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Mathematicians answer key question in ‘grand unified theory of pure mathematics’

Professor Jack Thorne will share the 2023 Frank Nelson Cole Prize in Number Theory for making progress on a decades-old problem related to the work of Ramanujan.

With Professor James Newton, who was a Junior Research Fellow at Trinity 2011-2014, he tackled a key question relating to the statistics of prime numbers and Ramanujan’s work on sums of squares.

The American Mathematical Society, which awards the Frank Nelson Cole Prize, called their work an ‘astonishing proof of a landmark, sought-after case of the Langlands Conjectures: namely the symmetric power functoriality for holomorphic modular forms.’

After studying maths at Trinity Hall, Jack Thorne completed his PhD at Harvard. He was a Clay Research Fellow, 2012-2017

Mathematicians such as Professors Thorne and Newton who work on number theory are concerned with questions about the distribution of prime numbers. Professor Thorne explained that a key factor governing the statistics of prime numbers is the Riemann zeta function, which can be used to estimate the number of primes up to a given bound.

The most optimistic estimates for this number are equivalent to the Riemann hypothesis, arguably the most famous unsolved problem in pure mathematics. This was a problem that the self-taught mathematician Srinivasa Ramanujan grappled with when he came to Trinity from India at the beginning of twentieth century, at the invitation of GH Hardy.

Professor Thorne, who is a Fellow of the Royal Society, said:

In 1916, during his time at Trinity, Ramanujan wrote down an analogue of the Riemann zeta function, related to the “sums of squares” problem (ie how many ways are there of writing a given number as a sum of squares).

Nowadays we see that Ramanujan’s analogue of the zeta function is one of an infinite family of objects that collectively govern the statistics of the sums of squares problem.

Ramanujan’s work in this area is now seen to fit into the framework of the Langlands Program, a series of conjectures made by this mathematician in the 1960s, which have been described as a ‘grand unified theory of pure mathematics.’

Professor Thorne, of Cambridge’s Department of Pure Mathematics and Mathematical Statistics, said:

The work recognised in our prize citation shows that these (infinitely many!) analogues of the Riemann zeta function have the same good properties, an expectation first made explicit by Langlands and Serre in the 1960s, and an open problem until now.

The award of the 2023 Cole Prize – to be made in January 2023 at the Joint Mathematics Meetings of the American Mathematical Association – follows Professor Thorne’s 2022 New Horizons Mathematics Prize, which he received in September 2021.

The Cole Prize in Number Theory recognizes notable research in number theory completed in the last six years and peer-review published. Read the full citation AMS :: News from the AMS

Read more Trinity Fellow Professor Béla Bollobás about Ramanujan at Trinity: Opinion: The man who taught infinity: how GH Hardy tamed Srinivasa Ramanujan’s genius | University of Cambridge

 

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