All Thinkers

Maryam Mirzakhani

Maryam Mirzakhani was an Iranian mathematician. She was the first woman ever to win the Fields Medal, the highest prize in mathematics. She was born in 1977 in Tehran, the capital of Iran. She grew up during the Iran-Iraq war of the 1980s, a difficult time for the country. Her parents encouraged her education despite the surrounding chaos. She was not interested in mathematics as a young child. She wanted to be a writer. She read novels constantly and dreamed of becoming a novelist. Her interest in maths grew slowly through middle school. By high school she was attending a special school for gifted girls in Tehran. She and her best friend Roya Beheshti became famous for being the first Iranian girls to win medals at the International Mathematical Olympiad. Mirzakhani won gold medals in 1994 and 1995, with a perfect score the second year. She studied mathematics at Sharif University in Tehran. In 1999 she went to the United States for graduate school at Harvard. She was supervised by Curtis McMullen, a Fields Medallist himself. Her doctoral work was already remarkable. She found new ways to count certain kinds of curves on curved surfaces. She continued at Princeton and then at Stanford as a professor. In 2014, aged 37, she became the first woman to win the Fields Medal. The medal is awarded only every four years and only to mathematicians under 40. The same year she was diagnosed with breast cancer. The cancer eventually spread to her bones and liver. She died in 2017, aged just 40. She left behind her husband and her young daughter Anahita.

Origin
Iran (later United States)
Lifespan
1977 - 2017
Era
Modern / 21st Century
Subjects
Mathematics Geometry Iranian Culture 21st Century Women In Science
Skills
Scientific Thinking Problem Solving Creative Expression Critical Thinking Research Skills
Why They Matter

Maryam Mirzakhani matters for three reasons. First, she did extraordinary mathematics. She worked on the geometry of curved surfaces, especially what mathematicians call moduli spaces. These are spaces of all possible shapes of a certain kind. Her work connected several different areas of mathematics that had seemed unrelated. Other top mathematicians used her results in their own work. Her thesis alone was published as three separate major papers in the leading mathematical journals.

Second, she was the first woman to win the Fields Medal. The medal had been given since 1936. For 78 years, every winner had been male. Mathematics had a long tradition of being treated as a male field. Mirzakhani's win in 2014 was a clear sign that this was wrong and was changing. Her win mattered to women mathematicians around the world. It also mattered to many girls who had never seen a major woman mathematician honoured at the highest level.

Third, she became a powerful symbol in Iran and around the world. She had been educated entirely in Iran until she went to graduate school. She showed what Iranian education could produce when given the chance. In a difficult political moment, she was a quiet bridge between Iran and the West. After her death, the Iranian government broke its own dress-code rules to allow newspapers to print her photograph without a hijab. Her funeral was attended by huge crowds. She remains an inspiration to young Iranians, especially Iranian girls.

