Including debt to Marina Ratner, 1938-2017

Maryam Mirzakhani won the Fields Medal in 2014. We and the whole community are grieving after losing her to breast cancer two weeks ago. She made several breakthroughs in the geometric understanding of dynamical systems. Who knows what other great results she would have found if she had lived: we will never know. Besides her research she also was the first woman and the first Iranian to win the Fields Medal.

Today we wish to express both our great sorrow and our appreciation of her work.

An article in 2014 by Jordan Ellenberg called her win a “cultural change in mathematics” not for her gender or nationality but for her field of dynamics. He called it “an infant compared to the other major branches of math.” Now dynamics has been studied since long before Isaac Newton, and we’ve covered the three-body problem among other topics. What he means is that abstraction away from physics was needed to boost mathematical tools of analysis and that this gained thrust only in the second half of the 1900s.

We can put it this way: Dynamics has always been a moving target. The work that Mirzakhani furthered gives it a fixed frame. Whole ensembles of possible motions can be represented by parameters to form a space—one like a manifold but with a quotient structure. This space becomes a single geometric object by which to analyze the dynamics. We can give a facile analogy to how Boolean circuits are often considered easier to analyze than Turing machines because they are fixed whereas Turing machines move. But there is a greater potential conduit to problems in complexity theory: both her work and the attack on P vs. NP by Ketan Mulmuley and co-workers involve orbits and their closures.

## Executive Toys in Flatland

Perhaps the best example of a dynamical system to play with is the familiar executive toy of metal balls on strings. Usually there are five identical balls as at left below, but let’s say a junior executive might start with just two as shown in the middle. Composite of various sources plus extra drawing.

Now let’s transport the company to Edward Abbott’s Flatland. Junior executives there have two balls that go back and forth along a line inside a confined area. We don’t know how gravity would work in Flatland—at least not classical gravity—but the edges of the line segment would propel a ball colliding with them back toward the center. Of course we assume all collisions are perfectly elastic, meaning in particular that they conserve momentum. Admittedly contrary to the illustrations, we also assume the “balls” are really point particles of vanishing radius.

We can now trade a ball for a dimension. We can represent configurations of the balls by points ${(x,d)}$ where ${x}$ is the displacement of the left ball from the left end and ${d}$ is the distance between the balls. These points form a triangle as shown, with left-right remaining the directions of the first ball and up-down corresponding to left-right for the second ball. The combined directions and velocities of the two balls become one direction and velocity of the blue ball shown in the triangle. The two balls collide—remember we made their radii infinitesimal—when the blue ball is on the hypotenuse.

The neat fact is that the dynamics of the two balls in 1D are faithfully represented by the Newtonian behavior of the one ball in the triangle. Collisions with the sides or with each other, at any velocities, become angle-preserving collisions with the sides. A proof may be found here (first pages) along with a representation of three constrained particles on a circle. The only thing we need to avoid is if the two balls hit the left and right sides simultaneously or hit each other against a side. That corresponds to the blue ball hitting a corner, a singular event we are entitled to ignore. Abracadabra, our executive is now gaming at billiards on a triangular table.

The last trick is the niftiest and works with any triangle—and more generally with polygons. We can reflect the triangle along one of its edges as diagrammed in a survey by Mirzakhani’s Stanford colleague Alex Wright which is a major source for our post:

The billiard trajectory becomes a straight line into the reflected copy. Obviously it would be nicer if we could analyze straight lines—that is, geodesics—in a larger space. When and how we can make the space may recall the tiling problems of our previous post but the rules are different. We need not tile in the plane but can use surfaces of arbitrary genus and metrics that allow angles greater than ${2\pi}$ around a conical point. This is where the special mathematical framework and tools for the work by Mirzakhani we are discussing enter in.

## Ultra-Simple Billiards Isn’t So Simple

We’ve exemplified that billiards can represent some other kinds of dynamical systems. Of course, billiards—even with just one ball—is interesting in itself. We can play it on tables shaped like other polygons besides triangles, or not polygons at all. Here are some questions we would like to answer:

1. If you shoot the ball from point ${x}$ in direction ${d}$, will it come back to point ${x}$ in that direction? That is, does its trajectory form a closed loop?

2. Does the trajectory come arbitrarily close to every point on the billiard table? Note that if it is closed then the answer is no.

3. Given points ${x}$ and ${y}$, can we shoot the ball from ${x}$ so that it goes to ${y}$?

4. How large—in a sense of measure or genericity—is the set of points and directions that form closed loops? Not to mention those that end in one of the (infinitesimally small) vertex “pockets”…

5. If ${x}$ and/or ${d}$ are changed slightly, how large and how easily computed are the effects on the trajectory as a function of elapsed time? That is, how “unpredictable” are the effects?

