The Mechanism
On 13 August 2014, at the International Congress of Mathematicians in Seoul, Maryam Mirzakhani became the first woman, and the first Iranian, to be awarded the Fields Medal — mathematics' highest honor, given every four years to mathematicians under 40 for outstanding contributions. She was 37. Mirzakhani had grown up in Tehran during the Iran-Iraq war, won gold medals at the International Mathematical Olympiad in 1994 and 1995 (the second with a perfect score), completed her PhD at Harvard in 2004 under Curtis McMullen, and held positions at Princeton and then Stanford. Her doctoral thesis, "Simple Geodesics on Hyperbolic Surfaces and Volume of the Moduli Space of Curves," answered a question that had been open for decades: how to *count* simple closed geodesics on a hyperbolic Riemann surface — that is, how to count the ways one could draw a closed loop on a curved surface that never crosses itself. Mirzakhani found a formula for how that count grows as the loop length increases, and in the process produced a new proof of the Witten-Kontsevich theorem on intersection numbers of moduli space (a deep statement linking string theory and geometry). Her work used dynamics — treating the space of all possible surfaces as itself a kind of phase space in which orbits could be studied — to answer questions in pure geometry, and the methods she developed have shaped the field since. Her process was tactile: she worked on enormous sheets of paper spread across the floor, drawing doodles and surfaces by hand for hours. She died of breast cancer on 14 July 2017, three years after the Medal, at age 40. In 2018 the Breakthrough Prize created a satellite prize for women in mathematics named in her honor.
Why It Matters
The Fields Medal is often described as mathematics' highest honor, and for decades every winner had been a man. Mirzakhani broke that pattern in 2014, but the reason she won was not a simple single calculation. Her work answered a long-open question about how the number of simple closed geodesics grows on a hyperbolic surface, then linked that problem to moduli spaces and even gave a new proof of the Witten-Kontsevich theorem. It is remarkable that a question about drawing loops on curved surfaces could connect to such deep parts of modern geometry and mathematical physics.
Wait — That's Not Quite Right
A common mistake is to think the Fields Medal is like a lifetime achievement award or that it goes to older mathematicians. In fact, it is awarded every four years to mathematicians under 40 for outstanding contributions. Another misconception is that Mirzakhani won for one flashy trick. Her prize recognized years of work that combined geometry, dynamics, and careful counting on curved surfaces.
Vocabulary
- Fields Medal
- Maryam Mirzakhani
- hyperbolic surface
- geodesic
- moduli space
- Riemann surface
- intersection numbers
- Witten-Kontsevich theorem
- dynamics
- Princeton
- Stanford
- International Congress of Mathematicians
Quick Quiz
5 questions · For classroom or kitchen table
The Experiment
Draw a Curved-Surface Loop Lab
Find a sheet of paper, a pencil, and a few round objects such as cups, bowls, or a balloon. Trace the objects to make curved shapes, then imagine you are trying to draw a closed loop on each shape that does not cross itself. On a flat circle, a loop is simple to sketch. On a curved surface, the idea gets trickier, which is one reason Mirzakhani's work was so important.
Now choose one shape and try to draw as many different non-crossing loops as you can, using different lengths of string or yarn laid on the paper. Count them by hand and compare your counts with a partner or family member. You are not doing real hyperbolic geometry, but you are practicing the same kind of thinking: noticing patterns, counting carefully, and asking how the number of possibilities changes as a shape gets larger or more complicated.
If you want, spread the paper on the floor the way Mirzakhani sometimes did and work for a few minutes at a time. Sketch arrows, loops, and labels next to your shapes. Then write one sentence about what changed when you made the loops longer or shorter.
paper, pencil or pen, string or yarn, cup or bowl, optional balloon, adult supervision recommended for floor work and to avoid tripping
Where this came from
- Maryam Mirzakhani, "Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces," *Inventiones Mathematicae* 167 (2007) 179-222. Fields Medal citation: International Mathematical Union, 2014 (https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2014). Profile: Erica Klarreich, "A Tenacious Explorer of Abstract Surfaces," *Quanta Magazine*, 12 August 2014 — https://www.quantamagazine.org/maryam-mirzakhani-is-first-woman-fields-medalist-20140812/
Want next week's entry in your inbox?
One short email a week with the latest field guide entry — the fact, the explanation, the quiz, and the activity. Free for parents and teachers.
For adults only · Unsubscribe anytime