Key Ideas
1
The Fields Medal
2
She Wanted to Be a Writer
3
Curved Surfaces
Key Quotations
"I don't think that everyone should become a mathematician, but I do believe that many students don't give mathematics a real chance."
— Maryam Mirzakhani, interview with the Clay Mathematics Institute, 2008
Mirzakhani said this in a 2008 interview, after she had become an established mathematician but before her Fields Medal. She was thoughtful about how mathematics was taught. She did not think everyone needed to study advanced maths. But she felt that many students gave up on maths too early. They decided they were not 'maths people' before they had really tried. She wanted them to know that maths could be enjoyable and beautiful, even for people who found it hard at first. She herself had not been an obvious maths child. She had been a reader of novels. The shift came slowly. For students, this line is encouraging. Maths is not just for people who 'get it' instantly. It rewards patience and effort. Many students give up too soon. Mirzakhani's own life is a counterexample to the idea that you have to be a born mathematician.
"The beauty of mathematics only shows itself to more patient followers."
— Maryam Mirzakhani, quoted in obituaries and tributes, 2017
This line summarises Mirzakhani's view of mathematical work. The beauty of maths, she said, does not appear quickly. It rewards patience. People who study maths for a few weeks may feel only confusion. People who stay with it for years start to see deep structures. The patterns that connect different areas. The unexpected connections between problems that seemed separate. Mirzakhani was famously patient. She would think about a problem for years. She drew slowly. She wrote slowly. Her best results came from long quiet effort. The quotation captures her attitude. For students, the line is useful. Most school subjects require patience to be enjoyed. Music, sports, languages, art, and writing all reward people who stick with them. Mathematics is the same. The first few weeks may be hard. The reward comes later. Mirzakhani thought this was true and lived as if it were true.
Using This Thinker in the Classroom
Scientific Thinking When introducing students to modern mathematics
How to introduce
Tell students about Maryam Mirzakhani. The first woman ever to win the Fields Medal, the highest prize in mathematics. She worked on the geometry of curved surfaces. She drew big pictures on paper to help her think. She died young of cancer at 40. Show students a photograph. Tell them she did not start out wanting to be a mathematician. She wanted to be a writer. The shift came slowly through middle and high school. For students just meeting maths as a serious field of study, her story is a powerful starting point. Maths is alive. Real people, including women and people from many countries, are doing important new work right now. Mirzakhani is one of the great recent examples.
Problem-Solving When teaching students about patience in difficult work
How to introduce
Read with students Mirzakhani's saying that the beauty of maths only reveals itself to patient followers. Discuss what this means. Mathematics is hard. The first weeks of a new topic often feel confusing. Many students give up at this stage. Mirzakhani thought this was a mistake. The reward comes later. Discuss with students whether this is true of other things they have learned. Music. Sport. Languages. Reading. Many skills feel impossible at first and become easier with time. The lesson is general. Patience pays. Mirzakhani lived this principle in mathematics. Students can apply it elsewhere.
Cultural Heritage and Identity When teaching students about science across cultures
How to introduce
Tell students that Mirzakhani was educated entirely in Iran until graduate school. She attended Iranian state schools. She studied at Sharif University in Tehran. She won her gold medals at the International Mathematical Olympiad while still at school in Iran. The world's first woman Fields Medallist was Iranian. Discuss with students what this tells us. Excellence in mathematics happens around the world. Wealthy Western countries do not have a monopoly on serious science. Iran has produced many great scientists, despite its political difficulties. So have many other countries that get less attention in Western media. Mirzakhani's example is a useful corrective to assumptions about which countries produce great minds.
Further Reading

For a first introduction, the 2020 documentary film Secrets of the Surface, directed by George Csicsery, follows Mirzakhani's life and work. It includes interviews with her family, colleagues, and former teachers. The film is suitable for general audiences and includes accessible explanations of her mathematical work. Quanta Magazine has published several excellent articles about her, freely available online. The Stanford University News Service produced detailed obituaries that explain her contributions in plain language.

