Peter Sarnak (Courant-NYU / Princeton University ):
Ramanujan Graphs

Highly connected but sparse ("expanders") have proven to be fundamental building blocks in many algorithms in computational complexity theory. We review the definition(in terms of eigenvalues of the adjacency matrix) and construction of Ramanujan Graphs, these being optimal in this connection. We then discuss recent developments related to the eigenvalues and random graphs and matrices as well as higher dimensional Ramanujan Buildings.