Bounded gaps between primes

November 17 to November 21, 2014

at the

American Institute of Mathematics, San Jose, California

organized by

John Friedlander, Dan Goldston, and Soundararajan

Original Announcement

This workshop will focus on the remarkable progress made in the last year on gaps between prime numbers. The breakthrough result of Zhang first established that there are bounded gaps between consecutive primes, and this was quickly followed by another extraordinary argument by Maynard (and discovered independently by Tao), establishing the existence of many primes in bounded intervals. Both results start from a method pioneered by Goldston, Pintz and Yildirim, but then proceed in very different directions. Zhang's breakthrough is based on an extension of the Bombieri-Vinogradov theorem (building on earlier work of Fouvry and Iwaniec, and Bombieri, Friedlander and Iwaniec), while Maynard's work re-examines the classical sieve method of Selberg and finds an astonishingly strong variant.

The workshop will discuss the ideas behind these breakthrough results, and explore possible applications to other problems. The classical Bombieri-Vinogradov theorem has numerous pplications, and one may hope that the partial extension of Zhang will also have more consequences. Likewise, the Selberg sieve is ubiquitous in number theory, and it would be worth exploring the flexibility of the weights in Maynard's work. Another natural focus of the workshop will be on obtaining sharper versions of the results of Zhang and Maynard, taking into account the subsequent improvements by Polymath and perhaps others, and on understanding the limitations of these methods. The workshop will bring together senior researchers in analytic number theory, sieve methods, additive combinatorics, together with many promising young researchers in these and related areas.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

A note on small gaps between primes in arithmetic progressions
by  Deniz Ali Kaptan,  Acta Arith. 172 (2016), no. 4, 351-375  MR3471048
Limit points and long gaps between primes
by  Roger Baker and Tristan Freiberg,  Q. J. Math. 67 (2016), no. 2, 233-260  MR3509992
Short intervals with a given number of primes
by  Tristan Freiberg,  J. Number Theory 163 (2016), 159-171  MR3459566