Key Ideas
1
Counting Closed Curves
2
How She Worked
3
Iran and the World
Key Quotations
"I find discussing mathematics with colleagues of different backgrounds one of the most productive ways of making progress."
— Maryam Mirzakhani, interview with the Guardian, 2014
After winning the Fields Medal, Mirzakhani gave several interviews. She often spoke about how she actually worked. One thing she emphasised was conversation. She did her best work, she said, when talking with mathematicians from different backgrounds. People who worked on different problems could see her own problems with fresh eyes. They asked questions she would not have asked herself. They suggested connections she had missed. Mathematics is sometimes imagined as a solo activity, with the lone genius working in silence. Mirzakhani's actual experience was different. She thrived on collaboration. Many of her major results came from working with other mathematicians. For students, this is a useful corrective to the lone-genius image. Most great work, in any field, involves serious conversation with others. Even people who work mostly alone often have a small group of trusted colleagues they talk with regularly. Mirzakhani is one of many examples.
"I think it's rarely about being smart. It's about working hard."
— Reported in interviews about Mirzakhani by colleagues, c. 2014-2017
Mirzakhani often said versions of this. She did not think she was specially smart in some inborn way. She thought she worked hard. The phrasing was modest, perhaps too modest. She was clearly extraordinarily talented. Other mathematicians have described her as one of the most original minds they had ever met. But the modesty had a real point. Many students assume mathematicians are different from them, born with some special gift. Mirzakhani thought this assumption was wrong and harmful. It led students to give up too early. They thought maths was not for them because they did not 'get it' immediately. She wanted to push back. Hard work mattered. Persistence mattered. Many talented students could go further than they thought, if they kept going. For students, this attitude is helpful. It is also realistic. Mirzakhani had unusual talent and worked extremely hard. Both contributed. Pretending only one mattered misses the truth.
Using This Thinker in the Classroom
Creative Expression When teaching students about the creative side of mathematics
How to introduce
Tell students about Mirzakhani's working style. She drew large pictures on big sheets of paper. Sometimes she covered her floor with diagrams. Her daughter described her work as 'painting'. Discuss with students how this contradicts common images of maths as cold and symbolic. Real mathematical thinking is often visual, creative, and slow. Mirzakhani drew because drawing helped her see. She thought through pictures. Other mathematicians work different ways: some scribble equations, some take long walks, some discuss problems aloud. The diversity of working styles in maths matches the diversity of styles in art, music, or writing. Maths is more creative than students often imagine.
Critical Thinking When teaching students about whose contributions get recognised
How to introduce
Tell students that the Fields Medal had been awarded since 1936. From 1936 until 2014, every winner was male. Discuss with students what this might mean. There had been many great women mathematicians during those 78 years. None had won. Why? The reasons are complex. Some women left the field because of harassment or unwelcoming culture. Some had their work ignored or stolen. Some were not nominated by male colleagues. The pattern of under-recognition has been studied carefully. Mirzakhani's win in 2014 was not because no woman before her had been good enough. It was because the field finally honoured what had been there all along. The case is a useful study in how prizes both reward excellence and reflect the prejudices of those who give them.
Further Reading

For deeper reading, the special issue of the Notices of the American Mathematical Society dedicated to Mirzakhani after her death (volume 65, 2018) includes essays by leading mathematicians on her work and life. The ICM 2014 lecture by Curtis McMullen (her doctoral supervisor) introduces her work for mathematicians. Erica Klarreich's 2014 Quanta Magazine profile, written when she won the Fields Medal, gives a clear introduction.