Some of these questions are challenging even for triangles. Every acute triangle has a closed loop that visits the three bases of the three altitudes, but it is not known whether every obtuse triangle has a closed loop at all. On a convex billiard table the answer to question 3 is immediately yes, but what about non-convex tables? If the edges are mirrors and ${x}$ is a candle, we are asking whether ${y}$ is illuminated—and how much if any of the surface remains in shadow. Although Wikipedia traces the question only to Ernst Straus in the early 1950s, I wonder if Newton thought of it during his work on multiple-prism arrays in his great treatise Opticks. This book by Serge Tabachnikov has more.

The questions become more attackable if we assume that every interior angle of the polygon ${P}$ is a rational multiple ${\pi}$. Then ${P}$ is called a rational polygon. There are only finitely many ways that ${P}$ can be iteratively reflected around one of its edges and the changes in orientation form a finite group ${G_P}$ that is dihedral. This is easy to visualize if the copies of ${P}$ tile the plane in the sense of the last post. Group theory and topology and abstract spaces extend our horizon because they can be used on polygons that don’t simply tile and allow us to apply the straight-line reflection trick.

## Translation Surfaces

A “clump” of non-overlapping polygons in the plane generates a translation surface if:

1. The set of edges of all the polygons has even size ${n = 2m}$ and can be partitioned into ${m}$ legal pairs.

2. A pair ${(e,e')}$ is legal if ${e}$ and ${e'}$ have the same size and orientation and the interior of the polygon ${e}$ belongs to is on the opposite side of the interior of the polygon ${e'}$ belongs to (which might be the same polygon).

3. If two polygons ${P}$ and ${P'}$ touch at an edge ${e}$ of ${P}$ then the coincident edge ${e'}$ of ${P'}$ must be paired with ${e}$. (This is automatically legal.)

For example, we can take a square and pair the opposite edges. Identifying them creates a torus. Besides the familiar 3D donut shape of a torus, we can picture it in 2D via how squares tile the plane. If we take a single octagon and identify the four pairs of opposite sides then all eight vertices become identified as shown below at left. We get a translation surface with angles summing to ${6\pi}$ at one vertex. This time octagons cannot tile the plane but we can still picture the space with algebraic help.

Two clumps generate the same space if one can be converted to the other by the operations of translating a polygon, bisecting a polygon along a diagonal, or doing the inverse of the latter to legally glue two polygons together. This equivalence relation is said to be difficult here but is evidently polynomial-time decidable.

We may also ignore interior edges; thus the reflections of the right triangle having smallest angle ${\pi/8}$—shown at right in our figure—are considered to yield the octagonal translation surface. Indeed, every translation surface can be presented by a single polygon (see section 12 of this) but not necessarily one that is convex.

Rotations and deformations of the polygons, however—shown in the middle of the figure—yield different spaces. We can describe those and other processes by groups acting on their coordinates. In the real plane there are two coordinates so we are talking about the general linear group ${GL(2,\mathbb{R})}$ of ${2 \times 2}$ matrices with real entries and its subgroups.

## The “Magic Wand” Theorem

The reflections of a rational polygonal billiard table yield a translation surface, but not every translation surface arises that way. What do we gain by the extra generality? What we gain are the algebraic tools and one more trick:

Instead of looking at different starting points for the billiard ball and rotating the direction in which it starts moving, we can look at rotations and linear stretchings of the translation surfaces. That is, instead of the orbit of the ball, we can study the algebraic orbit of the space under ${GL(2,\mathbb{R})}$ or some of its subgroups.

The orbits have their own spatial structure. This is one of the great features of representation theory conceived by Sophus Lie: groups of matrices acting on spaces ${T}$ form topological spaces ${T'}$ in their own right. Subgroups can be defined by parameters that act as coordinates for ${T'}$. So what happens when ${T}$ is a translation surface?

A simple answer was hoped for but experience with fractal behavior and chaos in related matters had restrained hopes of proving one. The answer by Mirzakhani in collaboration with Alex Eskin and joined by Amir Mohammadi was dubbed the “Magic Wand Theorem” in this survey by Anton Zorich:

Theorem 1 The closure of the ${GL(2,\mathbb{R})}$ orbit of a translation space is always a Riemannian manifold, moreover one definable by linear equations in periodic coordinates with zero constant term.

Despite the statement being simple and short the proof is anything but: almost half of the first paper’s 204 pages are devoted to approximation techniques employing random walks amid conditions of low entropy meaning low rate of divergence or “unpredictability.” Zorich says more about the wide panoply of techniques the proof brings together. Thus the ultimate dynamics were brainpower, knowledge, interaction, focus while assembling all the moving parts, and sheer hard work.

## What It Does

What does the “Magic Wand Theorem” do? To quote the title of a paper by Samuel Lelièvre, Thierry Monteil, and Barak Weiss, “Everything is Illuminated.” They solved question 3 above for rational polygons by showing that at most finitely many points remain in shadow—and illumination comes arbitrarily close to those points. It is just amazing that a simple question that Newton would have instantly understood needed such heft to answer. As they say in their abstract:

Our results crucially rely on the recent breakthrough results of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, and on related results of [Alex] Wright.

