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Features: Faculty Insights

 

Jack Thorne, Professor of Number Theory at the Department of Pure Mathematics and Mathematical Statistics (DPMMS), has been awarded a prestigious 2022 New Horizons in Mathematics Prize.

New Horizons Prizes are awarded annually to early career researchers, complementing the well-known Breakthrough Prizes. Up to three New Horizons prizes are awarded in mathematics each year, and each is worth US$100,000.

"It's really humbling to receive the Prize," says Thorne. "I'm familiar with the work of most of the former prize winners and I hesitate to compare myself to them — it's very nice to be counted amongst a group of people who I greatly respect."

Ramanujan and squares

Thorne's work, conducted in collaboration with James Newton, can be traced to the kind of questions anyone who likes playing with numbers might stumble across. You might ask yourself, for example, which integers can be written as a sum of square numbers: the number 13 is a sum of two square numbers because

13=22+33,

the number 30 is a sum of three square numbers because

30=12+22+52,

and so on.

Questions regarding sums of squares have fascinated number theorists throughout the ages, and the famous Indian mathematician Srinivasa Ramanujan was no exception. In 1916, during his time in Cambridge, he considered a very particular sums-of-squares problem: in how many ways can you write an integer as a sum of 24 square numbers?

I was able to make quite a few example sheet questions accessible to students that were also closely related to my work. The students responded very positively — I really appreciated that! Jack Thorne

Ramanujan came up with an answer that works for any integer, but it was only approximate. The difference between the true answer and his approximation was described by an error term. Out of this error term, Ramanujan built a mathematical object called an L function, about which he made a number of conjectures. And it turns out, there is actually a whole family of such functions, whose properties control the statistics of the sums of squares problem.

Specific as the initial problem about sums of 24 squares may seem, Ramanujan's conjectures had a far-reaching influence. "Amazingly, they motivated a lot of the most important work in number theory of the twentieth century," says Thorne. The proof of one of them, constructed by the Belgian mathematician Pierre Deligne as part of a bigger problem (the Weil conjectures), was even rewarded with a Fields Medal in 1978.

Riemann and primes

Ramanujan's functions were not the first of their kind, and they were not the first to have caused excitement among number theorists. Around sixty years before Ramanujan's work in Cambridge, the German mathematician Bernard Riemann was attracted by a class of numbers no number theorist can ignore: the primes. These are integers that are only divisible by themselves and 1 — and since every integer can be written uniquely as a product of primes, they count as fundamental building blocks of number theory.

We have known since the time of Euclid that there are infinitely many primes, but there isn't a formula that captures them all in the way that, say, the formula 2n captures all even numbers. One thing number theorists are particularly interested in is how the prime numbers are distributed among the other integers. We know that they get sparser as you move up the number line, but there's no obvious regularity to their appearance.

An important key to the statistical distribution of the primes lies with what is now called the Riemann zeta function, which is also an L function. "The properties of the Riemann zeta function, the values it takes and how quickly it grows, etc, seem to be very closely connected to questions about the distribution of prime numbers," says Thorne.

The values for which the Riemann zeta function is equal to zero play an especially important role in this context. Indeed, the Riemann hypothesis, which conjectures exactly what form these zeroes take, constitutes one of the biggest unsolved problems in all of mathematics.

New horizons

The work that won Thorne the New Horizons Prize involves the family of L-functions containing Ramanujan's L-function. "Together with James Newton we proved some fundamental facts about Ramanujan' functions which are very much analogous to what Riemann proved in his initial work about the Riemann zeta function in the 1850s," he says.

While the Riemann zeta function gives you information about the statistical distribution of the primes themselves, Thorne and Newton's work gives information about the statistics of slightly different questions that are related to primes. For example, given the equation

x2+y2=10,

you can ask yourself how many integer solutions there are: how many pairs of integers (x,y) satisfy the equation.

If you like working in the world of primes you can modify this question by imagining the integers as wrapping around a clock face with p hours, where p is a prime number. If p=13, for example, then you'd consider the numbers 1 to 13 to be distinct, but identify 14 with 1, 15 with 2, 16 with 3, and so on. The question then becomes how many solutions the equation has in this arithmetic mod p, as it is called.

You can pose this question for each prime number p, but the answer is likely to be different every time, which is why it makes sense to ask for general statistics. "For example, you can ask how many solutions there are to the equation modular p on average [over all primes p]," explains Thorne. That's the kind of question Thorne and Newton's work can provide an answer to.

This, however, is not the whole story.

Langlands programme

One of the many people who worked on Ramanujan's zeta functions over the years was the Canadian mathematician Robert Langlands. His aim was far grander, however, then understanding particular problems involving sums of squares. Instead, he was formulating in the 1960s a web of conjectures that link number theory to geometry and analysis. To be more precise they link objects in number theory that are comparatively basic (such as diophantine equations) to more complex mathematical objects from geometry, called automorphic forms.

"Automorphic forms are difficult to define and difficult to study, but if you can show that there is this link between the number theoretic world and the automorphic world, then you can often learn an awful lot about the number theoretic world you started off in," explains Thorne.

Langlands programme, as the set of conjectures has become known, is one of the biggest single projects in mathematics. "The programme has many different aspects and touches on almost all parts of number theory," says Thorne. "You can use it to make predictions about many different things, by taking different aspects of it."

Once all of Langlands are proved, the result will be akin to a grand unified theory for mathematics. Thorne's work, for which he has been awarded the New Horizons Prize, is an important step along the road: it establishes part of Langlands' conjectures for a class of objects including Ramanujan's Delta function.

Sharing vision

Thorne has been interested in this particular piece of research ever since he learnt, as Part III student at Cambridge, what an automorphic form was, but it wasn't until one day in 2019 that he realised how one might attack it. "I had this idea that two things which I hadn't thought were related might work together to make progress on this problem. I straight away picked up the phone and dialled James Newton's number and said 'do you think that might work?' It was a wonderful experience working with him."

Thorne has shared his vision, not just with Newton, but also with students at Cambridge who were taking the famous Part III Masters Programme. "Part III is a unique thing in the mathematical world — such a large masters programme with a huge number of courses being taught at a high level," he says.

"Something I really enjoyed last year was giving my Part III modular forms course. Because it was of such direct relevance to the research that I was doing I was able to make quite a few example sheet questions accessible to students that were also closely related to my work. The students responded very positively — I really appreciated that!"