(this article is jointly written with Norman Do)
I don’t think that everyone should become a mathematician, but I do believe that many students don’t give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.
— Maryam Mirzakhani, 2008
The recent passing of Maryam Mirzakhani came as a shock to many of us in the world of mathematics. Not only was she in the prime of her life, she had also been intensely active in her work, posting research articles online on the arXiv right up until November 2016.
Although we did not know Maryam Mirzakhani personally, we were both fortunate to have met her. Like many others, we were impressed by her friendliness and enthusiasm. One of us (Dan) met her as a prospective PhD student in 2004, just as she was finishing hers. Her PhD advisor Curtis McMullen was very keen to explain that his student had been doing great work. Both Mirzakhani and McMullen’s discussions of mathematics remained incomprehensible to this young student, but he was impressed by their excitement about her work. And for the other author (Norm), Mirzakhani’s work formed the basis of his PhD thesis.
We have both been inspired by her mathematics and her example, and heartened to see her work gain recognition. Mirzakhani was a deserving recipient of the Fields Medal in 2014. As the first woman to do so, she is a true trailblazer.
Both the authors were influenced by Mirzakhani’s work, or rather, that small portion of it which we understand. Even that small portion, sitting as it does on the cutting edge of research mathematics, towers in its abstraction — but it is not so far removed from everyday experience that we think it impossible to explain some small part of it for a general mathematical audience.
So bear with us as we attempt to share something about Mirzakhani’s work. As Einstein said, “Nature hides her secret because of her essential loftiness, but not by means of ruse.”
From the art of Escher to conformal geometry
A great deal of Mirzakhani’s work has illuminated our understanding of so-called moduli spaces, which are “spaces of shapes” important in mathematics and physics.
But first things first. Consider the picture below. It’s a 1956 lithograph by the Dutch artist M. C. Escher entitled Print Gallery.
This strangely distorted image contains several surprising features. It depicts a man looking at a picture, and in that picture is a city, and in that city is a gallery, and in that gallery is… a man, the very same man we started with!
The picture is delightfully self-referential. But the way the picture is distorted is particularly interesting. (For an extended discussion, see this article of B. de Smit and H. W. Lenstra Jr.) If we compare to a “straight” version, we see that all of the angles in Escher’s drawing are accurate! Although lengths have been distorted, angles have not.
Mathematicians have long been interested in this kind of geometry, in which angles are important, but lengths are not. It’s known as conformal geometry and has all sorts of applications throughout mathematics and physics.
Now let’s consider something simpler than Escher’s artwork. Let’s take a circular disc \(D = \{(x,y) \in \mathbb{R}^2 \colon x^2 + y^2 \leq 1 \}\) and ask about its conformal symmetries: how can the points of the disc move around, so that all angles between curves remain the same? More precisely, we ask for bijections \(f \colon D \rightarrow D\) that preserve angles.
Perhaps surprisingly, there are quite a few ways to do this, one of which is demonstrated below.
There is a conformal symmetry that takes the curves of constant radius and angle shown in the disk on the left to the curves shown in the disk on the right.
However, while there are many such transformations of the disc, there are not really that many. In fact, there are interestingly many.
If we consider the bijections \(D \rightarrow D\) which preserve distances as well as angles — also known as isometries — then the only such maps are the rotations. (We ignore reflections, since they reverse orientation and so negate angles). There is a 1-parameter family of rotations of \(D\).
On the other hand, if we simply consider smooth bijections \(D \rightarrow D\), there are many, many such maps. This set of maps is, in a suitable sense, infinite-dimensional.
As it turns out, the set of conformal symmetries \(f \colon D \rightarrow D\) is three-dimensional. Conformal geometry sits somewhere between the isometries of Euclidean geometry, and the smooth maps of topology: isometries are too rigid, smooth bijections are too flexible, but conformal symmetries are just right. Conformal geometry strikes an interesting balance between rigidity and flexibility.
