Equidistribution and Primes

Author: 
Peter Sarnak
Date: 
Sat, Sep 1, 2007
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS 10th Anniversary Lectures
Abstract: 
We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given.
Notes: