Abstract

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.



Information: