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 kroner (about US$1 million).
Speaking via webcast, Deligne said he was surprised to learn that he had won the prize this morning. 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.”
The Academy has rewarded Deligne, who works at the Institute for Advanced Study (IAS) in Princeton, New Jersey, “for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields”.
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 Oslo today.
“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. “Deligne is unusual in being both. He’s got a very special mind.”
Algebraic geometry explores the geometric objects that are sets of solutions to algebraic equations — for example, a circle of radius r can be described by x2+y2=r2. In modern mathematical parlance, these shapes are called algebraic varieties. Algebraic geometry 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 the points on algebraic varieties that have integer coordinates (in the case of the circle, x and y must be whole numbers). The number of such solutions — typically, there are only finitely many — can be calculated from a formula 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 says that this proof “completed a long-standing program” in mathematics. “By solving that”, adds 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 it, 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.
In 1978 Deligne’s proof won him the Fields Medal, the original ‘maths Nobel’, which can only be awarded to recipients under 40 years of age. The Abel Prize has no age limitations. 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.
“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.”