Wright’s survey also notes that Theorem 1 converts many results of the form ‘X happens in almost all cases (but we don’t know specifically which)’ into ‘X happens in all cases.’

The theorem also makes previous upper and lower bounds for certain counting problems coincide. Incidentally, one of the major results in Mirzakhani’s PhD thesis, cited in the article accompanying her Fields Medal, showed how to count simple closed geodesics in Riemannian manifolds as a function of their length ${L}$. The count can jump—e.g. when ${L}$ passes the length of a loop around a torus—but behaves nicely asymptotically.

## Marina Ratner’s Precursor Theorems

The amplification of previous knowledge also shows in the relation of Theorem 1 to a theorem by Marina Ratner that inspired it:

Theorem 2 Let ${G}$ be a Lie group and ${\Gamma}$ a finitely periodic structure within ${G}$—that is, a lattice. Let ${H}$ be a subgroup of ${G}$ definable by real matrices ${A}$ such that some power of ${A - I}$ is zero and the entries of ${A}$ are functions of one real parameter ${t}$. Then for every point in ${G/\Gamma}$, the closure of its orbit under ${H}$ is a manifold defined by homogeneous equations.

The Fields citation article calls Theorem 1 “a version of Ratner’s theorem for moduli spaces,” noting that the latter are “totally inhomogeneous.” It says Mirzakhani was thus able to transfer questions about dynamics on inhomogeneous spaces into nicer homogeneous cases. Other theorems by Ratner form a nexus that is all reflected in Mirzakhani’s work with Eskin and Mohammadi.

By sad coincidence, Marina Ratner also passed away earlier this month. Yesterday’s New York Times gave her a long obituary as well, noting how she did some of her best work after age 50 and that it was a basis for work by others including Mirzakhani. Jointly they provided much to inspire. Here are Mirzakhani in a still from a Harvard Math lecture video at the point where she introduced billiards and the illumination problem, and Ratner receiving an honorary doctorate from The Hebrew University in Jerusalem:

## Open Problems

Our most ambitious question is whether Mirzakhani’s work can be made to have a magic effect on orbit closure problems that some are trying to use to illuminate complexity theory.

Again we express our condolences to her family and colleagues.

Update 7/30: We received permission to use Assad Binakhahi’s beautiful memorial drawing titled “Unfinished Equation” from both the artist and Radio Farda, whose torch logo appears at its upper left. To them many thanks. The previous picture of Mirzakhani has been moved alongside the one of Marina Ratner, with an added sentence above them and some other word changes at top and here.margin

1. July 28, 2017 10:10 am

😥 such a tragedy that the 1st female winner of fields medal is no longer with us after only a few years winning the prize. she was clearly a private person, not revealing her health challenge at the time of the aware. a new valiant dimension of her character emerges in retrospect. coincidentally am very focused on dynamical equations wrt Collatz conjecture at the moment & feel have made great strides there, and that there is a lot of as-yet untapped possibility in this area, incl for other problems/ generalizations.

2. July 28, 2017 11:00 am

oh, and a strange irony Mirzakhani dying exactly at age 40, the cutoff age for fields medal, and the medals intention of being award to mathematicians with high future potential. reminds me of the expr “he who has the gold makes the rules” but have long had some mixed feelings on the 40 cutoff age for this award which does seem like a funky-bordering-on-questionable requirement sometimes wrt all other awards, which place no age limit. here is a case where it could be have said to have “seemingly backfired”…

• August 27, 2017 9:55 pm

vznvzn You commented that the age 40 cutoff point for eligibility to win the Fields Medal, “could be have said to have seemingly backfired” in the case of Maryam Mirzakhani.

Mirzakhani won the Fields Medal in 2014, when she was 37 (36?) years old. In his blog post about her life and death, Terrence Tao wrote,

it is remarkable how productive she still was mathematically in the last few years. Perhaps her greatest recent achievement has been her “magic wand” theorem with Alex Eskin…

Mirzakhani continued her work after winning the Fields Medal, and made substantial contributions, even prior to her untimely death at age 40. The decision to award her the Fields Medal in 2014, as a mathematician with high future potential, seems to have been a good decision, despite her death only a few years afterward.

3. July 28, 2017 12:30 pm

Re: Open Problems

Thoughts to the marriage of dynamics and logic —

&09758; Differential Logic

4. July 29, 2017 6:36 am

Aside from her day job, Maryam Mirzakhani also wrote a SODA paper: https://arxiv.org/pdf/1611.08339.pdf

5. September 8, 2017 5:26 am

Another Obituary:

Sameen Ahmed Khan,
Maryam Mirzakhani (1977-2017),
Current Science, 113 (5), 982-983 (10 September 2017).
(Fortnightly Publication of the Indian Academy of Sciences).

Sameen

Sameen Ahmed KHAN
Assistant Professor
Department of Mathematics and Sciences
College of Arts and Applied Sciences (CAAS)
Dhofar University
Post Box No. 2509
Postal Code: 211
Salalah
Sultanate of OMAN http://www.du.edu.om/