Key Ideas
1
Moduli Spaces
2
Why So Few Women in Maths?
3
The Photograph After Her Death
Key Quotations
"I will be happy if it encourages young female scientists and mathematicians."
— Maryam Mirzakhani, statement after winning the Fields Medal, 2014
Mirzakhani was the first woman to win the Fields Medal. The historical significance of this was huge. She was characteristically modest about her own role but clear about what she hoped her win would mean for others. She said she would be happy if it encouraged young women in maths and science. The line is short and direct. It does not claim too much. She does not say her win will change everything. She just says she hopes it will encourage others. The hope was reasonable. After her win, applications from women to graduate maths programs rose. Other women won the Fields Medal in later years. The change is slow but real. For advanced students, this kind of careful hope is worth thinking about. Mirzakhani did not pretend her win solved sexism in maths. She also did not deny it mattered. She held the right balance: clear about the achievement, modest about her role, hopeful about the future.
"It is fun. It is like solving a puzzle or connecting the dots in a detective case. I felt that this was something I could do, and I wanted to pursue this path."
— Maryam Mirzakhani, interview with the Clay Mathematics Institute, 2008
When asked why she became a mathematician, Mirzakhani gave answers like this. Maths was fun. The problems were like puzzles or detective cases. The work was finding clues, following them, slowly building a picture. The phrasing is plain. It might apply to many fields. The point is that for her, maths offered the kind of intellectual play that some people get from puzzles, mysteries, or strategy games. The play was the reward. The deep theorems came from following her interest, not from chasing prizes. For advanced students, the line raises a useful question. What kind of work feels like play to you? The work that feels like play, rather than effort, is often where you can go deepest. Mirzakhani found maths felt this way. Other people find writing, science, art, sport, or many other fields feel this way. The feeling is a clue worth following.
Using This Thinker in the Classroom
Research Skills When teaching students about how mathematical research actually works
How to introduce
Walk students through what Mirzakhani actually did. She worked on counting closed curves on curved surfaces. She used techniques from several different mathematical fields. She developed new methods that connected previously unrelated areas. Her doctoral thesis became three major published papers, each a significant contribution. Discuss with advanced students how this differs from school maths. Research mathematics is not about answering exercises in a textbook. It is about finding new questions, developing tools to answer them, and connecting your work to a wider picture. The work often takes years. Most papers go through many revisions. Most ideas do not work. Mirzakhani's career shows the patience required. She published relatively few papers. Each one was carefully developed. The work was thorough rather than fast.
Critical Thinking When teaching students about brain drain and global talent
How to introduce
Discuss with advanced students Mirzakhani's path. She was educated in Iran. She moved to the United States for graduate work and stayed there for the rest of her career. The pattern is common. Many talented young people from poorer or politically difficult countries move to richer Western countries for better research conditions. Discuss what this means for both kinds of countries. The home country invests in education and loses the eventual benefit. The host country gains talented people without paying for their early training. Mirzakhani spoke warmly of her Iranian education throughout her life. She did not reject Iran. She lived in America because that is where she could do her work. The case is a useful starting point for thinking about international migration of skilled people. The pattern affects medicine, science, engineering, and many other fields.
Common Misconceptions
Common misconception

Mirzakhani won the Fields Medal because she was a woman.

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What to teach instead

She did not. The Fields Medal is awarded by a committee of leading mathematicians who evaluate the depth and importance of the work. Mirzakhani's research had been recognised as outstanding for years before the medal. She had already been promoted to a full professorship at Stanford, one of the top positions in the field. Her papers were heavily cited and used by other top mathematicians. The medal recognised mathematical achievement at the highest level. She was the first woman to win because she was the first woman whose work the committee recognised at this level. The barrier had been there for decades. Once it broke, several other women won major prizes in the years that followed. Suggesting she got the medal for any reason other than her work misunderstands both her and the prize.

Common misconception

Her work has no practical use.

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What to teach instead

Some of it does and some of it has not yet been applied. This is normal in pure mathematics. Many results that seemed purely abstract have found applications decades or centuries after they were first proved. Mirzakhani's work on moduli spaces and dynamical systems already connects to questions in physics, including string theory. Her counting methods have been used in other areas of mathematics. Even when the immediate practical use is unclear, pure mathematical work creates tools and methods that other researchers build on. Calling deep mathematical research useless because it has no obvious immediate application misunderstands how mathematics actually develops over time.

Common misconception

She was always going to be a mathematician.

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What to teach instead

She was not. As a child she wanted to be a writer. She read novels constantly. Mathematics was something she came to slowly through middle school and high school. She was not the kind of child who solved hard maths problems for fun before age ten. Her shift to maths happened gradually. Her best friend Roya Beheshti and her teachers helped her see that maths could be exciting. By age 17 she was winning gold at the International Mathematical Olympiad. The path was not predetermined. Many great mathematicians and scientists have had similar paths, finding their field gradually rather than knowing from childhood. Children who do not show early interest in maths can still become serious mathematicians, as Mirzakhani's life shows.

Common misconception

Only people with extraordinary natural talent can do real mathematics.