One way to understand the degree of flexibility in conformal maps is as follows. Take any three distinct points \(p_1, p_2, p_3\) on the boundary of the disc \(\partial D = \{(x,y) \colon x^2 + y^2 = 1\}\). Take another three distinct points \(q_1, q_2, q_3 \in \partial D\). Then there is a unique conformal transformation \(f \colon D \rightarrow D\) such that \(f(p_i) = q_i\) for \(i=1,2,3\). This property is known as triple transitivity.
Triple transitivity means that conformal transformations are specified by three points. Being specified by three parameters, the set of conformal symmetries of \(D\) is indeed a three-dimensional space.
Tessellations of the hyperbolic plane by triangles
Hyperbolic geometry and the shape of space
It turns out that conformal symmetries of \(D\) preserve circles and lines: the image of any circle or line on \(D\) under a conformal symmetry is again a circle or line.
In fact, the situation is even better than that. One can put a metric on the disc — defining a new notion of distance, different from the standard Euclidean metric — so that all conformal transformations are isometries. This metric is known as a hyperbolic metric; it is a scalar multiple of the Euclidean metric, but the scalar depends on \(r\), the (Euclidean!) distance from the origin.
\(\displaystyle \text{Hyperbolic distance} = \frac{2}{1-r^2} \; \text{Euclidean distance}.\)
With this metric, the disc is known as the Poincaré disc model of the hyperbolic plane, and in both pictures above, all the triangles are congruent. The “tiny” triangles are the same size as the “big” ones; as \(r \rightarrow 1\), indeed \(\frac{2}{1-r^2} \rightarrow \infty\), and in fact \(\partial D\) is infinitely far away from the points inside \(D\)!
Between any two points on \(\partial D\), there is a unique circle or line perpendicular to \(\partial D\). These are in fact the “straight lines” or geodesics of hyperbolic geometry: with the hyperbolic metric, they are shortest distance curves between points.
By drawing such geodesics joining three points \(p_1, p_2, p_3 \in \partial D\), we have an ideal triangle. (Ideal just means that the vertices lie on \(\partial D\).) The triple transitivity of conformal symmetries means that if you take any ideal triangle, and specify where those vertices are to go on \(\partial D\), then the rest of the triangle, and in fact the whole disc, comes along for the ride — and in fact this transformation preserves hyperbolic distance.
Thus, from the point of view of conformal geometry, all triangles of this type have the same shape: all triangles with vertices on \(\partial D\), and sides given by lines or circles perpendicular to \(\partial D\), are conformally equivalent. From the point of view of hyperbolic geometry, all ideal triangles are congruent. But from either point of view, we are looking at the same thing: the conformal symmetries of the disc are the same thing as the isometries of the hyperbolic plane.
While ideal hyperbolic triangles are all congruent, the same could not be said of quadrilaterals. There are many different shapes of quadrilaterals that cannot be related to each other by conformal transformations. If we consider all possible shapes of quadrilaterals, then they form a space called the moduli space of quadrilaterals. Roughly and in short, a moduli space is a space of shapes.
Mathematicians are interested in moduli spaces because they describe all the possible shapes something can have. The moduli space viewpoint is not only crucial in understanding the mathematical nature of particular shapes, but also features in physics, such as when string theorists want to understand all the possible ways that a string can evolve in time.
In any case, the notion of moduli spaces extends far beyond triangles and quadrilaterals. We can instead consider much more complicated surfaces, such as the surface of a donut or a pretzel. The moduli space of pretzels is then the space of shapes of all pretzels, in the worlds of conformal or hyperbolic geometry.
A donut has one hole and a pretzel has three: we say the surface of the donut, or torus, has genus 1, and the surface of the pretzel has genus 3. There is a moduli space of tori and a moduli space of pretzels.
