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At the beginning of August the International Congress of Mathematicians (ICM) took place in Rio de Janeiro, Brazil. The ICM is the biggest maths conference of them all, attracting thousands of participants, and also sees the awards of some very prestigious prizes. The famous Fields medal is one of them: it is awarded every four years to four mathematicians up to the age of forty, “to recognise outstanding mathematical achievement for existing work and for the promise of future achievement.” Along with the Abel Prize the Fields medal counts as the highest honour in mathematics.

**Caucher Birkar**

This year’s laureates between them cover a range of mathematical areas. Caucher Birkar, of the University of Cambridge works in an area called *algebraic geometry*. Just as the equation *y=x* defines a straight line, so can more complex equations define more complex, and abstract, geometric objects, called *algebraic varieties*. There is an infinite zoo of algebraic varieties, so mathematicians try and classify them into families, just as one would classify a collection of butterflies.

For a large group of varieties (technically, smooth and projective ones) there’s a particularly nice method of classification: mathematicians believe, but can’t yet prove, that every such variety is related to one of a family of relatively simple varieties, each made up of just three basic building blocks. Sorting smooth, projective varieties into groups according to which of the possible three-building-block varieties they are related to is the over-arching aim of the field. Birkar is being honoured for the significant contributions he has made to this classification programme.

**Alessio Figalli**

The work of Fields medallist Alessio Figalli, of the ETH in Zurich, is inspired by a practical problem: how to best transport a distribution of “stuff” from one place to another. As an example, imagine you own a series of gold mines and want to transport the yield of each mine to a bank for safe deposit. You’ve got the same number of banks as mines, and your task is to decide which mine to match up with which bank. The optimal way of doing this depends on what you feel is best: you might want to minimised the overall distance travelled by the gold, or perhaps you’d prefer the total time it spends on the road.

Phrased in more general terms, this kind of optimal transport problem turns from something you might be able to solve using pencil and paper into a deep problem in the theory of mathematical functions. Figalli has been honoured for his contributions to optimal transport theory and the way he has applied it to real-life problems. For example, the formation of clouds can be viewed as an optimal transport problem, leading to applications in meteorology.

**Peter Scholze**

Peter Scholze, of the University of Bonn, is the person everyone guessed would receive a Fields medal. He already enjoyed superstar status at the last ICM in 2014, and at the tender age of 30 is the youngest of this year’s laureates.

Scholze’s work involves number theory, algebra and geometry. Number theorists are very fond of prime numbers, because they can be viewed as the building blocks from which all other integers can be built: that’s because

every integer can be written as a product of prime numbers, for example 30=2x3x5. Given some prime number *p*, number theorists often like to know how other integers relate to *p*: are they divisible by *p* or some power of *p*, and if not, what is the remainder? In this context it is useful to come up with a new notion of distance, which relates to the prime *p*: two integers *x* and *y* are considered close if their difference is divisible by a large power of *p*. Extending this notion gives a so-called *p-adic field, *a sort of analogue of the real numbers, which relates to the prime number *p*.

Doing algebra and algebraic geometry on such p-adic fields raises some very subtle problems. Scholze received his medal for coming up with new tools for dealing with such problems: *perfectoid spaces*. His idea has revolutionised the area in to a growing new area of research, has already helped solve open problems and opened up new avenues for research.

**Akshay Venkatesh**

Most of Akshay Venkatesh’s career has involved exploring the borders of number theory with different areas of mathematics. He has brought together concepts and tools from different areas with great impact on solving open problems and pointing the way to future progress. An example is Venkatesh’s resolution of the *sub convexity* problem in number theory.

As mentioned above, number theorists are interested in the prime numbers, in particular, they would like to know just how the prime numbers are distributed among the other numbers. One of the most famous open problems in mathematics, the *Riemann hypothesis*, relates to this question. The hypothesis claims that a certain type of function, the *Riemann zeta function*, is equal to 0 along a critical line in the plane. The Riemann zeta function is related to the prime numbers, which is why knowing where it is equal to zero gives us information of the distribution of primes.

Venkatesh’s work involved, not the Riemann zeta function specifically, but a larger class of functions called *L*–*functions*, which are also related to prime numbers. The sub convexity problem is concerned with tightening a bound on the size of L-functions along a critical line, and has been of interest for a century. Venkatesh resolved it in 2010, generalising all previous work in the area and also bringing together approaches from algebraic geometry, dynamics and number theory, proving significant new results in these areas along to way to a final proof.

To find out more about the work of these four Fields medallists, and about the other prizes that were awarded at the ICM, see here.

Diretor Geral do IMPA Marcelo Viana. Winners of SBM Journalism Prize ICM Closing Ceremony – ICM 2018.

**The theft of a Fields medal**

While mathematicians at the ICM focused (mostly) on the maths, the international media focused on one of the Fields medallists and a scandalous crime: Caucher Birkar, already of interest because of his personal history as a Kurdish refugee in the UK, had his Fields medal stolen just minutes after receiving it. As people came up to congratulate him, someone stole his briefcase, containing the medal, from the chair directly behind him. The Brazilian organisers of the conference went to great lengths to help local police find the thief, so far to no avail.

When Birkar received a new medal a few days after the theft he made up for the shock with a brief but moving re-acceptance speech. “This has been widely covered in the media, and I am now more famous,” he joked. “And the number of people who know what a Fields medal is, is way more than it was last week.”

“In the beginning it was a shock, it happened so fast, but very soon I recovered,” he went on. “In the grand scheme of things this is a really, really small thing. I have seen much worse things in my life, and if I was discouraged by such small things I wouldn’t be here today.”

Originally from Kurdistan, now living in the UK after being accepted as a refugee, Birkar explained the importance of the award to his home. “The Kurdish people are very happy, and in the region, in Iran, Iraq and Turkey. This is a positive thing and I hope it will inspire people, that they can succeed.”

Birkar grew up in a city on the border between Iran and Iraq during the war: “My city is right on the border. You can imagine we were an easy target. The threat was always everywhere.” He explained how he went on to become one of today’s greatest mathematicians: “There are a few reasons. It as to do with the way of life of the region. When I was born in my village life was traditional – we produced everything we needed. The other reason is that Kurdish culture has been conditioned for difficulties by its history.” And the final reason for Birkar’s success was family: “When I was 10,11,12 my brother helped me a lot to learn mathematics. Family helps enormously – directing, not pushing.”

And what helped Birkar carry on during his career, moving far from home to a new life in the UK? “At some point I realised that learning maths is another way of enjoying life. That’s why I never stopped.”

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