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What to teach instead

Real talent matters, but so does sustained hard work. Mirzakhani herself emphasised hard work over innate ability. She thought many students gave up on maths too early because they assumed they did not have the gift. She believed that many talented students could go much further with patience and effort. Her own experience supports this. She was a reader before she was a mathematician. The shift came through interest and persistence, not through some pre-existing label of being a 'maths kid'. Mathematics rewards people who keep working when problems seem hard. Pretending only special people can do it discourages many talented students who would benefit from staying with the subject.

Intellectual Connections
Develops
Emmy Noether
Noether, the great early 20th-century German mathematician, was the most important woman mathematician of her era. She faced enormous obstacles, including being barred from formal university positions for years because she was a woman. Her work in algebra and theoretical physics was foundational. Mirzakhani worked in a different field of mathematics nearly a century later, but in a tradition Noether helped open. Reading them together gives students a sense of how women in mathematics have built on each other's work over generations, often with great difficulty. Noether's struggle to get recognised at all preceded Mirzakhani's recognition at the highest possible level.
Develops
Srinivasa Ramanujan
Ramanujan, the great early 20th-century Indian mathematician, came from a country and culture not usually recognised as producing world-class mathematicians. Mirzakhani came from Iran, which was similarly underestimated in Western mathematical circles. Both demonstrated that great mathematics could come from anywhere given the right conditions. Both also died young, leaving careers cut short at moments of great promise. Reading them together gives students a sense of how mathematical talent has emerged from many countries and cultures, despite Western assumptions about where major work would come from.
Complements
Ada Lovelace
Lovelace, the 19th-century English mathematician and early computing pioneer, was an early woman in fields that were largely closed to women. She faced the assumption, common in her time, that women could not do serious mathematics. She demonstrated otherwise through her work. Mirzakhani worked in a very different time, but in a field where the same assumptions had taken longer to break down. Reading them together gives students a sense of how women have done mathematics across centuries, often against considerable resistance. The breakthrough Mirzakhani represented in 2014 had roots running back through Lovelace and many others.
Develops
Euclid
Euclid, around 300 BCE, established geometry as a deductive science. Mirzakhani worked in the long tradition of geometry that descended from Euclid's foundation. Modern geometry has gone far beyond Euclid in many ways. Mirzakhani worked on curved surfaces, where Euclid's parallel postulate fails and the geometry is non-Euclidean. But the basic idea of geometry as a careful study of shapes through proof comes directly from Euclid. Reading them together gives students a sense of how mathematics builds on itself across many centuries. The field Mirzakhani worked in did not exist in Euclid's time. The way of thinking about it descends from him.
Complements
Dorothy Hodgkin
Hodgkin, the great British chemist, was one of the most important women scientists of the 20th century. She worked in chemistry rather than mathematics, but the parallels with Mirzakhani are striking. Both were exceptional women in male-dominated scientific fields. Both did patient, careful work over many years. Both eventually received their fields' highest honours. Both were modest about their own achievements while serving as important models for younger women in science. Reading them together gives students a sense of how women have made foundational contributions to many scientific fields, often against substantial obstacles.
Complements
Tu Youyou
Tu Youyou, the Chinese pharmacologist who won the 2015 Nobel Prize in Medicine, came to Western recognition late in her career. Like Mirzakhani, she was a major scientist from a country sometimes underestimated in Western scientific circles. Like Mirzakhani, she received the highest honour in her field after years of work that had been quietly important. Reading them together gives students a sense of how the global geography of scientific recognition is changing. Major science is being done in many countries. Western prizes are slowly catching up with this reality. Both Mirzakhani and Tu Youyou represent moments in this longer shift.
Further Reading

For research-level engagement, Mirzakhani's published papers are available on the arXiv preprint server and in journals including Inventiones Mathematicae, the Annals of Mathematics, and the Journal of the American Mathematical Society. Her 2014 ICM Plenary Lecture, available online, surveys her own work. Curtis McMullen, Howard Masur, Alex Eskin, and others have written technical commentaries on her contributions. The Maryam Mirzakhani Memorial Lecture series at various universities continues to honour her work.