Dr. Mirzakhani was one of four Fields winners in 2014, at the International Congress of Mathematicians in South Korea. Until then, all 52 recipients had been men. She was also the only Iranian ever to win the award.
President Hassan Rouhani of Iran released a statement expressing “great grief and sorrow.”
He wrote, “The unparalleled excellence of the creative scientist and humble person that echoed Iran’s name in scientific circles around the world was a turning point in introducing Iranian women and youth on their way to conquer the summits of pride and various international stages.”
Dr. Mirzakhani’s mathematics looked at the interplay of dynamics and geometry, in some ways a more complicated version of billiards, with balls bouncing from one side to another of a rectangular billiards table eternally.
A ball’s path can sometimes be a repeating pattern. A simple example is a ball that hits a side at a right angle. It would then bounce back and forth in a line forever, never moving to any other part of the table.
But if a ball bounced at an angle, its trajectory would be more intricate, often covering the entire table.
“You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union to profile the 2014 Fields winners. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.”
In work with Alex Eskin of the University of Chicago, Dr. Mirzakhani examined billiards tables of more complicated shapes, and in fact considered the dynamics of balls bouncing around all possible tables that fit certain criteria.
It was a challenging problem that had been attacked by many prominent mathematicians. That included Curtis T. McMullen, her thesis adviser at Harvard and also a Fields medalist, who had solved a special case. But no one had a good idea of the path toward a more encompassing solution.
Amie Wilkinson, a mathematics professor at the University of Chicago, recalled sitting in on a meeting with Dr. Mirzakhani and Dr. Eskin. Whereas Dr. Eskin tended to be pessimistic, seeing all the potential pitfalls that could scuttle a proof, Dr. Mirzakhani was the opposite.