JAMES MAYNARD TO RECEIVE 2014 SASTRA RAMANUJAN PRIZE

The 2014 SASTRA Ramanujan Prize will be awarded to Dr. James Maynard of Oxford University, England, and the University of Montreal, Canada. The SASTRA Ramanujan Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The prize will be awarded during December 21-22 at the International Conference on Number Theory at SASTRA University in Kumbakonam (Ramanujan’s hometown) where the prize has been given annually. Dr. Maynard has made revolutionary contributions to prime number theory in his doctoral thesis of 2013 and in his post-doctoral work in the last twelve months. Some of these results relate to the great unsolved problem “the prime twins conjecture” and its generalization to the “prime k-tuples conjecture” for admissible sets of k linear functions. In his doctoral thesis at Oxford University written under the direction of Professor Roger Heath-Brown, he obtained a significant improvement of the Brun-Titchmarsh inequality for the distribution of primes in arithmetic progressions, by establishing a uniform upper bound for arithmetic progressions whose common difference does not exceed the eighth root of the largest term, improving all known previous bounds on this problem in this situation. This work appeared in Acta Arithmetica in 2013. Also in his doctoral thesis, he obtained bounds for the total number of prime factors in the product of k admissible linear functions, again improving all known earlier bounds on this problem, except in one case, that of three admissible linear functions, where his result was just as strong as the 1972 theorem of Porter. This work of Maynard appeared in Mathematika in 2014. But in a separate paper that appeared in the Proceedings of the Cambridge Philosophical Society in 2013, he improved Porter’s upper bound as well. In the last twelve months, while on a post-doctoral position at the University of Montreal under the mentorship of Professor Andrew Granville, Maynard stunned the world by making major advances on both the large gap and small gap problems for primes. With the average gap between the n-th prime and the next prime being asymptotically log n as given by the Prime Number Theorem, two questions arise: (i) How small can this gap be infinitely often (the small gap problem), and (ii) How large can this gap be infinitely often (the large gap problem)? The Prime Twins Conjecture asserts that the gap is 2 infinitely often. The first major advance was by Goldston-Pintz-Yildirim a few years ago when they showed that the average gap between the n-th prime and the next is much smaller than log n infinitely often. And last year, Y. Zhang produced the astounding result that the gap is no more than 70 million infinitely often, producing for the first time a bounded gap! , leading the Polymath project, reduced Zhang’s bound to 4680. But soon after, by an entirely different, simpler, and more elegant technique, Maynard surprised everyone by proving that the gap is no more than 600 infinitely often. This landmark paper is soon to appear in the Annals of Mathematics. In that same paper, Maynard shows that for any given integer m, the gap between the n + m-th prime and the n-th prime can be bounded by a prescribed function of m, for infinitely many n. Thus Maynard is making his mark on several long standing problems in prime number theory in rapid succession. His latest acievement is his announcement of

1 the solution to the $10,000 problem of Paul Erd¨os concerning large gaps between primes (also announced independently and simultaneously by Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao). In addition to his individual papers, Maynard has begun major collaborations with a number of researchers. His recent paper with William Banks and Tristan Freiberg establishes a surprising result on the limit points of the set obtained by taking ratios of the gap between the n-th prime and the next and the average gap log n. Thus by his own spectacular results and by his collaborations, Maynard has become a new dominant force in prime number theory. The 2014 SASTRA Ramanujan Prize citation reads as follows: “James Maynard is awarded the 2014 SASTRA Ramanujan Prize for his revolutionary contributions to prime number theory, for making the strongest advances thus far on vari- ous long standing problems on primes, and for the ingenious techniques he has introduced which will influence future research in the field. The prize recognizes his fundamental doc- toral thesis at Oxford University and his sensational post-doctoral work at the University of Montreal. In particular, the prize is for his 2013 paper in Acta Arithmetica in which he establishes the strongest known form of the Brun-Titchmarsh inequality, his 2103 paper in the Proceedings of the Cambridge Philosophical Society and his 2014 paper in Mathe- matika in which he improves all previously known results on almost prime k-tuples. The prize recognizes his recent paper on small gaps between primes to appear in the Annals of Mathematics in which he establishes the sensational result that the gap between consecu- tive primes is no more than 600 infinitely often. The prize also makes note of his recent announcements on the solution of the large gaps problem on primes due to Erd¨os, and his joint work with Banks and Freiberg on the limit points of the sequence of normalized prime gaps.” James Maynard was born in Chelmsford, England on June 10, 1987. He obtained his BA and Masters in Mathematics from Cambridge University in 2009. He then joined Balliol College, Oxford University, where he received his Doctorate in Philosophy in 2013. During 2013-14 he was a post-doctoral fellow at the University of Montreal, Canada. Thus within a period three years, Maynard has startled the mathematical world with deep results in rapid succession. The SASTRA Ramanujan Prize will be his first major award in recognition of his many fundamental results in prime number theory. The 2014 SASTRA Ramanujan Prize Committee consisted of Professors Krishna- swami Alladi - Chair (University of Florida), Roger Heath-Brown (Oxford University), Winnie Li (The Pennsylvania State University), David Masser (University of Basel), Barry Mazur (), Peter Paule (Johannes Kepler University Linz), and Michael Rapoport (University of Bonn). Previous winners of the Prize are and in 2005 (two full prizes), Terence Tao in 2006, Ben Green in 2007, in 2008, Kathrin Bringmann in 2009, Wei Zhang in 2010, Ro- man Holowinsky in 2011, Zhiwei Yun in 2012, and Peter Scholze in 2013. The award of the 2014 SASTRA Ramanujan Prize to James Maynard is a fitting recognition in the tenth year of this prize and in keeping with the tradition of recognizing path-breaking work by young mathematicians.

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