Distinguished Lecture Series 2024: James Maynard (Oxford)
20 – 22 March 2024
Venue: Lecture Theatre G.10, Ground Floor, School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG
Organised in collaboration with the School of Mathematics, University of Bristol, UK
James Maynard, Mathematical Institute, University of Oxford, UK
James Maynard is a professor of Number Theory at the University of Oxford. He works in analytic number theory, particularly the study of prime numbers. He is a fellow of the Royal Society and has been awarded numerous prizes, including the SASTRA Ramanujan Prize, an LMS Whitehead Prize, an EMS Prize, the Compositio Prize, the AMS Cole Prize and a New Horizons Prize.
James was awarded the Fields Medal in 2022 for “contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation”.
Lecture 1: Classical Sieve Methods (Colloquium Style)
We will give an overview of ‘standard’ sieve methods: what are they? And what are they good for? What can they (and can they not) say about prime numbers? Classical sieve methods are an exceptionally versatile set of techniques that are ubiquitous in analytic number theory, but often fall just short of the task which they were designed for: finding prime numbers. Sometimes these limitations can be side-stepped allowing us to prove results about the existence of primes, such as in work on bounded gaps between primes.
Lecture 2: Primes and Sieves II: Prime Detecting Sieves
We give an overview of how the limitations of ‘standard’ sieves are overcome by introducing extra arithmetic information into the method, which in principle can detect prime numbers and achieve the original goal of sieves. This offers a possible attack to many famous open problems about prime numbers, but unfortunately can currently only be made to work in ‘nice’ situations. Nevertheless, there is a general approach to trying to count primes in sets which are ‘not too sparse’, such as sets with digit restrictions.
Lecture 3: Primes and Sieves III: Optimality of Prime Detecting Sieves
We will talk about some of the key open questions in sieve theory, and what we would need to have an efficient tool-kit to answer questions about primes. Work-in-progress (with Kevin Ford) allows us demonstrate a provably optimal version of prime-detecting sieves in various settings, as well as demonstrations of the limitations of the current prime-detecting setup.
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