In recent years new and important connections have emerged between
discrete subgroups of arithmetic groups, automorphic forms and
representations on the one side, and questions in discrete mathematics,
combinatorics, and graph theory on the other side.
One of the main examples of this interaction is the abstract and
explicit construction of Ramanujan graphs and hypergraphs, using the
Jacquet-Langlands correspondence and Deligne's theorem (number field case
for rank one groups), resp. Lafforgue's theorem (function field case for
higher rank groups) on Hecke eigenvalues (Ramanujan conjecture). First I
will give a panorama about Langlands functoriality and in particular,
I'm going to explain two important problems in arithmetic number theory;
namely the Jacquet-Langlands correspondence and the Ramanujan
conjecture. I shall devote the rest of my talk to the explicit
construction of Ramanujan graphs and hypergraphs. There are great many open problems in these areas, some of which will be addressed in the body of my talk.
|Wednesday, Jan. 3, 2007, 14:00-16:00
|Niavaran Bldg., Niavaran Square, Tehran, Iran