# James Maynard

Affiliation: University of Oxford

Email: james.alexander.maynard@gmail.com

Title Of Talk: The distribution of prime numbers

Abstract:

Talk 1 (Ramanujan Colloquium): Linear equations in primes

Abstract: Many of the most famous and most important questions on the distribution of primes can be cast as solving systems of linear equations with prime variables. The twin prime conjecture, Goldbach's conjecture, $k$-term arithmetic progressions of primes and most questions about small gaps between primes can all be seen in this manner, as well as several questions with applications to diophantine geometry or cryptography.

We will describe some of the progress on these questions, with a particular emphasis on establishing weak forms of some of these questions which has led to new results on bounded gaps between primes and large gaps between primes, amongst other things.

Talk 2: Prime values of polynomials

It is believed that any irreducible integer polynomial with no fixed prime divisor and positive lead coefficient should represent infinitely many prime numbers. This is well beyond current techniques, but Friedlander,Iwaniec and Heath-Brown have shown the multivariate polynomials $X^2+Y^4$ and $X^3+2Y^3$ represent infinitely many primes, despite representing a thin set of integers. We will describe recent work showing the existence of many multivariate polynomials which each represent infinitely many primes but only a thin set of integers.

Talk 3: Primes with restricted digits

Numbers for which one digit does not occur in their digital expansion are rare - there are $O(x^{1-\epsilon})$ such integers less than $x$. We will talk about recent work showing that there are infinitely many prime numbers where one digit does not occur.

RAMANUJAN COLLOQUIUM POSTER

Last update made Sat Mar 12 10:16:27 PST 2016.