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Many students decide early that they are "not a maths person". Maryam Mirzakhani was told something similar by a teacher in her first year of middle school in Tehran, and lost interest for a while. Years later she became the first woman to receive the Fields Medal. This post explains, in school-level language, what she actually studied, gives a genus example you can check with a pencil, and pulls out what her own words say about how to learn hard mathematics.
Who was Maryam Mirzakhani?
The MacTutor biography of Mirzakhani records that she was born on 12 May 1977 in Tehran and died on 14 July 2017 in California. She attended the Farzanegan school for girls. In her first year there a teacher told her she was not particularly talented in mathematics; a different teacher the next year encouraged her. She and her friend Roya Beheshti then asked the principal for the kind of olympiad problem-solving classes that boys' schools had, and the principal arranged them.
The official IMO results for Iran show what followed. In 1994 Mirzakhani scored 41 out of 42 and won a gold medal. In 1995 she scored 42 out of 42, a perfect paper, and won gold again.
She studied at Sharif University of Technology in Tehran and took her PhD at Harvard in 2004 under Curtis McMullen. The International Mathematical Union's account of her work adds that she was a Clay Research Fellow and assistant professor at Princeton from 2004 to 2008, then a professor at Stanford. On the IMU's Fields Medals 2014 page, her citation reads: "for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
What is the genus of a surface?
The IMU release starts with an idea you can picture. Surfaces can be classified, if you ignore their exact shape, by a single number: the number of handles, called the genus. A sphere has genus 0, the surface of a coffee cup has genus 1, and the surface of a pretzel with three holes has genus 3.
You can compute the genus with Euler's formula for a surface divided into vertices, edges and faces: V − E + F = 2 − 2g.
Example 1: a cube. V = 8, E = 12, F = 6. So V − E + F = 8 − 12 + 6 = 2, which gives 2 − 2g = 2 and g = 0. A cube can be squashed into a sphere without tearing.
Example 2: a torus. Draw a 3 by 3 grid of squares on a doughnut, wrapping around both ways. Each square has one corner, two edges and one face that it does not share with the next one, so V = 9, E = 18, F = 9. Then V − E + F = 0, so 2 − 2g = 0 and g = 1. One handle.
A common mistake is to count shared edges twice. Each edge belongs to two faces, so count edges once, by walking around the grid.
What did Mirzakhani prove?
A Riemann surface is a surface with extra geometric structure; the IMU explains that one important version lets you measure angles, lengths and areas in hyperbolic geometry. On such a surface you can look for closed geodesics: loops that cannot be shortened by sliding them around. The number of closed geodesics shorter than a length L grows exponentially with L.
Mirzakhani asked what happens if you count only simple ones, loops that never cross themselves. The IMU release explains that she showed the count grows like a constant times L^(6g−6), a power of L rather than an exponential.
Here is what that means for a genus 2 surface. Then 6g − 6 = 6, so for long loops the count of simple closed geodesics grows like L⁶. Doubling L multiplies the count by about 2⁶ = 64, while the count of all closed geodesics grows far faster.
The same number 6g − 6 appears in another way. The IMU notes that the shapes a genus g surface can take depend on 6g − 6 parameters, called moduli, so the moduli space for genus 2 has 6 dimensions. Mirzakhani proved her counting result by computing volumes in this space. That route also gave a new proof of a conjecture of Edward Witten. Later, with Alex Eskin and Amir Mohammadi, she showed that certain paths in moduli space are far more regular than anyone expected.
What did she say about learning mathematics?
In an interview with the Clay Mathematics Institute in 2008, Mirzakhani said: "I am a slow thinker, and have to spend a lot of time before I can clean up my ideas and make progress." Asked for advice to young people, she said: "I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it." She added: "The beauty of mathematics only shows itself to more patient followers."
What can a student take from Mirzakhani's way of working?
A slow start is not a verdict. She struggled in middle school and won two IMO golds a few years later. A bad term tells you what to fix, not who you are.
Ask for the class you need. She and her friend asked for olympiad training their school did not offer. If you want to prepare for olympiads, our guide to the IOQM maths olympiad is a place to start.
Give ideas time. If a proof will not come out cleanly, leave it and return the next day. Thinking slowly is still thinking.
Draw. Her work was about shapes. Sketch the surface, the loop or the graph before writing equations.
When a new idea such as genus or a geodesic is the sticking point, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can learn one idea at a time.
Try it now: count V, E and F for a triangular prism, check that you get g = 0, and then explain why in one sentence.
