Maryam Mirzakhani is worldwide famous for having been the first woman, as well as the first person from Iran, to be awarded the Fields Medal at the International Congress of Mathematicians in Seoul in 2014. A beautiful video, by the Simons Foundation and the International Mathematical Union, where she is talking about her work, can be found here.

EWM received with heavy hearts the notice of her untimely death on July 14, 2017. Most famous for being the first female mathematician to win the Fields medal,  “For her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces”,  since 2008 she was a full Professor at Stanford University. By all accounts a modest and generous person, she will be sorely missed by the mathematical community. We at EWM offer our sincerest condolences to her husband and daughter, and to all her family, friends and collaborators.

More information on her life and career can be found  following the links  on Quantamagazine, at Stanford University, on Terry Tao’s blog, in The Newyorker or in The guardian, as well as many many other places.

Celebrating Maryam Mirzakhani

Ami Radunskaya (President of the Association for Women in Mathematics (USA) and Lily Khadjavi

In July 2014, Maryam Mirzakhani made history as the first woman to win a Fields Medal, the most prestigious award in mathematics. Ruth Charney, then President of the Association for Women in Mathematics (AWM), remembered her feelings as she sat in the audience at the ceremony in Seoul, Korea: “At the end of the ceremony, all the Fields medalists, as well as the people presenting the awards were lined up on stage. In the centre of the stage were three women: Park Geun-hye, the first female President of South Korea, Ingrid Daubechies, the first female President of the International Mathematics Union, and Maryam Mirzakhani, the first female Fields Medalist. I tend to be rather blasé about such things, but this one sent chills up my spine!”

Professor Mirzakhani’s achievements had repercussions around the globe. She became a celebrity in her birth-country, Iran. Hassan Rouhani, the President of Iran, tweeted two photographs of Maryam, one with, and one without a hijab: “Congrats to #MaryamMirzakhani on becoming the first ever woman to win the  #FieldsMedal, making us Iranians very proud”.[1]  When something so momentous and joyous occurs, we all feel a part of it: we were all very proud!  When the news of Maryam’s death on July 14, 2017, reached us, we were devastated.

Kristin Lauter, past-President of the AWM, was sitting next to Ruth Charney at the ceremony in Seoul: “Many of us were cheering loudly from the audience and there was a sense of a real breakthrough and potential for change, and certainly an outpouring of emotion. The same outpouring of emotion has been palpable at the terribly sad news of her death these last few days.”  In fact, Maryam had been diagnosed and treated for cancer before the ceremony in Seoul. Although she was able to attend the ceremony with her husband, Jan Vondrák, and daughter, Anahita, she left the event early, before delivering her lecture.

The Fields Medal was not the first of Mirzakhani’s accolades. She was a Clay Mathematics Research Fellow, she won the Clay Research Award and the AMS Blumenthal award, and was elected a member of the French Academy of Sciences, the National Academy of Sciences and the American Academy of Arts and Sciences. She was a Professor in the Mathematics Department at Stanford University, where she inspired students and colleagues alike.

Maryam Mirzakhani was born in Tehran, Iran, in 1977. In a film made by the Simons Foundation and the International Mathematical Union[2], she describes herself as the “lucky generation” because, by the time she was a teenager, the Iran-Iraq war was over and opportunities were open to young people. She tells us that, as a girl, she was more excited about reading novels than doing mathematics, and thought she would be a writer. She was first drawn to mathematics because of its challenge, but then found that it was “really nice”, and that she enjoyed doing it. She was the first student in her all-girls high-school in Tehran to participate in the International Math Olympiad (IMO), where she became the first Iranian to win two gold medals, achieving a perfect score in 1995. By the time she was an undergraduate at the Sharif University of Technology in Tehran, Maryam must have begun to identify as a mathematician: she published three papers as an undergraduate, one of which is cited regularly by combinatorists.[3]  She went on to receive a PhD from Harvard University, working under the supervision of Curtis McMullen. Her 2004 thesis quickly brought her recognition as a creative, even visionary mathematician. In this work, Mirzakhani solved two longstanding open problems, connecting the two together in a thesis that experts in the field describe as “truly spectacular”.[4]

