An Affine Sieve

Princeton University, Princeton NJ

Investigators

Abstract

The proposal is concerned with applying classical and more recent forms of the combinatorial sieve, in the setting of an orbit of a group of morphisms of affine n-space,which preserves integer points. The setting unifies and generalizes the problem of finding points at which a polynomial takes on values which are prime or has few prime factors.This "affine sieve" has numerous applications to both classical and novel diophantine problems. The methods used to develop an effective sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic and additive combinatorics. The Twin Prime Conjecture asserts that there are infinitely many pairs of prime numbers which differ by two. It is one of the longest standing unsolved problems in mathematics. While such problems are driven first by curiosity, the techniques that have been invented for their study have proven to be fundamental more broadly. The proposal is concerned with far-reaching generalizations of the twin prime conjecture and with developing new techniques to prove parts of these general conjectures. The interplay between number theory, combinatorics and theoretical computer science has been a very active one in recent years. Many times in this context the applications have been of number theoretic ideas to the other two fields. In the present project the reverse application is also critical.