Clearly discs are one type of surface, donuts are another type, and pretzels are another type again. They are different in a topological sense — strictly speaking, they are not homeomorphic. (That is, there is no bijection \(f \colon \text{Donut} \rightarrow \text{Pretzel}\) with \(f\) and \(f^{-1}\) both continuous.) A classical theorem of topology says that a surface is specified topologically by its genus \(g\) and its number of boundary components \(n\). The case \((g,n) = (0,1)\) is the disc; \((1,0)\) is the torus; and \((3,0)\) is the pretzel.
A moduli space consists of surfaces of a given topology, up to conformal symmetry. Slightly more precisely, consider the set of all surfaces of genus \(g\) with \(n\) boundary components. Consider two such surfaces \(S\) and \(S’\) to be equivalent if there is a conformal bijection \(f \colon S \rightarrow S’\). The set of all the equivalence classes of such surfaces is the moduli space \(\mathcal{M}_{g,n}\).
Thus, the moduli space of pretzels \(\mathcal{M}_{3,0}\) is the set of all pretzels, but we consider two pretzels equivalent if they are related by a conformal bijection.
As it turns out, the moduli space of pretzels is 12-dimensional. In other words, there is a 12-dimensional space of shapes of pretzels. In general, the moduli space \(\mathcal{M}_{g,n}\) has dimension \(6g-6+2n\). This was essentially discovered by Bernhard Riemann in the 19th century.
As mathematicians continued to explore these moduli spaces, they discovered that they in turn have their own natural geometry, given by the so-called Weil–Petersson metric.
This geometry — the shape of moduli space — is the shape of a space of shapes!
Mirzakhani investigated the geometry of moduli spaces, and made a raft of discoveries. The moduli spaces \(\mathcal{M}_{g,n}\) of different types of surfaces — i.e. for different \(g\) and \(n\) — are all related to each other in an intricate way. Introducing brilliant new methods to study moduli spaces, she was able to prove several results about not only the geometry of moduli spaces, but also elementary questions about curves and surfaces.
One such theorem concerns simple closed geodesics on a hyperbolic surface. A hyperbolic surface is a surface which at every point looks locally like (i.e. is locally isometric to) the Poincaré disc. A geodesic, or “hyperbolic straight line” on the surface, is closed if it goes around in a loop (i.e. begins and ends at the same point, without a corner), and simple if it has no self-intersections. So a simple closed geodesic is a straight non-intersecting loop.
Now, there are uncountably many simple closed curves on a surface, but if we “straighten them” into geodesics, the number of simple closed geodesics is countable. And if we count all the simple closed geodesics with length up to some value \(L\), the number of simple closed geodesics is finite and we can find it.
By the works of Delsarte, Huber and Selberg that started in the 1940s, we know that on a hyperbolic surface, the number of closed geodesics of length at most \(L\) is approximately \(e^L/L\).
\(\displaystyle \# \{ \text{closed geodesics of length at most L} \} \sim \frac{e^L}{L}\)
(Strictly speaking, we mean oriented, primitive closed geodesics.) This theorem has a fascinating analogy with number theory, where it is known that the number of primes that are at most \(e^L\) is also approximately \(e^L/L\). Indeed, the theorem above is often known as the prime number theorem for hyperbolic surfaces.
Note, however, that this “classical” theorem says nothing about simple closed geodesics. Does the number of simple closed geodesics of length at most \(L\) also grow like \(e^L/L\)? Mirzakhani answered this question with a definite no: in fact, she showed that on a closed hyperbolic surface of genus \(g\),
\(\displaystyle \# \{ \text{simple closed geodesics of length at most L} \} \sim L^{6g-6}.\)
(Actually, this was essentially known previously, depending on what \(\sim\) means here…) That is, while the closed geodesics grow exponentially with length, simple closed geodesics grow polynomially, and the degree of the polynomial depends on the genus.
