On 16 June 2014, I received an email from the President of the International Mathematical Union.
The subject of the email was a question “Will you be at ICM?” and the body of the email consisted of just two lines: “And will you have some disposable time? I have a favour to ask … Best, Ingrid.”
The sender was Professor Ingrid Daubechies and ICM is the International Mathematical Congress, which is held every four years. The most anticipated event at each ICM is the award of the Fields Medal, arguably the most celebrated prize in mathematics. In 2014, the ICM was to be held in Seoul, Korea.
When I answered Ingrid’s email, I learnt that I was to be part of a small group of female mathematicians entrusted with a special job in Seoul, involving such amazing news that it set a bell ringing in my heart — a bell that is still ringing today.
This group learnt that a Fields medal was to be awarded to a female mathematician for the first time in its long and luminous history. We were asked to be “on call” to provide support and help for the recipient at ICM.
The prestigious medal, often dubbed the Nobel Prize for mathematics, is awarded to recognise extraordinary results in mathematics by a mathematician under the age of 40.
Before that news arrived in my inbox, 52 Fields medals had been awarded, all of them to men.
Now, the established ranks of the mathematical world were going to acknowledge not only that women can do maths, but that their achievements can be as brilliant as the highest achievements ever recognised in recorded mathematical history.
One of the four brilliant mathematicians who were awarded Fields medals at the 2014 ICM was Maryam Mirzakhani, who was born and educated in Iran, and completed a PhD in mathematics at Harvard University. Behind that brief outline, there was an exceptional person with an astonishing history.
Iran admitted girls to their International Mathematics Olympiad (IMO) team for the very first time in 1994. The contest is difficult, involving six problems, each worth seven marks and distributed over two consecutive days. Maryam was a member of that first team to include girls.
Maryam did not just participate; her performance was spectacular. She dropped only one mark in 1994, and in the next year, she achieved a perfect score. She was the first Iranian ever to have been awarded two gold medals and a perfect score in the IMO.
An observer who witnessed the IMO medal ceremonies in 1994 and 1995 wrote to me that: “The two girls in the Iranian team were completely covered in black — except for their feet. When the Iranians went to collect their medals, one had to infer what was underneath from the confident way the girls’ trainers strode visibly across the stage. It was very, very impressive — if a rather unusual way to witness/infer youthful ambition.”
Much later, that drive and ambition led Maryam to move to the US to pursue graduate study in Mathematics at Harvard University, where her PhD advisor, Curtis McMullen, also a Fields medallist, said she showed “determination and relentless questioning.”
Most research mathematicians spend their working lives as highly accomplished technical specialists working within one subfield of mathematics, much like a performer might interpret and play glorious music on just one instrument.
To give an example: Imagine wrapping a ribbon around a doughnut. You could put the ribbon around a longitudinal cross-section of a doughnut or you could put it around its equator, or combinations of these. You might be able to shuffle the ribbon around, but you can’t shorten the length of a ribbon without it falling off or cutting into the doughnut.
The closed curve represented by each ribbon is what’s known as a geodesic. On Earth, the geodesics are given by great circles (the largest circles you can possibly draw on a sphere).
Imagine you can describe geodesics even as a surface deforms continuously into other shapes, for example, as a doughnut becomes a coffee mug.
Maryam studied geodesics on hyperbolic surfaces, which are surfaces that are shaped like saddles. That is — in one direction the surface folds down and away from you (for example, saddle fitting around the belly of a horse) while in the other direction it folds upwards (the length of the saddle curving up a horse’s spine).
The question was: can we estimate the number of geodesics on different types of hyperbolic surfaces, as the length of the geodesic grows?
In her PhD thesis, Maryam proved a sharp estimate — the most precise estimate possible — of this number on hyperbolic spaces, by studying all possible deformations of a surface.
One of the stunning results that came out of her work was a proof of a famous result in the entirely different field of string theory — that is, a proof of Witten’s conjecture, which concerned the physics of two different models of two-dimensional quantum gravity.
To study such problems, Maryam used the mathematical setting of moduli spaces, which has always been regarded as tricky and mysterious.
It was amazing that Maryam could delve not only into its mysteries, but actually pull out deeply held treasures she discovered in such spaces. Each time she proved something, it created ramifications that travelled far and wide into other areas.
I met Maryam and her husband Jan and their little daughter Anahita at the 2014 ICM in Seoul. Maryam was a devoted mother, always holding Anahita and paying close attention to her reactions and emotions.
The year prior to the award of the Fields medal, Maryam had been diagnosed with breast cancer. The group of women told of her award — nicknamed the “MM shield” — was formed to give her support and protection from possibly excessive media intrusions and to allow her the space and time to rest.
We were delighted that she had recovered enough to attend the ICM and accept her Fields medal, although she could not stay to deliver her invited lecture.
So it was a shock to hear the news on the 14 July, 2017, that Maryam Mirzakhani had been overcome by her illness. As a female mathematician myself, I felt like a close family member passed away.
But I know that in the world somewhere, now or in the future, there is a little girl or a woman, brave, persistent, driven to discovery through relentless questioning, who is deeply talented, who may win a Fields medal again.
The bell is still ringing in my heart.