this construction with ideas used in earlier constructions due to Lubotzky, Phillips, and Sarnak and, independently, Margulis leads one to expect that there is a theory of "Ray Class Ideals" in Quaternion Algebras with associated Hecke operators, theta series and modular forms.

The first half of the first talk will deal mostly with easy (though non standard) graph theory and should be very accessible (even to undergraduates).