Maryam Mirzakhani was the remarkable mathematician who, in 2014, became only the first woman—and first Iranian—to receive the Fields Medal, often regarded as the Nobel Prize of mathematics. But more than just a giant milestone, Mirzakhani’s journey was unusual, and her particular mathematical vision was visually striking, and became the subject of a documentary Secrets of the Surface that chronicles both her life and ideas. It is a story that’s both uplifting and touching, as Mirzakhani was struck down by cancer at the prime of her career in 2017, only 40 years old.

Beginnings

Born in 1977 at a difficult time in an Iran which was about to enter into war with Iraq (from 1980 to 1988). Maryam nevertheless had a family who supported her curiosity in science.

I grew up in a family with three siblings. My parents were always very supportive and encouraging. It was important for them that we have meaningful and satisfying professions, but they didn’t care as much about success and achievement. In many ways, it was a great environment for me, though these were hard times during the Iran-Iraq war. My older brother was the person who got me interested in science in general. He used to tell me what he learned in school.

From a young age, Maryam had a great imagination and, when she was eight years old, she would make up stories about a girl who achieved great things, such as becoming mayor or travelling the world. About the time the war ended, she completed her studies at elementary school and sat an examination for the Farzanegan middle school for girls in Tehran. This school, administered by Iran’s National Organization for Development of Exceptional Talents, aimed to educate the brightest pupils. Maryam did not do particularly well in this first year at Farzanegan middle school, and she was being told by her teacher that she was not particularly talented in that subject. This was a blow to Maryam’s confidence and for a while, she lost interest in mathematics and toyed with the idea of becoming a novelist. As it turned out, she had a different and more encouraging mathematics teacher in her second year, who made revived her interest and confidence in the subject.

Success followed quickly as she and a friend made it to the Iranian Mathematical Olympiad team in 1994. The international competition was held that year in Hong Kong and Mirzakhani scored 41 out of 42 and was awarded a gold medal, while her friend was awarded a silver medal. Again, in following year, Maryam joined the Iranian Mathematical Olympiad team, this time to compete in Toronto, Canada. Mirzakhani scored 42 out of 42 and was once again, won a gold medal.

Graduate Whizz Kid

Shortly after her second Olympiad medal, Maryam began her study of mathematics at Sharif University of Technology, the leading university in Iran for the physical sciences. She did well and even published papers while as an undergraduate. After obtaining her degree from Sharif University, Maryam went to the United States where she attended graduate school at Harvard University. There she started to attend seminars given by Curtis McMullen, a Fields Medallist. McMullen became her doctoral advisor. He said of her:

She had a sort of daring imagination. She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, “Is it right?” I was always very flattered that she thought I would know.

Her work at Harvard involved studying closed curves (known as closed geodesics) on a hyperbolic surface. Roughly speaking, a geodesic is a path between two points on the surface, whose length cannot be shortened by deforming it. A closed geodesic a geodesic that starts and ends at the same point, the simplest example of which is… you guessed it: a circle. It was the peculiar properties of closed geodesics that Maryam was primarily interested in (I will go no further on the technical but see Further Study at the end of this blog).

Harvard University awarded Maryam a Merit fellowship in 2003, and a doctorate in the following year for her 130-page thesis Simple Geodesics on Hyperbolic Surfaces and Volume of the Moduli Space of Curves. For this outstanding thesis, she was also awarded the Leonard M and Eleanor B Blumenthal Award for the Advancement of Research in Pure Mathematics in 2009.

At Princeton

Maryam turned down Harvard’s offer of a junior fellowship in 2004, at Harvard, since something better awaited her. True enough, in that year she was awarded a Clay Research Fellowship and was appointed as an Assistant Professor of Mathematics at Princeton University. She said, quite modestly, that:

The Clay fellowship] was a great opportunity for me; I spent most of my time at Princeton which was a great experience. The Clay Fellowship gave me the freedom to think about harder problems, travel freely, and talk to other mathematicians. I am a slow thinker, and have to spend a lot of time before I can clean up my ideas and make progress. So I really appreciate that I didn’t have to write up my work in a rush.

At Stanford

Her years at Princeton led to many published papers, mostly on closed geodesic spaces. When the Clay Fellowship ended in 2008, she left Princeton and was appointed as Professor of Mathematics at Stanford University. There, she had met Jan Vondrák, a theoretical computer scientist and applied mathematician who was a postdoctoral teaching fellow at Princeton University from 2006 to 2009. She married Vondrák and they had a daughter Anahita born in 2011.

At Stanford, Maryam began working with two other mathematicians, Alex Eskin (University of Chicago) and Amir Mohammadi (University of Texas at Austin) on one of the hardest problems in their area, which roughly speaking involved determining whether “complex geodesics and their closures in moduli spaces”, which are known to be highly erratic, can be “tamed”.  The problem was so hard that at many points along the way, her collaborators felt like giving up. But not Maryam. “I don’t give up easily”, she recalled in an interview on this work. With her persistence, they soldiered on and complete the proofs. The paper was published in 2015 in the prestigious journal, Annals of Mathematics. Eskin had collaborated with Maryam before, resulting a paper published in 2013 on the dynamics and geometry of moduli spaces where they presented a proof of their famous “magic wand theorem.” That result allows for the ability to calculate the diffusion rates of a wide range of systems (a trivial example being the motions of billiard balls) without having to compute their individual trajectories.

For her outstanding contributions to these areas of mathematics, Maryam was honoured in 2014 with a Fields Medal presented to her by the International Mathematical Union. The citation states that the award was: ” ... for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.”

A Star is Struck

Even before she was awarded the Fields Medal in 2014, Maryam had been diagnosed with breast cancer. But she continued working on mathematics, producing not only results of great significance but developing tools along the way that will be used by researchers in the field as they continue to push forward. Unfortunately, by 2017, the cancer spread to her liver and bones and in July that year, she died, robbing mathematics of one of its brightest stars at the age of 40.

Stanford President Marc Tessier-Lavigne said this of Maryam following her death:

Maryam is gone far too soon, but her impact will live on for the thousands of women she inspired to pursue math and science. Maryam was a brilliant mathematical theorist, and also a humble person who accepted honours only with the hope that it might encourage others to follow her path. Her contributions as both a scholar and a role model are significant and enduring, and she will be dearly missed here at Stanford and around the world.

Further study

The mathematics of Maryam Mirzakhani: https://chalkdustmagazine.com/features/mathematics-maryam-mirzakhani/