This spring, not long after Caucher Birkar learned that he would be receiving the Fields Medal, the highest honor in mathematics, he shared a memory from his undergraduate years. Even by that time he had come a long way. Born and raised in a rural subsistence farming village in the Kurdish region of western Iran, Birkar had made his way to the University of Tehran, one of the pre-eminent universities in the country. There, at the math club, he recalled studying the pictures of Fields medalists lining the walls. “I looked at them and said to myself, ‘Will I ever meet one of these people?’ At that time in Iran, I couldn’t even know that I’d be able to go to the West.”

There was a lot about his future that Birkar couldn’t have predicted at that time: his flight from Iran, his request for political asylum, the push to rekindle a nearly abandoned field of mathematics. And today, at a ceremony in Rio de Janeiro, the honor of being selected as one of four winners of the Fields Medal, a prize conferred every four years by the International Mathematical Union on the most accomplished mathematicians in the world under the age of 40 at the beginning of the year in which the prize is awarded. “To go from the point that I didn’t imagine meeting these people to the point where someday I hold a medal myself — I just couldn’t imagine that this would come true,” said Birkar, who turned 40 in July.

As a mathematician, Birkar has helped bring order to the infinite variety of polynomial equations — those equations that consist of different variables raised to various powers. No two equations are exactly alike, but Birkar has helped reveal that many can be neatly categorized into a small number of families. In two papers published in 2016 he showed that an infinite number of different polynomials can be defined by a finite number of characteristics — a result which demonstrated that this bewildering array of seemingly unrelated algebraic equations shares something in common.

Birkar’s recent work on Fano varieties is one of his most influential mathematical results, but it sprang from an impulse that has governed him since his self-guided initiation into mathematics almost three decades ago: an impulse to create something new.

The First Revolution

When Birkar was very young, he liked to be near his mother, Sakina, while she baked bread. She remembers a scene from when Birkar was about three. “He was sitting next to me, trying to get very close to me, trying to let his feet touch my feet,” she said.

Birkar’s mother told me this story over Skype. He and I were sitting in the living room of his small house in Cambourne, a community 10 miles outside of Cambridge, England, that was built from scratch two decades ago. His mother was in the family’s house in Marivan, a Kurdish village in a mountainous region along the Iran-Iraq border where Birkar’s family has lived for generations. She wore a white headscarf and was sitting on a couch between Birkar’s oldest brother and his father, Majid.

Caucher Birkar was born in 1978, the third of six children. His family lived on a subsistence farm growing rice, wheat and vegetables. They also kept cows and the occasional horse. The family was self-sufficient in a way that helped shelter Birkar from the worst of the tumult that engulfed the region during his childhood — the Islamic Revolution of 1979 and the brutal eight-year war with Iraq that soon followed.

“Just after I was born there was a revolution and war, but the fact that we produced everything we needed meant we could survive,” he said.

Birkar’s father had attended school for a few years. His mother had no formal education at all. But Birkar and his siblings went to the village school. Around fifth grade, Birkar started to notice mathematics. “I felt something. Nothing serious, it was just a little feeling that I’m good in mathematics,” he said.

Early on, the main steward of Birkar’s mathematical interest was Haidar, his oldest brother, who introduced him to the basic concepts of calculus. “He was very curious about mathematical and physical questions,” Haidar said over Skype. “I remember he’d pick up my textbooks and try to solve the problems.”

Birkar remembers his brother teaching him something else as well: that knowledge can be exquisite.

“He was probably the only person I knew who followed his interests, not only to do good in school. It was much more than that, he really was interested in certain things,” Birkar said. “He would do things no one else would do in that environment.”

By high school Birkar had outrun his older brother’s mathematical knowledge and was left to learn the subject on his own. He took out books from the local library with titles like Men of Mathematics and What Is Mathematics. His family remembers him reading late into the night, often while listening to music — a habit he maintains today.

Even in his first brushes with professional mathematics, Birkar wanted to do more than just admire other people’s discoveries. “I read all these books and I had the feeling that just reading things is not enough. I also wanted to create my own stuff, to create something new,” he said.

As a high school student he started writing his own mathematical proofs. In college he began to submit them to mathematical journals. Later on, after he’d had some formal training, he realized that the proofs he’d found had been discovered long ago. “Maybe I didn’t prove anything significant, but just the experience, that kind of attitude, proved to be useful in the later stages of my education,” he said.

Birkar tested into the University of Tehran. In his final year as an undergraduate he traveled to England. While there, he sought political asylum in response to the “political problems in Iran for people like me,” he said, referring to his identity as a Kurd, a minority ethnic group that has often suffered state-sponsored repression. The British government placed him in Nottingham, a city in central England. During the year it took for the government to process his case, Birkar met faculty members at the University of Nottingham, where he enrolled after his asylum request was granted.