Canadian mathematician Robert Langlands has been named as the winner of the 2018 Abel Prize - one of mathematics’ most-prestigious awards—for discovering surprising and far-ranging connections between algebra, number theory and analysis, the Norwegian Academy of Science and Letters announced on 20 March.
Langlands, who is known to be skeptical of math awards, greeted news of the Abel Prize with hesitation.
“I do not know what to say in response to the prize,” he told the Star in an email. “I had to ask the gentleman who called to give me a half-hour to think things over.”
Langlands said he accepted the award after consulting his wife, Charlotte, who noted that “one or several people must have made a considerable effort on my behalf.” He added he is “still not sure what to think, except that I am grateful to these people for their good opinion.”
The 81-year-old mathematician, an emeritus professor at the Institute for Advanced Study in Princeton, would receive the financial award of six million Norwegian kroner ($777,000) from Norway's King Harald V at an award ceremony in Oslo on May 22.
Named after the 19th-century Norwegian mathematician Niels Henrik Abel, the prize has been awarded since 2003 for “outstanding scientific work in the field of mathematics.”
Previous winners include Andrew J. Wiles, a mathematician now at the University of Oxford who proved Fermat’s Last Theorem; Peter D. Lax of New York University; and John F. Nash Jr., whose life was portrayed in the movie “A Beautiful Mind.”
‘Grand unified theory of math’
“Langlands program,” explores a deep connection between two pillars of modern mathematics: number theory, which studies arithmetic relationships between numbers, and analysis, which is an advanced form of calculus. The link has far-reaching consequences that mathematicians have used to answer centuries-old questions about the properties of prime numbers.
Langlands first articulated his vision for the program in 1967 — when he was 30 — in a letter to the famed mathematician André Weil. He opened the 17-page missive with a now-legendary stroke of modesty: “If you are willing to read it as pure speculation, I would appreciate that,” he wrote. “If not — I am sure you have a waste basket handy.”
His starting point was the theory of algebraic equations (such as the quadratic, or second-degree, equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially.
But Galois also showed that solutions to such equations must be linked by symmetry. For example, the solutions to x5 = 1 are five points on a circle when plotted onto a graph comprised of real numbers along one axis and imaginary numbers on the other. He showed that even when such equations cannot be solved, he could still glean a great deal of information about the solutions from studying such symmetries.
Inspired by subsequent developments in Galois’s theory, Langlands’ approach allowed researchers to translate algebra problems into the ‘language’ of harmonic analysis, the branch of mathematics that breaks complex waveforms down into simpler, sinusoidal building blocks.
Today mathematicians working in the Langlands program are trying to prove that relationship and many other related conjectures. At the same time, they’re using Langlands-type connections to solve problems that would otherwise seem out of reach. The most celebrated result in this regard is Andrew Wiles’s proof in 1995 of Fermat’s Last Theorem. Wiles’s proof depended in part on exactly the type of relationship between number theory and analysis that Langlands had predicted decades earlier.