Maryam Mirzakhani, A Candle Illuminating The Dark
One of the most brilliant mathematical minds of this century lost her life to cancer at only 40. Here’s what she illuminated.
“I think it’s rarely about what you actually learn in class. It’s mostly about things that you stay motivated to go and continue to do on your own.”
-Maryam Mirzakhani, on success in mathematics
Abstract mathematics sometimes has surprisingly practical applications, to not only physics but other arenas as well. Take, for example, the work of extraordinarily innovative mathematician Maryam Mirzakhani, whose recent death at the age of 40 has been mourned around the world. One of the theorems she co-developed sheds light on several related longstanding physics quandaries having to do with ricocheting and diffusion — of light, billiards, the wind, and other entities. Undoubtedly, given its generality, it will find many uses in science, sports, and beyond, for years to come.
The class of problems Mirzakhani was interested in dates back more than a century. In 1912, Austrian statistical physicist Paul Ehrenfest and his wife, the Russian mathematician Tatjana Afanassjewa-Ehrenfest, proposed the ‘wind-tree’ model as a way of trying to understand how impediments in a system affect diffusion. (In this context, diffusion means the spreading out of particles, light, gases, etc., due to their natural motion.) They imagined a bounded forest that was empty except for regularly spaced trees — symbolized as rectangles forming a periodic pattern within a square lattice. Imagine the wind entering the forest from a certain direction and scattering off the various trees according to the law of reflection (incoming angle equals outgoing angle). How quickly, they wondered, would nearby streams of air particles separate from each other and spread throughout the entire forest?
The wind-tree problem is similar to a game of billiards in which obstacles are placed at regular intervals on a table. Neighboring billiard balls that happen to hit different sequences of obstacles might end up, after a short period of time, in widely separated positions on the table. Tracking such disparate behavior would require precise knowledge of initial conditions and the configuration of the obstacles and walls. Even the slightest changes in initial conditions could lead to tremendous differences in the paths and final positions that the billiard balls take.
Even if the obstacles are removed, the problem remains complex. A slight perturbation to a ball’s initial launch angle might result in drastically different behaviors. For some angles, for instance, it might bounce off successive walls in a simple, closed pattern, forming a triangle or another closed shape, before returning to its starting position. It would repeat that closed trajectory again and again like an orbiting planet. However, for other angles, it might continue its bouncing for a much longer period of time, etching out crosshair patterns — until either it ends up stuck in a corner or the entire table is traversed and it reaches the starting point again. In other words, slight differences might convert the system from simply periodic to chaotic.
In the early 1950s, German-American mathematician Ernst Straus, one of Albert Einstein’s assistants at the Institute for Advanced Study in Princeton, suggested yet another variation of this theme, called the “illumination problem.” He wondered if a completely mirrored room, of any shape, could always be illuminated by a single candle placed somewhere in the room. That is, under what circumstances could the light from the candle bounce off the mirrored walls again and again in such a way that the room is full of light? Roger Penrose, in 1958 just at the start of his career, showed how elliptical mirrors could be configured into a room that, for a single candle placed anywhere within, must always have unreachable regions. Such a configuration, known as the “Penrose unilluminable room,” is doomed to have dark patches. In 1995, George Tokarsky extended the work of Penrose by designing a 26-sided, polygonal unilluminable room. (In 2007, the Tokarsky unilluminable room was a key element in the “Trust Metric” episode of the mathematical detective TV series Numb3rs.)
Replace light rays with billiard balls or streams of air particles and it is easy to realize deep connections between these predicaments. All involve geodesics: the shortest possible paths through certain spaces. All involve the law of reflection. All depend on the geometry and topology (i.e., the number of obstacles) of the situation. Replace the flat room or surface with an even more complicated surface, such as a hyperboloid (saddle-shape), and such problems become even more intriguing.
It takes a brilliant mind to take in the entire picture and look at such situations in their most general forms. That was the genius of Maryam Mirzakhani, who was unafraid to tackle major questions in mathematics — rather than cranking out solutions to simpler, more specific problems, a temptation for any researcher trying to build a resume.
Born in Tehran, Iran in 1977, Mirzakhani was a young girl during revolutionary times, centered on the fall of the Shah and the rise of a new regime. Soon thereafter, neighboring Iraq invaded Iran and began a brutal war that lasted for eight years, until 1988, and cut short more than a million lives. Despite the turbulent times, she remained focused on her schooling and developed a strong passion for literature. Her family lived near a street with many bookshops, which fed her voracious reading habit. At that point she imagined herself becoming a writer.
