On Monday 20 August my daughter Daisy (not a mathematician) and I went to a commemoration event for Maryam Mirzakhani organised by the School of Mathematics and Physics at the University of Queensland.
In 2014 Mirzakhani became the first woman ever to win a Fields Medal (the maths equivalent of a Nobel Prize). In 2017 she died of cancer.
Cecilia González Tokman gave the event an idea of Mirzakhani’s areas of research. Cheryl Praeger, herself one of the first women ever to serve on the executive committee of the International Mathematical Union, and the second woman ever to become a professor of mathematics in Australia, talked about Mirzakhani’s work fitted in with the battle for equality for women in mathematics.
Azam Asanjarani (now in New Zealand) and another Iranian woman mathematician (now in Australia) talked about their memories of Maryam Mirzakhani as a teenager in Iran, where she did high school and university before going to Harvard University in 1999. Mirzakhani worked in the USA, at Harvard, then Princeton, then Stanford University, until her death.
In middle school Mirzakhani was more interested in reading novels and becoming a writer than in maths. In one round of tests, she got 20/20 in every subject, except maths, where she got 16/20. Annoyed, she declared in future that she “wouldn’t even try” in maths.
But she did. Gradually she realised that there are stranger riches, greater beauties, in maths than the routine school syllabus suggests. On one account, she was first inspired by learning about Gauss’s solution to the problem of adding all the whole numbers from 1 to 100.
Solution: Write those numbers in a column. In another column, next to it, write those numbers in reverse order. Then each row – 100 + 1, 99 + 2, 98 + 3, etc. – adds up to 101. There are 100 rows. So the total of the whole numbers from 1 to 100 is ½ (101 × 100), or 5050.
She became one of the first Iranian girls to go to the International Mathematical Olympiad, and did well there. But even up until shortly before she went to university she would say: “I love maths, but I’m afraid I’m not good enough at it for university”.
She gained confidence, and had her first research paper published even before she finished her first university maths course. It was about “graph theory”, that is, the maths of networks studied as consisting of vertices connected by edges, without regard to the length or shape of edges.
It was about “complete tripartite graphs”, that is, graphs whose vertices can be divided into three groups so that every vertex in every group is connected to every vertex in every other group but not to any vertex in its own group. When, it asked, can those “complete tripartite graphs” be made up by putting together “5-cycles”, that is, sets of five vertices connected one after another in a loop.
Below: a complete tripartite graph where the 3 groups are (1) the single vertex on the left (2) the two vertices in the middle (3) the four vertices on the right.
That problem belongs to “discrete mathematics”, the maths of whole numbers and of objects and patterns which can be encoded in whole numbers rather than by the uncountable “real” number line which includes √2 and π and such, and which we need to do calculus. In fact, according to her CV, Mirzakhani with her friend Roya Beheshti (who came to the USA to do maths postgrad work at the same time as her, and also stayed) published a book or pamphlet on “Elementary Number Theory” before leaving Iran. “Number theory” means theory to do with the counting numbers 1, 2, 3, …
Mirzakhani stood out, mathematically, for her ability to link different areas of maths. Her Fields Medal was for work on Riemann Surfaces. Riemann Surfaces came out of work on “complex analysis”, i.e. calculus of functions w = f(z) where w and z are both complex numbers. Click here for a summary of how Riemann Surfaces come from complex analysis.
Mirzakhani, however, who seems to have had an uncanny ability to visualise these Riemann Surfaces (2-dimensional surfaces sitting in four-dimensional space), and get an idea of them by sketches, would use them to solve problems of counting possibilities (combinatorics) in discrete maths.
To students at Stanford, she described herself as a “slow mathematician”. Hardly. But what really set her out was not quickness in calculation, or instant insight, but, as Curtis McMullen, her Ph D supervisor at Harvard, put it, “relentless questions” and “fearless ambition”.
McMullen has recounted that she would come to his office and pose questions that were “like science fiction stories… vivid scenes… some unexplored corner of the mathematical universe… strange structures… beguiling patterns…”. “Is it right?” she would ask. “As if I might know the answer”, comments McMullen.
“I believe that many students don’t give maths a real chance”, Mirzakhani later commented. After completing one of her research projects, she said to a co-worker: “If we knew things would be so complicated, I think we would have given up”. Then she added: “I don’t know; actually, I don’t know. I don’t give up easily.”
Click here to watch a video of Mirzakhani lecturing on “dynamics of moduli surfaces of curves”. (A “Riemann surface” is also a “complex curve”). She is lecturing for researchers in a specialised field of maths, so do not expect to understand the detail. You can, however, get a taste of what sort of ideas Mirzakhani was working with, her enthusiasm, her humour.
One set of problems she mentions in the lecture are billiard-table problems. If you shoot one ball at another on a square billiard table, for example, can you protect the target ball by having only a finite number of guard balls around it?
One guard ball will protect the target ball from a direct shot. One will protect it from a shot bouncing directly off one wall. But then there are an infinite number of paths with more and more bounces which the shot can take.
It turns out, however, that you can protect the target ball with only a finite number of guard balls.
The proof (click here) uses the idea that you can map the bouncing paths as straight lines if you draw copies of the square adjacent to it, and rewrite the bouncing path as going straight into the neighbouring copies.
The fundamental idea here is equating a square (actually, in this case, a 2×2 square, the original square plus three adjacent squares which allow us to model the first bounces as continuations along straight lines) with a torus (a ring donut) by gluing together the opposite pairs of edges of the square. An ordinary flat space is equated with a Riemann surface.
This is the short video produced to mark Mirzakhani’s Fields Medal: click here.
One of her other notable achievements is to have become the first woman, since the 1979 “Islamic revolution”, to appear in the Iranian press without a headscarf. Apparently the Supreme Leader, Ayatollah Khamenei, was so taken with Iranian national pride when Mirzakhani won her Fields Medal that he allowed a picture of her. Like other Iranian women emigre mathematicians, she didn’t wear a headscarf after she left Iran and no longer was compelled to.
The Iranian women mathematicians who knew Mirzakhani recounted that when she and Roya Beheshti went to the International Mathematical Olympiad, they had to work in full, all-swathing, loose-fitting religious dress. Their enthusiasm and dynamism could be detected only by the quick movements of their trainers beneath their robes.
Look at the video of Mirzakhani lecturing, and you can see she couldn’t do as she did if swathed in robes on the pretext that God would be angered by seeing her expound mathematics more vividly and directly.