Dear NTG students and post-docs,
    The course of Shalom, described below, should be a great
    introduction to some fundamental number theory and its applications.
    This work on Ramanujan graphs, due to Margoulis,  as well as Lubotzky,
    Phillips, Sarnak, and others, is quite fundamental and it has stimulated
    a huge amount of research. Both it and the Ruzwiewicz Problem use basic
    deep results in the theory of automorphic forms. Attendance is strongly
    recommended to number theorists!
       best,
           Don

	   Course Description - 285I: Discrete groups, expanding graphs and invariant
	   measures. Instructor: Yehuda Shalom

	   In the course we present the solution to two seemingly unrelated problems:
	   one, in combinatorics, is the construction of ``optimal" expander graphs
	   (so called, Ramanujan graphs), which are highly connected sparse finite
	   graphs, of fundamental importance in combinatorics and computer science.
	   The other, in measure theory, is the Ruzwiewicz problem, concerning
	   finitely additive invariant measures on the sphere. We shall
	   understand why these two problems are essentially the same, and what is
	   the deep mathematics quite amazingly involved in their solutions.

	   In fact, as beautiful and elementary to state as they are (they will be
	   fully explained in our first meeting this Friday without any background
	   assumed), more importantly these problems will serve as motivation to
	   discuss fundamental objects in the theory of algebraic and arithmetic
	   groups, number theory and infinite dimensional representation theory,
	   as well as basic notions in group theory such as amenability and Kazhdan's
	   property (T). All of this will be explained without assuming any graduate
	   level background.

	   The course could appeal to students who are interested in any of the
	   areas mentioned above, as well as to students who would appreciate
	   an opportunity to expand their mathematical horizons in what will aim to
	   be a fun and cost effective way. The course has to do with functional
	   analysis (its official title) as much as it has to do with combinatorics,
	   number theory, algebraic groups theory, geometric group theory and more.
	   It will based on a 1994 book by Alex Lubotzky with the same title,
	   although as we shall see, much has happened since.