Under 40

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The age of 40 is considered special in mathematics because it’s an ad-hoc criterion for deciding whether a mathematician is young or old. This idea has been well established by the under-40 rule for Fields Medal, based on Fields‘ desire that:

while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others

Though it must be noted that this criterion doesn’t claim that after 40 mathematicians are not productive (example: Yitang Zhang).  So I wanted to write a bit about the under 40 leading number theorists which I am aware of (in order of decreasing age):

• Sophie Morel: The area of mathematics in which Morel has already made contributions is called the Langlands program, initiated by Robert Langlands. Langlands brought together two fields, number theory and representation theory. Morel has made significant advances in building that bridge. “It’s an extremely exciting area of mathematics,” Gross says, “and it requires a vast background of knowledge because you have to know both subjects plus algebraic geometry.” [source]
• Melanie Wood: Profiled at age 17 as “The Girl Who Loved Math” by Discover magazine, Wood has a prodigious list of successes. The main focus of her research is in number theory and algebraic geometry, but it also involves work in probability, additive combinatorics, random groups, and algebraic topology.  [source1, source2]
• James Maynard:  James is primarily interested in classical number theory, in particular, the distribution of prime numbers. His research focuses on using tools from analytic number theory, particularly sieve methods, to study primes.  He has established the sensational result that the gap between two consecutive primes is no more than 600 infinitely often. [source1, source2]
• Peter Scholze: Scholze began doing research in the field of arithmetic geometry, which uses geometric tools to understand whole-number solutions to polynomial equations that involve only numbers, variables and exponents. Scholze’s key innovation — a class of fractal structures he calls perfectoid spaces — is only a few years old, but it already has far-reaching ramifications in the field of arithmetic geometry, where number theory and geometry come together. Scholze’s work has a prescient quality, Weinstein said. “He can see the developments before they even begin.” [source]