Expander graphs, in general, and Ramanujan graphs, in particular, have been objects of intensive research in the last four decades. Many application came out, initially to computer science and combinatorics and more recently also to pure mathematics (number theory, geometry, group theory ). In recent years, there has been an interest in generalizing this theory to higher dimensional simplical complexes. We plan to survey first the classical theory and then describe the more recent developments.

With the notion of interaction with oracles as a unifying theme of much of my dissertation work, I discuss novel models and results for property testing and computational learning, with the use of Fourier analytic and probabilistic methods.

I will describe some recent results and problems regarding influence of sets of variables on Boolean functions: In 1989 Benny Chor conjectured that a balanced Boolean function with n variables has a subset S of size 0.4n with influence (1-c^n) where c0 follows from a theorem by Kahn, Kalai and Linial (KKL).I will present a recent counterexample by Kahn and me showing that up to the identity of c, the KKL bound cannot be improved.