Peter Sarnak

Abstract:   We review briefly various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the developments towards resolving these long standing questions. We then put these problems in a natural and geometric context of actions by morphisms on affine n-space. We outline a theory that has been developed in this context and some its applications to classical problems .The methods used to develop a combinatorial sieve in this setting involve automorphic forms ,expander graphs and unexpectedly arithmetic combinatorics.