Title: Ramanujan Conjectures, Density Hypotheses and Applications for Arithmetic Groups.
Abstract: The Generalized Ramanujan Conjecture (GRC) for GL(n) is a central open problem in modern number theory. Its resolution is known to yield applications in many fields, such as: Diophantine approximation and arithmetic groups. For instance, Deligne's proof of the Ramanujan-Petersson conjecture for GL(2) was a key ingredient in the work of Lubotzky, Phillips and Sarnak on Ramanujan graphs.
One can also state analogues (Naive) Ramanujan Conjectures (NRC) for other reductive groups, G, whose validity would imply various applications for the arithmetic congruence subgroups associated to G. However, already in the 70's Howe and Piatetski-Shapiro proved that the (NRC) fails even in the case of classical split groups.
In the 90's Sarnak-Xue put forth the conjecture that a Density Hypothesis version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications.
In this talk I will describe how to prove such Density Hypotheses for certain classical groups, by invoking deep and recent results coming from the Langlands program. Finally, we shall end with some applications of these Density Hypotheses, such as bounding the betti numbers of congruence hyperbolic manifolds, proving a strengthened version of a conjecture attribute to Gromov.