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‘Grand unified theory of maths’ nets Abel Prize

Robert Langlands’ ideas unearthed connections within mathematics that have helped to solve centuries-old problems and aided researchers in disparate fields.

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Robert Langlands stands in front of a blackboard covered with equations.

Robert LanglandsCredit: Rich Schultz

The Canadian mathematician Robert Langlands has won 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.

At 81, he is still an active member of the Institute for Advanced Study (IAS) in Princeton, New Jersey, where he occupies the office that was once Albert Einstein’s.

The mathematician outlined what became known as the Langlands programme in 1967 and carried out parts of it himself. The programme is a sort of Rosetta stone that allows researchers to translate between different fields of mathematics. That way, a problem that seems unsolvable in one language can become more approachable in the other. And this connection reveals two seemingly different concepts to be two aspects of a deeper truth.

Other researchers have gone on to greatly expand the scope of the programme. At least three mathematicians have won Fields Medals for confirming small parts of the grand scheme. Over time, researchers realized that some older problems of mathematics were actually special cases of the extended programme. One, called the Weil conjectures, was solved by the Belgian mathematician Pierre Deligne, who received the 2013 Abel Prize for that work. Another was a problem cracked in the 1990s by British number theorist Andrew Wiles and a coauthor: that work led them to solve Fermat’s last theorem, earning Wiles the Abel Prize in 2016.

The span of the connections was so broad — earning the description ‘grand unified theory of maths’ — that they often baffled Langlands himself. “It’s almost like you are an archaeologist and you dig up a stone in the desert — and it turns out to be the top of a pyramid,” says mathematical physicist Robbert Dijkgraaf, who heads the IAS.

The Abel Prize is modelled after the Nobel prize and has been given out annually since 2003. It carries an award of 6 million kroner (US$777,000).

Langlands outlined the first version of the programme in 1967, when he was a young mathematician visiting the IAS. 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.

In the 1980s, Vladimir Drinfel’d, a Ukrainian-born mathematician now at the University of Chicago in Illinois, and others proposed a similar connection between geometry and harmonic analysis. Although this idea seemed to be only loosely inspired by the Langlands programme, mathematicians subsequently found stronger evidence that the two fields are connected. (Drinfel’d received a Fields Medal in 1990.)

This geometric Langlands programme encompassed an older conjecture that also related certain equations to harmonic analysis, and which was confirmed in Wiles' proof of Fermat's last theorem, a problem in number theory that had been unsolved for more than 300 years. “It was a great pleasure for me, but also a great surprise,” Langlands wrote in 2007, when Wiles incorporated some of his work into their proof.

The field that blossomed from the Langlands programme has become so broad that Langlands has said that he does not fully understand all of the work that goes on in it, and in particular, some of the implications that the geometric version might have in physics. His IAS colleague Edward Witten, a theoretical physicist and a winner of the 1990 Fields Medal who investigated those connections in the 2000s, has said, “I personally only understand a tiny bit of the Langlands programme.”

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