# Maryam Mirzakhani, the Only Woman to Have Won Math's Highest Honor

In mathematics, the highest award you can receive is the Fields Medal. Created by Canadian mathematician John Charles Fields, it is only awarded once every **four years**, to a maximum of **four** mathematicians, and all must be under the age of **40-years-old**.

While the first Fields Medal was first awarded in 1936, the medal has only been continuously awarded every four years since 1950. Famous recipients include the physicist Edward Witten in 1990 and Russian mathematician Grigori Perelman in 2006, who ultimately declined the medal.

From 1936 and 2014, all the recipients of the Fields Medal had been men, that is until 2014. That's when an Iranian woman along with Brazilian Artur Avila, Canadian Manjul Bhargava, and Austrian Martin Hairer received a Fields Medal. Meet Maryam Mirzakhani.

### A remarkable mind

Maryam Mirzakhani was born in 1977 in Tehran, Iran. As a child, she dreamed of becoming a writer. During high school, Mirzakhani met her lifelong friend, Roya Beheshti Zavareh, who now teaches mathematics at Washington University in St. Louis. The two girls approached the principal of their all-girls school and asked that the same math problem-solving classes that were taught at the all-boys school be taught at their school.

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In 1994, when Mirzakhani was 17, she and Zavareh made the Iranian Mathematical Olympiad team, and Mirzakhani received a gold medal. The following year, Mirzakhani achieved both another gold medal and a perfect score at the International Mathematical Olympiad, which was held in Toronto, Canada that year.

Mirzakhani received a bachelor's degree from Sharif University in Tehran in 1999, and she and Zavareh wrote a book together entitled, *Elementary Number Theory, Challenging Problems,* which was published in 1999.

For graduate school, Mirzakhani traveled to Harvard University in the U.S. where she became interested in *hyperbolic surfaces*, the strange doughnut-shaped objects with two or more holes and saddle points.

A curved surface can have a "straight" line segment, known as a *geodesic*, which is the shortest path between two points. On a hyperbolic surface, some geodesics are infinitely long, while others close up into a loop like the circles on a sphere.

Most geodesics intersect themselves many times before they close up, but a tiny fraction which are called "simple" geodesics, never intersect themselves. Mathematicians had been unable to determine how many simple closed geodesics of a given length a hyperbolic surface could have.

Mirzakhani solved that problem in her 2004 Ph.D. dissertation in which she developed a formula for how the number of simple geodesics of length *X* grows as *X* gets larger. She also determined a formula for the volume of *moduli space*, which is the set of all possible hyperbolic structures on a given surface.

If that weren't enough, Mirzakhani also provided a new proof of an old conjecture that had been proposed by Edward Witten concerning the topological measurements of moduli spaces as it related to string theory.

Mirzakhani's dissertation generated three influential papers which were published in the top mathematics journals. After a stint as a professor at Princeton University, in 2009, Mirzakhani became a professor at Stanford University.

### A game of billiards

In 2006, Mirzakhani began a collaboration with the University of Chicago's Alex Eskin. Eskin is a 2019 recipient of the Breakthrough Prize. He and Mirzakhani began analyzing the range of behaviors of a ball on a polygon-shaped billiard table, where the table's angles are a rational number of degrees. A rational number can be expressed as the quotient of two integers where the denominator is non-zero.

Mirzakhani and Eskin imagined deforming a billiard table by contracting it along the direction of a billiard ball's trajectory. This transformed the original table into a succession of new ones which is the moduli space that is comprised of all possible billiard tables having a specific number of sides.

At the 2014 Fields Medal award ceremony, American mathematician Jordan Ellenberg explained Mirzakhani's research as:

"... she studies billiards ... She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn't directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way;

... the table itself moves like a strange planet around the universe of all possible tables ... it's what you need to do in order to expose the dynamics at the heart of geometry; for there's no question that they're there."

### Mirzakhani's Legacy

In 2008, Mirzakhani married fellow professor of mathematics at Stanford Jan Vondrak, and the two went on to have a daughter, Anahita. Mirzakhani's and Vondrak's home was littered with large sheets of paper on which she would draw physical representations of the concepts she was thinking about. Her daughter called this "painting".

In 2013, Mirzakhani was diagnosed with breast cancer, and on July 14, 2017, she died of the disease. Mirzakhani was only 40-years-old.

Following her death, Iranian newspapers, along with Iranian President Hassan Rouhani, broke with tradition and published photos of Mirzakhani where she isn't wearing a head covering.

To allow Mirzakhani's daughter to be able to visit Iran, the Iranian parliament sped up passage of an amendment that allows the children of Iranian mothers who are married to foreigners to receive Iranian nationality.

Before her death, Mirzakhani had become a member of the prestigious National Academy of Sciences, and following her death, the International Council for Science declared May 12th, which is Mirzakhani's birthday, International Women in Mathematics Day.

Tehran's Sharif University of Technology named their College of Mathematics library after Mirzakhani, and a conference hall in the Iranian city of Isfahan is named after her. The satellite firm Satellogic named a satellite after her.

In November 2019 the Breakthrough Prize Foundation announced a **$50,000** prize in Mirzakhani's honor to be awarded to outstanding women mathematicians. Asteroid 321357 Mirzakhani was named in her memory.

In February 2020, on International Day of Women and Girls in STEM, Maryam Mirzakhani was honored by UN Women as one of **seven** female scientists, dead or alive, who have shaped our world.

Had she lived past the tender age of 40, who knows what Maryam Mirzakhani could have accomplished. Still, she has left a lasting legacy for all women and girls in the fields of mathematics and science.

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