Structure in the Primes, with applications

ended

Questions about prime numbers make up several of the oldest and most important open problems in mathematics. Unfortunately our techniques for solving these problems are very limited; even some of the most basic and simple to state questions about primes are well beyond current techniques. This project studies several different questions related to the distribution of the primes, with the aim of developing new flexible techniques for studying the primes in general. Such new techniques would then give insight to the fundamental problems at the heart of the subject. The only general approach we have to counting primes is via variants of ‘Type I’ and ‘Type II’ arithmetic information. There have been several remarkable developments in sieve methods in recent years, which have greatly enhanced the utility of Type I information. Without establishing some sort of Type II information, however, it seems unlikely that one can fully solve the most important problems in the subject. This proposal seeks to develop both our Type I techniques and our Type II techniques, as well as the interactions between them. A common theme throughout the proposal is to identify and classify potential obstructions to traditional methods, and then overcome these obstructions using a combinations of new ideas. Often these new ideas will come from other areas of mathematics, such as combinatorics, geometry, probability, automorphic forms or harmonic analysis. This approach has already led to significant advances in our understanding of primes in recent years, most notably in the gaps between primes. The proposal is based around several intermediate problems for developing these connections further, giving opportunities for proof-of-concept results of such new ideas overcoming old barriers.