In 2014, Maryam Mirzakhani, then 37-years-old, a mathematician working at Stanford University, became the first woman to win the Fields Medal since its inception in 1936. The Fields Medal is awarded to mathematicians under 40 who have made outstanding contributions to mathematics that hold future promise. Her death, on July 14, due to breast cancer when she was only 40, saddened the entire mathematics community.
Why is she important?
Mirzakhani’s work straddled several branches of mathematics. She was intuitive and persistent. She would often take large sheets of paper and draw geometric patterns on them as she thought of the problem, prompting her daughter to think she was “painting again.”
Manjul Bhargava of Princeton University told The Hindu that Mirzakhani was a “master of curved spaces.” As he explained in an email, “Everyone knows that the shortest distance between two points on a flat surface is a straight line. But if the surface is curved — for example, the surface of a ball or a doughnut — then the shortest distance [along the surface] between two points will also be along a curved path, and can thus be more complicated. Maryam proved many amazing theorems about such shortest paths — called geodesics — on curved surfaces, among many other remarkable results in geometry and beyond. Her work, and the research programmes she started, will have an impact on mathematics and physics for years to come.”
She had an uncanny intuition for geometric problems, Professor Bhargava said, which she would solve through drawings that looked like beautiful doodles “but were in fact profound geometric insights that she would then make rigorous later on.” Mirzakhani was interested in complicated curved surfaces or hyperbolic geometry: for example, studying loops that don’t intersect; probing deeper to answer how many loops, which don’t intersect, are there of less than a given length on a curved surface.
What interested her?
She worked with Alex Eskin to find a solution to the problem of understanding the trajectory of a billiard ball as it bounces around a table. In this, she generalised a work done by her doctoral adviser Curtis McMullen of Harvard University, also a Fields medallist.
A consequence of her work was to give an entirely new proof of a conjecture made by leading string theorist Edward Witten (a 1990 Fields medallist). The first proof of this conjecture was given by Maxim Kontsevich in 1992; it was such a difficult thing to prove that this work itself, in part, won him the Fields Medal in 1998. Mirzakhani’s proof of Witten’s conjecture was by relating it to an “elementary problem of counting the number of geodesics on individual surfaces.” Mirzakhani worked aggressively and went for deep and fundamental problems, never reaching for the “low-hanging fruit.” It was her strong geometric intuition and her fluency in a range of diverse techniques that made it possible for her to tackle these problems. In the words of Professor McMullen, “Maryam was a brilliant mathematician who has left us far too soon, and who will continue to inspire others to follow in her path.”
What was her field of work?
A tribute on the Stanford University website says she specialised in an area of mathematics that “read like a foreign language to those outside mathematics — moduli spaces, Teichmüller theory, hyperbolic geometry, Ergodic theory and symplectic geometry.” Each of these terms comes with a well-developed theory backing it and Mirzakhani’s work was both breaking new ground and building a bridge across these areas. In an interview to Quanta magazine, her collaborator Professor Eskin said her doctoral thesis was such that you could immediately recognise that it belonged in a textbook.
What do peers remember?
An Iranian citizen, Mirzakhani grew up in Tehran in the post Iran-Iraq war period, “the lucky generation,” she said in a rare interview she gave to Quanta magazine. A private person, she was known to be humble and creative. “We overlapped at Harvard University early in our careers, and became instant friends then… Her work has just been so remarkable then and ever since; it was indeed a great honour for me to receive [the Fields Medal] together with her,” said Professor Bhargava.
