Secrets of the Surface introduces Maryam Mirzakhani to a wider public, telling the story of her life as recounted by friends and colleagues, looking at her mathematical work through the words of collaborators, and gauging her impact on future generations.

The film opens with a scene filmed at the Kherad Institute in Tehran in March 2019. A group of high school girls are hard at work on a blackboard crowded with geometric shapes and equations. One of them, Delara Jandaghi, explains that “there is a very good feeling behind solving the problem. You know, when you try hard and you can’t find the solution, and suddenly… ‘Ah! That’s it.’ And I think Maryam Mirzakhani could show this passion to everyone.”

Maryam Mirzakhani’s mathematics is often described by her colleagues as being deeply beautiful. In the film we raise the possibility that her approach to shapes and surfaces is somehow linked to the spectacularly beautiful patterns and motifs in Iranian art. As we look at elaborate tiles and structures around the Jameh Mosque in Isfahan, mathematician Ali Rejali tells us there is a mathematical basis for the patterns created by craftsmen and architects a thousand years ago.

The stories of Maryam’s childhood in Iran are recounted by recordings of her own voice from a 2014 film produced by the Simons Foundation, and interviews with her husband Jan Vondrak and classmates. We learn about her generous, caring nature, and her initial aversion to math. She wanted to be a writer.

According to her lifelong friend, mathematician Roya Beheshti Zavareh, it was not until they were in seventh grade that Maryam’s interest in mathematics took off. Teachers and classmates were soon struck by her speed at solving problems, and a prodigious ability to concentrate. Yasaman Farzan, a physicist and researcher at the Institute for Research in Fundamental Sciences (IPM) in Iran, remembers that Maryam “made a point to solve any problem in hand several different ways. It was even a joke that Maryam has solved this problem four different ways.”

Maryam and Roya were fortunate to attend one of the elite Farzanegan schools. Under the guidance of inspiring teachers and an enlightened principal, Kheirieh Beigom Haerizadeh, the girls advanced at their own pace. When Mrs. Haerizadeh heard about a summer program for high school sophomores at Sharif Technical University, she persuaded the administrators to admit the two girls. “We participated in that program when we finished ninth grade. That was the first time that we were mixed with boys,” Beheshti says.

The mathematically talented pair were soon the first girls to be part of Iran’s Math Olympiad team, winning medals in Hong Kong in 1994 and Toronto in 1995. Their success was widely publicized in the media and Mirzakhani became an instant celebrity in Iran. By the time Maryam and Roya were enrolled at Sharif University as undergraduates, Maryam had co-authored a paper with Professor Ebadollah Mahmoodian, and published another on her own.

The mathematical program at Sharif was designed to allow the most talented students to explore at their own pace. Yahya Tabesh was department head. He had known Maryam and Roya from their days on the Iranian Math Olympiad team, and noted Mirzakhani’s abilities at engaging with problems. He is proud of the environment at Sharif. “We made a center of excellence,” he says. “Let them grow up by their own pace. Not control them, not push them.”

Siavash Shahshahani, another Sharif professor, describes the math program as an ideal place to foster the talents of students of Mirzakhani’s generation. “We had a curriculum that was very open. Students could move up easily. They took graduate courses while they were undergraduates, as Maryam did. We gave them more freedom than other universities,” he says.

In 1999, Mirzakhani and five of her classmates graduating from Sharif were admitted to six different top schools in the United States. Maryam chose Harvard, and with an active Iranian student-community support network, she and Roya quickly assimilated. Her advisor, Curtis McMullen, noticed her during her second year in his seminar. He gained an appreciation for her powers of concentration and the depth of her understanding when he asked her to work on an area called the McShane identity.

Mirzakhani moved from this problem to her thesis, which consisted of three proofs that attracted wide attention. The connections she made between seemingly unrelated fields opened up new areas of research. She was famous before her thesis was even published.

During the year before she graduated, Mirzakhani met Jan Vondrak, a computer science student from the Czech Republic. They married just as Maryam started a prestigious job at Princeton. Before long they both landed positions at Stanford, where Maryam’s teaching load was reduced so she could pursue research. She had already met Alex Eskin at Princeton, and the two started a collaboration that would lead to their groundbreaking result years later.

The two started working on Ratner’s theorem for moduli space. The exposition of Mirzakhani’s work with Eskin and Mohammadi for the film was co-written by mathematicians Erica Klarreich and Jayadev Athreya, who also scripted the animations created by Andrea Hale. Erica Klarreich prepared a layman’s explanation of the problem tackled by Mirzakhani this way. “If you hit a ball on a billiard table, what are the possible ways that it could travel? Maybe it’s going to roll into a corner, or maybe it’s going to bounce around in a cycle, or maybe it’s going to follow a more complicated trajectory. And here we might be talking about our usual rectangular billiard table, or we might take a more complicated shape like a triangle or a hexagon. And for most of these different-shaped billiard tables, it’s really hard to understand what the different possible trajectories are.”

Mirzakhani and Eskin worked for years, their progress often stymied by some minute detail, before they finally came up with a proof. Their path to a solution is described by Eskin and mathematician Anton Zorich as a harrowing climb to a summit that nobody knew really existed or not.

The proof by Mirzakhani and Eskin was called the “magic wand theorem” by mathematician Anton Zorich, because it has since been applied to so many other unrelated problems with success. In 2019, Alex Eskin was awarded the Breakthrough Prize in mathematics for his work with Mirzakhani on the problem.

While working with Eskin, Maryam and Jan had their daughter Anahita, and then Maryam was diagnosed with breast cancer. The cancer was successfully treated. During her recovery, Mirzakhani learned that she would be awarded the Fields Medal at the International Congress of Mathematics (ICM), to be held in Seoul, South Korea, in 2014.

Publicity shy by nature, Mirzakhani accepted the award while trying to avoid the limelight as much as possible. Still weak from her medical treatments, but also wary of the responsibilities thrust upon her as the “first woman” and the “first Iranian” to receive mathematics’ highest honor, she navigated the waters of celebrity as best she could.

Ingrid Daubechies, president of the International Mathematical Union (IMU) at the time, organized a protective shield for Mirzakhani at the Seoul meeting, while others parried the flurry of requests for interviews and appearances. Maryam returned to her work, and to spending as much time as possible with her daughter Anahita.

In 2016, the cancer returned, and Maryam succumbed to it the following year. The accolades and tributes in Iran were monumental. Universities, high schools, and math centers renamed programs and buildings for Maryam Mirzakhani. The film covers just a few of the many Iranian gestures in her honor, including a specially issued postage stamp.

All of the testaments to Mirzakhani’s mathematical legacy are too numerous to include, but the film touches on one conference held on her birthday at MSRI in Berkeley, and another commemorative symposium organized at the Fields Institute in Toronto. As Jayadev Athreya points out near the film’s end, mathematicians “expect to see her work popping up in unexpected places. It’s that kind of work. It’s going to show up in places that I can’t foresee right now.”