Weil to go (OK, so you have to know the pronunciation)

Here’s the initial draft (sort of) of my story for Nature news on the Abel Prize. I blanched when I read the award citation, but in the end this was fun.

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Proof of a deep conjecture about algebra and geometry nets Abel Prize

It has been four decades since Belgian mathematician Pierre Deligne completed the work for which he became celebrated, but that fertile contribution to number theory has now earned him the Abel Prize, one of the most prestigious awards in mathematics.

Given annually by the Norwegian Academy of Science and Letters and named after the famous Norwegian mathematician Niels Henrik Abel, the prize is worth 6 million Norwegian krone, or about US$ 1m.

The Academy has rewarded Deligne, who works at the Institute for Advanced Study in Princeton, New Jersey, “for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields.”

Speaking via webcast on Wednesday, Deligne said he was surprised to learn that he had won the prize. Despite having won major prizes before, he said, he did not spend much time wondering about when the next one would come. “The nice thing about mathematics is doing mathematics,” Deligne said. “The prizes come in addition.”

Deligne has made “many different contributions that have had a huge impact on mathematics for the past 40-50 years”, says Cambridge mathematician Timothy Gowers, who delivered the award address in Olso.

“Usually mathematicians are either theory builders, who develop tools, or problem-solvers, who use those tools to find solutions”, says Peter Sarnak, also at the IAS in Princeton. “Deligne is unusual in being both. He’s got a very special mind.”

Algebraic geometry explores the links between geometric objects and the algebraic equations that describe them – for example, the expression for a circle of radius r, x2+y2=r2. It has proved to have deep connections to many areas of mathematics, particularly the properties of pure integers (number theory).

This last connection is evident in the analogy between the Riemann hypothesis, which describes a relationship between prime numbers, and the so-called Weil conjectures, which were proposed by mathematician André Weil in 1949 – the subject of Deligne’s most famous result.

The Weil conjectures concern objects in algebraic geometry called algebraic varieties, which are the set of solutions of algebraic equations. The number of such solutions can be found from a function called the zeta function. While Riemann’s hypothesis concerns the nature of the Riemann zeta function, which determines how prime numbers are distributed among all the integers, the Weil conjectures specify some of the properties of zeta functions derived from algebraic varieties.

There are four of these conjectures. The first three were proved to be true in the 1960s, but the fourth and hardest – and the direct analogue of the Riemann hypothesis – was proved by Deligne in 1974. The Riemann hypothesis itself remains “the most famous unsolved problems in mathematics”, says Gowers – which is in itself an indication of the significance of Deligne’s proof.

Gowers adds that this proof “completed a long-standing programme” in mathematics. “By solving that”, says Sarnak, “he solved a whole lot of things at once”. For example, the solution also proved a long-standing, recalcitrant conjecture by the famous Indian mathematician Srinivasa Ramanujan.

In finding his proof, Deligne built on the work of his mentor, the German-born mathematician Alexander Grothendieck, who proved the second Weil conjecture in 1965. That work introduced a crucial concept called l-adic cohomology.

The general notion of cohomology, which concerns the topological properties of spaces described by algebraic equations, was itself first developed in the 1920s and 30s, and Weil recognized that it would be needed to prove his hypotheses. Grothendieck laid the foundations for finding the right cohomology, but his student Deligne found the final proof alone – and in a different way from what Grothendieck had imagined.

Deligne’s proof won him the Fields Medal, the “other maths Nobel” besides the Abel Prize, in 1978, and in 1988 he shared the Crafoord Prize with Grothendieck – making him an obvious candidate for the Abel. Since completing the work that secured his reputation, he has applied tools such as l-adic cohomology to extend algebraic geometry and to relate it to other areas of maths. For example, because much of his work is concerned with so-called finite fields – basically modulo arithmetic – it can be applied to the kind of digital logic used in computing. “People in computer science are using his results without even knowing it”, says Sarnak.

“Even if you took away his most famous result on the Weil conjectures”, says Gowers, “you would still be left with a great mathematician.”

Deligne said he had not thought yet about how he would spend the money that came with his Abel Prize, but that he would like to find a way to make it useful for mathematics. “To some extent, I feel that this money belongs to mathematics, not to me.”