# Top thinker Caucher Birkar talks maths and migration

Tens of thousands voted to choose the world’s top thinker from the 50 names we presented in our last issue. Here we speak to the winner, Cambridge mathematician Caucher Birkar

There has been phenomenal interest in our list of the World’s Top 50 Thinkers, with tens of thousands voting. After the ballot closed, I’ll confess to jitters on opening the results. In these polarised times, what sort of mind would rally the most online support? One of the more trenchantly opinionated thinkers on the list seemed a safe bet, a name that would probably alienate one set or another of Prospect readers.

In the event, though, the runaway winner was a thinker of a different sort: the 41-year-old Cambridge mathematician Caucher Birkar. It’s hugely satisfying to see work of pure thought honoured in this way—and doubly so because this is a Fields Medal winner with quite a backstory.

Birkar is a Kurd from Iran, one of four states that plays home to the stateless Kurdish people, and one where the traditional answer to their national aspirations has been an iron fist. Born in 1978, he grew up right by the border with Iraq; the bloody war raged throughout his childhood in the 1980s.

Speaking over a crackly connection from China, where Birkar is giving a lecture, he tells me how his family made a living off the land in western Iran: “My parents were farmers, as all my ancestors were farmers. We had a piece of land around the village and we—including myself and my brother—were charged with going to work on these fields, and grow all kinds of vegetables… wheat, and barley… We essentially produced almost everything we consumed.”

Did mathematical know-how lurk in this unlikely setting? “My mother never attended any school,” Birkar replies. “My father attended school up to primary school, and then he did not continue because it just wasn’t really practical.” Birkar’s talent for numbers was nurtured by his brother, six years older and then in secondary education. From the age of 10 or 11, he was introduced to “more advanced maths and physics… things like calculus.” In the UK, that’s the kind of thing you study at A-Level. Birkar, however, wasn’t studying maths to pass an exam: the great thing, he says, was realising you “could learn things just for fun, not just to get good marks in school.”

Birkar went on to study maths at the University of Tehran, and in 2000 arrived in the UK to take part in an academic competition. He never went home: instead, he claimed asylum, changing his name from Fereydoun Derakhshani to Caucher Birkar, which in Kurdish means “migrant mathematician.”

But what of his award-winning mathematical work? In journalism, a bit of bluffing comes with the territory. But when you talk to Birkar, it’s soon clear that there’s no bluffing your way through his field of contemporary “algebraic geometry.” The phrase itself doesn’t seem too daunting: Pythagoras’s rule, which most of us studied at school, expresses geometric relations as algebra. But when I cheerfully inquire about whether his work could be thought of as advanced Pythagoras, I am corrected. It would be “misleading” to suggest that mathematicians were “just doing anything like Pythagoras… after almost 3,000 years!”

So what does Birkar’s work involve? The simplest—if semi-metaphorical—answer turns out to be finding connections between complex shapes.

But when I ask for a pithy summary, he first offers this explanation: “The solution of polynomial equations.” These are a class of equations with multiple terms (X, Y) typically raised to varying powers (X2, X3, etc). So where does the geometry come in? Such equations are often used to describe shapes. Birkar explains that his thought proceeds in “two different languages: on one side, we have algebra, which is the equation and functions; on the other side, we have the geometric thing.”

Notably careful and precise, Birkar uses the imprecise word “thing” here, I think, because “shape” won’t quite do. Ordinarily he’s working in many more than three dimensions, with constructs that defy visualisation. But if he’s no longer dealing in shapes you can visualise, is this geometry at all? Yes, he tells me: “there is still intuition” that makes the “spaces” that “you can’t really draw” more mentally tractable than would be the case if you were doing equations without thinking visually. Algebra “gives you the techniques to criticise it and formulate it”; “the geometry,” the “intuition” about what “to do, what direction you should take.”

But what are you actually trying to do with these “things” or “spaces?” Birkar offers a pleasing comparison with biology, where “you’re putting living creatures into a group and you call them… mammals, [then] call another [group] fish or whatever,” not because all fish or all mammals are “the same,” but because “they share many similarities.” He says that what he is trying to do is, in essence, group the “spaces” that come out of his equations: “to put them in different groups, so that those in the same group will share similar properties.”

The roots of Birkar’s field can be traced back to the 19th century. By the early 20th century two-dimensional spaces were sorted out, while the third dimension was nailed between the 1970s and 90s. The difficulties multiply with dimensions four and up, which is where Birkar continues the work.

Is this really mathematical thinking for its own sake? Much of the impetus comes from other sub-disciplines, Birkar says, suggesting that his own work has “very deep connections” with number theory. Different branches inspire one another, and also devise new problem-solving tools for each other. But there are also “connections with other types of science,” most especially physics (whose close two-way connection with maths Marcus Chown explores on p69), and more applied fields like computer science and cryptography. There are practical applications, too. “When someone gets cash out of the ATM machine,” he says, “it’s likely algebraic geometry is being used.”

Safely brought back down to three-dimensional Earth, I ask Birkar about politics. This certainly impedes on his personal life. Although he lives in Cambridge with his wife and son, where he will soon join Trinity College, which has a long mathematical pedigree, he can no longer go back to Iran to see his family, and it’s extremely difficult for them to visit him.

But he remains sanguine about the broader picture. He finds the rise of the right “worrying, but it’s not the end of the world… look at the last 1,000 years and you see a lot of ups and downs.” And while politics thwarted science in centuries past, Birkar argues that this is less likely now, given that “young people know more and more about what’s going on in the world.” Brexit is certainly a bump in the road, “because there will be probably less money… less movement of people and so on.” But on a “100-year” view, he is confident that the march of understanding will continue.

He also insists on the connection between his discipline and the wider world of thought—and challenges the caricature of mathematicians as a breed apart, “crazy people who don’t care about anything but their equations.”

In truth, Birkar argues, “mathematics is part science, part art,” and often goes hand-in-hand with other interests. In his own case, these include “psychology, human history, natural history and music.” (He often does his brain-stretching work while listening to western classical or Kurdish music.) His hero is Alexander Grothendieck, the stateless father of algebraic geometry, who fled the Third Reich as a child, quit mathematical institutions in France because of their military funding, and went to Vietnam to give lectures on category theory in the forests surrounding Hanoi as a protest against American bombing.

Even so, Birkar says, some still think of an intellectual as “someone working in literature,” and “struggle to digest the idea that a mathematician can be a top thinker.” In Kurdish, however, it transpires that “Birkar” doesn’t just mean “mathematician”—it also means “thinker.”

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3/9/2019 19:30:00