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Number-theory prodigy among winners of most coveted prize in mathematics

The Fields Medals have been awarded to four researchers who work on number theory, geometry and differential equations.

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Fields Medal winners Akshay Venkatesh, Peter Scholze, Alessio Figalli and Caucher Birkar

The year’s Fields Medal-winners (left to right): Akshay Venkatesh, Peter Scholze, Alessio Figalli and Caucher Birkar.Credit: Pablo Costa/ICM2018

Number theorist Peter Scholze, who became Germany’s youngest ever full professor at the age of 24, and geometrician Caucher Birkar — a Kurdish refugee — are among the winners of this year’s Fields Medals, the most coveted awards in mathematics. The medals, which are given out every four years, were also awarded on 1 August to Alessio Figalli, a network-analysis researcher and Akshay Venkatesh, who also works on number theory. Their names were announced in Rio de Janeiro, Brazil, at the opening of the International Congress of Mathematicians.

The Fields Medals, given out by the International Mathematical Union, are awarded to up to four mathematicians aged 40 or younger. For the first time in the medals’ 82-year history, none of the awardees are citizens of the United States or France — two countries that together have netted nearly half of the medals so far. Maryam Mirzakhani, a winner in 2014, remains the only woman ever to receive the prize. (Mirzakhani died of cancer in 2017.)

Few observers doubted that Peter Scholze deserved a Fields Medal, or that he would win one this year — to the point that the question “Who else do you think will win, apart from Scholze?” has been often heard in the mathematical community. The 30-year-old became famous at 22, as a graduate student, for finding a way to drastically shorten a book-length proof in arithmetic geometry.

Scholze is now a professor at the University of Bonn in Germany, and a director at the Max-Planck Institute for Mathematics in the same city. Most of his work has connections to ‘p-adic fields’, exotic extensions of the ordinary number system that are useful tools for studying prime numbers. On the p-adics, he has built fractal-like structures called perfectoid spaces, which have helped him and others to solve problems across several fields of mathematics, including geometry and topology.

In a 2016 profile, a colleague described Scholze’s ability to zero in on the essence of a problem as evoking “a mixture of awe and fear and exhilaration”. In recent months, Scholze has been checking a gigantic proof of the abc conjecture, one of the biggest unsolved problems in number theory. In 2012, the enigmatic Japanese mathematician Shinichi Mochizuki posted a proof online, but no one has yet been able to say definitively whether it checks out. Now, Scholze and a colleague, Jacob Stix, are said to have found a significant gap in the proof.

Caucher Birkar, 40, has made breakthroughs in the classification of algebraic varieties — geometric objects that arise from polynomial equations, such as y = x2. He was born in 1978 in a region of western Iran dominated by the Kurdish ethnic group. Birkar recalls his childhood in one of several videos of the Fields medallists made available ahead of the announcements by the Simons Foundation in New York City, which funds mathematics and basic-science research: “My parents are farmers, so I spent a huge amount of time actually doing farming,” he says. “In many ways, it was not the ideal place for a kid to get interested in something like mathematics.”

After studying at the University of Tehran, in 2000 Birkar moved to the United Kingdom, where he got refugee status and, eventually, UK citizenship. He is now a researcher at the University of Cambridge. In the Simons video, Birkar says he hopes that his Fields Medal will put “just a little smile on the lips” of the world’s estimated 40 million Kurds.

Birkar’s win made headlines for more than just his research: before the award ceremony was over, his briefcase was stolen with his medal, which is made of 14-karat gold and is worth about US$4,200. The organizing committee of congress presented him with a replacement medal in a special ceremony on 4 August.

The Fields Medal

Credit: Stefan Zachow/IMU

Akshay Venkatesh, who is 36, works, among other things, on classical problems in number theory, including number systems that consist of fractions of whole numbers and roots such as √2. He is among the few mathematicians who have made substantial progress on a question formulated by mathematician Carl Friedrich Gauss in the nineteenth century. Venkatesh was born in New Delhi and raised in Australia, and is currently at the Institute for Advanced Study in Princeton, New Jersey.

Compared with the other three medallists, 34-year-old Alessio Figalli works in an area that is closer to the real world: optimal transport, which seeks the most efficient ways to distribute goods on a network. Figalli, who is Italian and works at the Swiss Federal Institute of Technology in Zurich, applies the field to partial differential equations, which have several variables and most often arise in physics.

Nature 560, 152-153 (2018)

doi: 10.1038/d41586-018-05864-w
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Updates & Corrections

  • Update 07 August 2018: This story has been updated to include details about the theft of Caucher Birkar’s Fields Medal.