Obituary: Maryam Mirzakhani died on July 14th

IMAGINE a frictionless ball rolling around a billiard table. Next, work out, on variously shaped tables, which set of ricochets would merely repeat a pattern, and which would eventually cover the whole surface. Full answers are still elusive, but it is the sort of mathematical puzzle that outsiders can at least imagine.

By Maryam Mirzakhani’s standards, such problems were mundane. In her world, the billiard tables were abstract geometric objects which stretched and warped. The problems involved not just one table but a “moduli space”, of all possible such surfaces. Fans called her work on these mind-spinning abstractions the “theorem of the decade”.

Until the joy of maths claimed her, she wanted to be a novelist. Books cost next to nothing in the Iran of her childhood, and her earliest ambition was to read everything. Later, her maths had a literary tinge. She thrilled to the unfolding plot lines in the problems she studied—though unlike in literature, she said, they evolved like live characters. “Just as you start getting to know them, you look back and realise your first impression is mistaken.”

By her own account she was a “slow” mathematician, both in the time it took her to get started (her first teacher in Tehran thought she lacked aptitude) and in the way she approached problems: teasing out solutions by doodling for hours on vast sheets of paper. These would swathe the floor of their home, to the delight of her toddler, and to the amused bewilderment of her tidy-minded Czech husband. The point, she said, was not to write down all the details, but to stay connected with the problem. She also likened mathematical inquiry to being lost in a forest, gathering knowledge to come up with some new tricks, until you suddenly reach a hilltop and “see everything clearly”.

But she was quick on other fronts. Encouraged by her teachers and older brother, she soared through the Iranian education system. She was the first girl to represent the country in the mathematical Olympiad, winning gold medals in two successive years. Her beloved abstract surfaces can be described geometrically, with angles, lengths and areas, or algebraically, with equations. She was fluent in both: a mathematical polyglot. She found it “refreshing” to cross what she dismissed as the “imaginary” boundaries between different branches of the subject.

After Harvard and a stint at Princeton, she ended up at Stanford, winning the Fields medal—broadly the maths equivalent of a Nobel prize—in 2014, the first woman to do so since its inception in 1936. Her doctoral thesis alone was an academic earthquake, leading to papers published in the three most-admired mathematical journals. Of her great breakthroughs, perhaps the most easily explained involves hyperbolic surfaces: roughly, doughnuts with two or more holes, but where each point on the surface curves upwards, like a saddle. These exist, in theory, in infinite varieties. A big puzzle involves “geodesic” lines: the shortest distances between two surface points. Some may be infinitely long; others are “closed”, forming loops with no endpoints. A fascinating and tiny handful, known as “simple”, never cross themselves. Her thesis revealed a formula for how the number of simple closed geodesics of a given length rose as that length increased. Such work might seem abstruse to outsiders, but uses abound, from cosmology to cryptography.

She belied stereotypes. To Americans, she had to explain that in her native Iran (unlike Saudi Arabia) women’s education and careers were not just tolerated but encouraged: her girls’ high school was run by a national organisation responsible for hothousing young talent. She was not only the first woman to win the Fields medal, but the first Iranian, making her a celebrity there. Some media flinched piously from portraying her without a headscarf, a taboo which frayed after her death. Her marriage to a non-Muslim was not recognised, hampering family visits. Many also bemoaned her emigration, part of a debilitating brain drain. She moved to America for postgraduate study in 1999, a time when today’s anti-Muslim immigration policies were unimaginable.

Drawing a line

She quailed only before the limelight. She ignored a friend’s e-mail telling her of the Fields award, assuming it was a practical joke. In remission from the cancer that would eventually kill her, she worried that chemotherapy had left her too weak to attend the awards ceremony.

Men have roughly five in every six maths-heavy academic jobs in America, part of a wider puzzle that neither nature nor nurture fully explains. One reason may be that maths talent and female fertility flower in the same crucial years. She acknowledged the problem of discouragement, but resisted pressure to be a role model; other women were doing great things too, and anyway research mattered more. At conferences, female colleagues, working in pairs, helped her dodge media inquiries. While one distracted the journalist, the other let her ricochet to a more familiar plane of being.

^{Liu Xiaobo, the subject of the cover story in last week’s issue, died shortly after it went to press. The Chinese dissident’s obituary is at economist.com/liu}