The Fields Medals have been awarded to four researchers who work on number theory, geometry and network analysis

Number theorist Peter Scholze, who became Germany’s youngest ever full professor aged 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 today to Alessio Figalli, a network-analysis researcher and Akshay Venkatesh, who also works on number theory. Their names were announced in Rio de Janeiro, at the opening of the International Congress of Mathematicians.

The Fields Medals, given out by the International Union of Mathematics, are awarded 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 to ever 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 “Who else do you think will win, apart from Scholze?” became an oft-heard question 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.

Most of Scholze’s work has connections to ‘*p*-adic fields’, exotic extensions of the ordinary number system that are useful tools for studying prime numbers. Upon the *p*-adics, he 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 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 had been able to say definitively whether it checked out. Now Scholze and a colleague are said to have found a significant gap in the proof. Scholze is a professor at the University of Bonn and also directs the Max-Planck Institute for Mathematics in the same city.

**Caucher Birkar**, 40, has made breakthroughs in the classification of algebraic varieties—geometric objects that arise from polynomial equations, such as *y *= *x*2. He was born in 1978 in an ethnic-Kurdish region of southern Iran. Birkar recalls his childhood in one of several videos made available by the Simons Foundation, which funds mathematics and basic-science research, ahead of the announcements: “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. 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 40 million Kurds.

**Akshay Venkatesh**, who is 36, works 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 Carl Friedrich Gauss in the 1800s. 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, called optimal transport, which seeks the most efficient ways to distribute goods on a network. Figalli applies it to partial differential equations, which are the equations with several variables that most often arise in physics. Figalli is Italian and works at the Swiss Polytechnic Institute (ETH) in Zurich.

*This article is reproduced with permission and was first published on August 1, 2018.*