This theorem was obtained by a deep understanding of moduli spaces and their geometry. Mirzakhani even went so far as to calculate the volume of moduli spaces. Since \(\mathcal{M}_{g,n}\) has dimension \(6g-6+2n\), this amounts to calculating a \((6g-6+2n)\)-dimensional volume. She showed that these volumes possess a rich structure. For instance, the volume of the moduli space of pretzels is precisely
\(\displaystyle \text{vol} (\mathcal{M}_{3,0}) = \frac{176557}{1209600} \, \pi^{12}.\)
Actually, this particular volume was known prior to Mirzakhani. But Mirzakhani found a way to calculate the volumes of all \(\mathcal{M}_{g,n}\) by an intricate recursive method. Moreover, she showed that if you fix the lengths of the boundary components to be \(L_1, L_2, \ldots, L_n\), then the volume of \(\mathcal{M}_{g,n}\) is a polynomial in \(L_1, L_2, \ldots, L_n\) of degree \(6g-6+2n\).
Mirzakhani showed us how to understand all this, and a whole lot more.
Magic wands and billiards
One theorem in particular, proved by Mirzakhani along with collaborators Alex Eskin and Amir Mohammadi (see here and here) has been described as a “magic wand” which can be used to attack a vast range of problems.
The magic wand theorem describes a surprisingly simple and rigid structure underlying certain group orbits in certain moduli spaces. A rigorous statement would take us too far afield, but it does have some more down-to-earth applications. For instance, this work has been used to significantly advance our understanding of mathematical billiards.
In mathematical billiards, just as in real billiards, we have a table, and we consider a ball rolling on the table, bouncing off the walls as it goes. (No jumps or spin allowed!) However, unlike standard billiards, we don’t just consider rectangular tables — we consider tables of whatever shape we like, limited only by our imagination.
The goal of mathematical billiards isn’t so much to get the ball into the pocket, but to understand the dynamics of where a ball can go, and how. For instance, if the ball starts in a particular position, can you hit it to any other position?
Now, for most billiard tables you can think of, you can probably find a way to get the ball from one point to any other point. (At least in theory, if you have better skills than us!) But in general it is a much more difficult problem than it seems.
In 1958, New Scientist published a Christmas puzzle column written by two people. One was an esteemed psychiatrist and geneticist; the other was to become an esteemed mathematician and mathematical physicist. They were the father–son duo of Lionel and Roger Penrose. They posed the following puzzle: can you design a billiard table on which you cannot hit the ball from every point to every other point?
Since a billiard ball bounces off a wall in the same way that a light beam reflects off a mirror, one can equivalently consider the following “illumination problem”: can you design a room with mirrored walls, in which a candle can be placed without illuminating the entire room?
The diagram above depicts a solution that involves a combination of straight sides and arcs of ellipses. But let’s now put ellipses aside, and only consider billiard tables that have straight sides. Can you design a room with straight mirrored walls, in which a candle can be placed without illuminating the entire room?
This problem was only resolved in 1995, when George Tokarsky successfully designed one with 26 sides.
Still, in Tokarsky’s construction you can illuminate almost the entire room. As it turns out, a candle placed at the red point on the left will illuminate every point apart from the red point on the right.
So we might ask: can you design a room with straight mirrored walls, in which a candle can be placed that leaves a whole region in the dark?
Mirzakhani’s work on moduli spaces has shed light, so to speak, on this problem. As it turns out, moduli spaces are deeply connected to mathematical billiards — because moduli spaces are spaces of shapes, and they can encompass the possible shapes of billiard tables.
The mathematicians Samuel Lelièvre, Thierry Monteil and Barak Weiss were able to apply the magic wand theorem of Eskin–Mirzakhani–Mohammadi to the illumination problem. They showed that, as long as the walls of the room meet at fractional numbers of degrees, then Tokarsky’s construction is as bad as it gets: a candle in the room will illuminate all but a finite number of points in the room. In other words, “almost everything is illuminated”.
And thanks to Mirzakhani’s work, a great deal more mathematics is now illuminated.