Maryam Mirzakhani attacked tough problems that required combining ideas from many mathematical areas: analysis, differential geometry, topology, dynamical systems and ergodic theory. In graduate school, she became interested in the geometry of the moduli space of hyperbolic surfaces. In her PhD thesis, she calculated their volume, shedding insight on the size of the set of possible geometries on these surfaces. This work had far-reaching implications; for example, it led to a new proof of Witten’s famous conjecture about quantum gravity. In her 2013 paper “Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus”[5], she relates the behaviour of lengths of simple closed geodesics on a hyperbolic surface to properties of the moduli space of such surfaces. In March of 2013, Maryam talked about this work in her plenary lecture at the second Association for Women in Mathematics (AWM) Research Symposium in Santa Clara, California, and at the January Joint Mathematics Meetings, she was awarded the Ruth Lyttle Satter Prize by the American Mathematical Society for “her deep contributions to the theory of moduli spaces of Riemann surfaces.”  In 2013, two papers appeared on the arxiv, co-authored by Alex Eskin and Amir Mohammadi, that explored geodesics on flat surfaces, rather than hyperbolic ones. They looked at translation surfaces, a generalization of unfoldings of a rational polygon, and studied orbits that are preserved under shearing, stretching and compressing (GL(2, R) actions)[6]. In their “breakthrough” theorem, the authors showed, surprisingly, that these orbits are always “nice” sets: manifolds that are described locally by linear equations with real coefficients. This theorem has been called the “magic wand” theorem because it has many applications in physics and mathematics. It has been predicted that many more applications for Mirzakhani’s theorems and techniques will be found in the years to come. She has given us an example of the passionate, private genius; she liked to tackle the more difficult problems, without fanfare. As a young colleague writes: “In defiance of the (sound) advice given to most young researchers, Mirzakhani declared that she wished to ‘avoid the low-hanging fruit,’ instead finding ambitious problems she could ponder for years at a time. Her unflinching eagerness to take on the most daunting problems, and her great success, was all the more notable given her unfailingly good-natured and humble personality.”[7]

When the 2014 Fields Medalists were announced, Tim Gowers, a Fields medalist at Cambridge, wrote: “I am thrilled that this day has finally come. Although women have contributed to mathematics at the highest level for a long time, this fact has not been visible to the general public. I hope that the existence of a female Fields medalist, who will surely be the first of many, will put to bed many myths about women and mathematics, and encourage more young women to think of mathematical research as a possible career”.

Yes, certainly, there exist many myths about women and mathematics. One such myth is that women are under-represented in mathematics competitions such as the IMO or the Putnam because they lack ability or talent. In fact, a cross-cultural study by Andreescu and colleagues[8] shows that participation and performance by women in these contests is highly dependent on their country of origin. The authors conclude that some Eastern European and Asian countries consistently produce young women with profound ability in mathematical problem solving (as evidenced by success in international competitions), while most other countries, including the USA, do not. Interestingly, children of immigrants to the USA and Canada from some of the countries that do well in the IMO, regardless of gender, are overrepresented among students identified as very gifted in mathematics; USA-born girls from all other ethnic/racial backgrounds, including white, are highly underrepresented. The take-away message is that young women with talent and interest in mathematics exist, but they are not always recognized due to socio-economic and cultural factors. In another study, Kane and Mertz consider several other myths, such as the greater male variability hypothesis or the theory that quantitative fields are not in line with a female disposition, which is drawn towards more nurturing fields such as teaching small children or nursing. Looking at data from many countries, educational systems and cultures, these authors conclude that “gender equity and other sociocultural factors, not national income, school type, or religion per se, are the primary determinants of mathematics performance at all levels for both boys and girls.”[9]  Signs of progress towards gender equity are apparent: in the 1960’s, only 6% of mathematics PhDs in the US were given to women, in 1990 that percentage had risen to 18%, and by 2015 it was 31%.[10] The fraction of female Professors in mathematics departments at PhD granting institutions in the US is now up to 11%. While progress is evident, there is still a ways to go.

Maryam Mirzakhani wrote the following in 2013 in response to being awarded the Satter prize:[11]

“… in my opinion, the situation of women in math is far from ideal. The social barriers for girls who are interested in mathematical sciences might not be lower now than they were when I grew up. And balancing career and family remains a big challenge. It makes most women face difficult decisions which usually compromise their work. However, there has been a lot of progress over the years, and I am sure this trend will continue.”

Let us celebrate this remarkable woman who courageously broke through the barriers to the pinnacle of achievement in mathematics. One way to celebrate Maryam Mirzakhani’s life is to encourage and mentor young women pursuing mathematics. We hope that all of us in the mathematics community will support, protect, and uplift one other.

[1] pic.twitter.com/oVL98NRdVF

[2] https://www.youtube.com/watch?v=qNuh4uta8oQ&feature=youtu.be

[3] Ed Dunne, https://blogs.ams.org/beyondreviews/2017/07/17/maryam-mirzakhani/

[4] Benson Farb, quoted in the Stanford News http://news.stanford.edu/, July 15, 2017

[5] J. Differential Geometry 94 (2013)

[6] For more details, see the expository article by Alex Wright “From rational billiards to dynamics on moduli spaces”, Bulletin of the American Mathematical Society, 53(1), 2016

[7] Alex Wright, “Maryam Mirzakhani 1977-1027”, Science 25, Aug 2017

[8] Andreescu, Gallian, Kane and Mertz “Cross-Cultural Analysis of Students with Exceptional Talent in Mathematical Problem Solving”, Notices of the AMS  55, No. 10 (2008)

[9] J. M. Kane and J. E. Mertz “Debunking Myths about Gender and Mathematics Performance”, Notices of the AMS 59 No. 1 (2012)

[10] 2015 Annual Survey of the Mathematical Sciences

[11] Notices of the AMS, 60 No. 4 (2013)