Maryam Mirzakhani is known for her work on moduli spaces of Riemann surfaces. Some of her most cited work looks at the moduli space of a genus Riemann surface with geodesic boundary components. In two of her papers, she computes the volume of these moduli spaces, with respect to the Weil-Petersson metric (see below). In another, she provides a means for counting the number of simple closed geodesics of length at most . Mirzakhani is also known for her work on billiards (see the review of her paper with Eskin and Mohammadi below), a subject closely related to moduli space questions. Teichmüller theory and the geometry of moduli spaces are famously deep subjects. Making progress requires mastering large areas of analysis, dynamical systems, differential geometry, algebraic geometry, and topology. I can only appreciate Mirzakhani’s work superficially, as I have not mastered those subjects. Instead, some reviews of her work are reproduced below.
1. One of her biggest projects, joint work with Eskin studying the action of SL(2,R) on moduli space, is not published yet. So there is no item in MathSciNet for it. You can read the latest version on the arXiv.
3. I started writing this post back in March, when I was highlighting the work of some remarkable mathematicians. It was delayed because describing her work is not simple: it is substantial and uses deep and difficult tools from several areas. Her papers are quite well written, with accessible introductions. However, the genius is in the details, which require real commitment to understand. The video produced for the ICM where she won her Fields Medal allows her to present something of her work. Amie Wilkinson describes Mirzakhani’s working style in this article in the NY Times. In a recent blog post, Terry Tao comments on how Mirzakhani was able to see disparate mathematical results through the lens of the mathematics she was developing herself.
4. Thank you to Tom Ward who spotted an inequality that was reversed in the original version of this post.
I am totally stunned to learn that Maryam Mirzakhani died today yesterday, aged 40, after a severe recurrence of the cancer she had been fighting for several years. I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.
My first encounter with Maryam was in 2010, when I was giving some lectures at Stanford – one on Perelman’s proof of the Poincare conjecture, and another on random matrix theory. I remember a young woman sitting in the front who asked perceptive questions at the end of both talks; it was only afterwards that I learned that it was Mirzakhani. (I really wish I could remember exactly what the questions were, but I vaguely recall that she managed to put a nice dynamical systems interpretation on both of the topics of my talks.)
After she won the Fields medal in 2014 (as I posted about previously on this blog), we corresponded for a while. The Fields medal is of course one of the highest honours one can receive in mathematics, and it clearly advances one’s career enormously; but it also comes with a huge initial burst of publicity, a marked increase in the number of responsibilities to the field one is requested to take on, and a strong expectation to serve as a public role model for mathematicians. As the first female recipient of the medal, and also the first to come from Iran, Maryam was experiencing these pressures to a far greater extent than previous medallists, while also raising a small daughter and fighting off cancer. I gave her what advice I could on these matters (mostly that it was acceptable – and in fact necessary – to say “no” to the vast majority of requests one receives).
Given all this, it is remarkable how productive she still was mathematically in the last few years. Perhaps her greatest recent achievement has been her “magic wand” theorem with Alex Eskin, which is basically the analogue of the famous measure classification and orbit closure theorems of Marina Ratner, in the context of moduli spaces instead of unipotent flows on homogeneous spaces. (I discussed Ratner’s theorems in this previous post. By an unhappy coincidence, Ratner also passed away this month, aged 78.) Ratner’s theorems are fundamentally important to any problem to which a homogeneous dynamical system can be associated (for instance, a special case of that theorem shows up in my work with Ben Green and Tamar Ziegler on the inverse conjecture for the Gowers norms, and on linear equations in primes), as it gives a good description of the equidistribution of any orbit of that system (if it is unipotently generated); and it seems the Eskin-Mirzakhani result will play a similar role in problems associated instead to moduli spaces. The remarkable proof of this result – which now stands at over 200 pages, after three years of revision and updating – uses almost all of the latest techniques that had been developed for homogeneous dynamics, and ingeniously adapts them to the more difficult setting of moduli spaces, in a manner that had not been dreamed of being possible only a few years earlier.
Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers. Today was a huge loss for Maryam’s family and friends, as well as for mathematics.
[EDIT, Jul 16: New York times obituary here.]
[EDIT, Jul 18: New Yorker memorial here.]
New York Times:
Maryam Mirzakhani, an Iranian mathematician who was the only woman ever to win a Fields Medal, the most prestigious honor in mathematics, died on Friday. She was 40.
The cause was breast cancer, said Stanford University, where she was a professor. The university did not say where she died.
Her death is “a big loss and shock to the mathematical community worldwide,” said Peter C. Sarnak, a mathematician at Princeton University and the Institute for Advanced Study.
The Fields Medal, established in 1936, is often described as the Nobel Prizeof mathematics. But unlike the Nobels, the Fields are bestowed only on people aged 40 or younger, not just to honor their accomplishments but also to predict future mathematical triumphs. The Fields are awarded every four years, with up to four mathematicians chosen at a time.
“She was in the midst of doing fantastic work,” Dr. Sarnak said. “Not only did she solve many problems; in solving problems, she developed tools that are now the bread and butter of people working in the field.”
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.”
and at last a video the although it is not new! but I love it.