Later, when she was a teenager, Mirzakhani was drawn to another intellectual pursuit, the study of mathematics. Her talent in that field was so outstanding that she was invited as the first girl to represent her native land in two International Mathematical Olympiads. She won gold medals in each: the first held in Hong Kong in 1994, where she scored 41 out of 42 points, and the second held in Toronto, Canada in 1995, where she achieved a perfect score of 42 out of 42. Her performance was so impressive that she received permission to skip the national qualifying exam for university admission.
In 1999, after receiving a B.S. degree in Mathematics from the well-regarded Sharif University, she was admitted to Harvard’s graduate program in mathematics. She completed a Ph.D. there in 2005, under the supervision of Curtis “Curt” McMullen. McMullen, a 1998 Fields medalist, specialized in dynamical systems (including the famously intricate fractal, the Mandelbrot set), hyperbolic geometries, Teichmüller space (pertaining to mappings, or connections between surfaces), and related fields — in short, how particles and fluids flow through various types of spaces and surfaces, leading to either periodic or chaotic behavior.
After several years doing research at Princeton, Mirzakhani was appointed to Stanford’s mathematics department, where she achieved the rank of Full Professor in 2008 at the unusually young age of 31, and where she remained for the rest of her career. Extending her masterful, widely praised dissertation work, she continued to pursue the study of geodesics, or most direct paths, in hyperbolic geometries. In a series of key papers she developed theorems to characterize when such geodesics would be closed paths, like the route of a airplane circumnavigating Earth, or open, like the zigzag motion of light through a fiber optic cable. Such curious types of patterns were deeply connected with the situations mentioned earlier, wind-tree, billiards, the illumination problem, and diffusion in general.
Hyperbolic geometries (also known as spaces of negative curvature) differ from more traditional Euclidean geometries in fascinatingly odd ways. For example, if a triangle is embedded in a hyperbolic surface, such as a saddle, the sum of its three angles is less than 180 degrees. Furthermore, in defiance of Playfair’s axiom (related to Euclid’s parallel postulate, which states that given a line and a point not on it, one and only one line parallel to the first can be drawn through that point), each line in hyperbolic space has an infinite number of lines parallel to it that pass through any given point not on that line. Therefore, geodesics in hyperbolic geometries tend to fan out, with separation rates measured by parameters called Lyapunov exponents. The pattern in which these lines fan out strongly resembles the physical process of diffusion; hence the deep connection between this abstract mathematical field and statistical mechanics in physics.
Perhaps Mirzakhani’s most famous contribution, in tandem with her frequent collaborator Alex Eskin of the University of Chicago (and later with Amir Mohammadi as well), was the so-called “Magic Wand Theorem,” published in 2013. Painstakingly developed over several years by piecing together and extending the work of many other mathematicians, it offered a litmus test for determining the orbital dynamics of systems, given certain properties of the space housing them and any obstacles within. Very quickly, the theorem found applications to the billiard problem, the wind-tree puzzle, and the illumination problem (showing when patches of darkness are inevitable). Like magic, one could calculate the diffusion rates of a wide range of systems, given the overall shapes and configurations involved, instead of having to compute the individual trajectories. As Russian mathematician Anton Zorich remarked:
The theorem proved by Alex Eskin and Maryam Mirzakhani is so beautiful and powerful that, personally, I have no doubt that it would find numerous applications far beyond our current imagination.
For unknown reasons that have often been speculated about, there has never been a Nobel Prize in mathematics. The choice was that of Alfred Nobel himself, who willed his fortune to fund prizes that he believed would support human progress. Unfortunately, he overlooked mathematics, perhaps deliberately, when writing his will.
The highest honor for mathematicians, therefore, is the Fields Medal, bestowed every four years (since 1936) to up to four young (under 40) mathematicians. In 2014, Mirzakhani became the first woman (and, to date, the only woman thus far) to receive that prize. It was an incredible breakthrough for the quest for equality in STEM disciplines. Despite the magnitude of the honor, she remained as humble and hard-working as ever.
In an interview published in Stanford News, Mirzakhani described her methods:
I don’t have any particular recipe [for constructing novel proofs] … It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.
Mirzakhani enjoyed a happy family life, delighting in spending time with her husband Jan Vondrák and her daughter Anahita. Her death on July 14, 2017, after several years undergoing treatments for cancer, has been heartbreaking for her family, friends, Stanford, and the mathematics community in general. A flickering candle, touched by the wind, lost its warm, shining flame.
As Iranian-born scientist Firouz Naderi, former Director of Solar System Exploration at NASA’s JPL, wrote:
A light was turned off today …. far too soon. Breaks my heart.
Paul Halpern is the author of fifteen popular science books, including The Quantum Labyrinth: How Richard Feynman and John Wheeler Revolutionized Time and Reality.
