{\huge Maryam Mirzakhani, 1977--2017}
Including debt to Marina Ratner, 1938-2017
\vspace{.25in} 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 condolences 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 calls it ``an infant compared to the other major branches of math.'' This sounds at first silly---dynamics has been studied since long before Isaac Newton. We have covered the mathematical techniques that were used over a hundred years ago to show how Newtonian dynamics of three or more bodies is unsolvable. What Ellenberg means is that abstraction away from physics was needed to foster mathematical tools to analyze and that this took root only in the second half of the 1900s.
We can put it this way: Dynamics has always been a moving target for mathematics, hard to pin down amid the tendency of many systems to show chaotic behavior. There are senses in which Mirzakhani and others have made it more of a fixed target. 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 fixed geometric object by which one can 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 for each input length $latex {n}&fg=000000$ whereas Turing machines move, but there is potentially a more powerful 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 balls on strings shown at left. Usually there are five identical balls as at left, but let's say a junior executive might start with just two like in the middle.
\includegraphics[width=5in]{ExecutiveToys.png}
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 $latex {(x,d)}&fg=000000$ where $latex {x}&fg=000000$ is the displacement of the left ball from the left end and $latex {d}&fg=000000$ 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:
\includegraphics[width=2.5in]{TriangleReflection.png}
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 $latex {2\pi}&fg=000000$ 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:
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 $latex {x}&fg=000000$ is a candle, we are asking whether $latex {y}&fg=000000$ 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 $latex {P}&fg=000000$ is a rational multiple $latex {\pi}&fg=000000$. Then $latex {P}&fg=000000$ is called a rational polygon. There are only finitely many ways that $latex {P}&fg=000000$ can be iteratively reflected around one of its edges and the changes in orientation form a finite group $latex {G_P}&fg=000000$ that is dihedral. This is easy to visualize if the copies of $latex {P}&fg=000000$ 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.
Translational Spaces
A ``clump'' of disjoint polygons in the plane generates a translation surface if:
For example, we can take a square and pair the opposite edges. Identifying them creates a torus. Or we can take four squares in a $latex {2 \times 2}&fg=000000$ box and choose to pair a vertical edge on the left either with its same-level counterpart on the right or ``switch'' to the other right-side edge. If we similarly ``switch-pair'' the top and bottom edges then we get an identification that cannot be pictured via a tiling in the plane. If we take a single octagon and identify the four pairs of opposite sides then all eight vertices become identified. We get a translation surface with angles summing to $latex {6\pi}&fg=000000$ at one vertex---again this is impossible to do in the plane but can be neatly described algebraically.
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 hard here but is evidently polynomial-time decidable.
\includegraphics[width=4.5in]{Octagons4.png}
We may also ignore interior edges; thus the reflections of the right triangle having smallest angle $latex {\pi/8}&fg=000000$---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 $latex {GL(2,\mathbb{R})}&fg=000000$ of $latex {2 \times 2}&fg=000000$ 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 $latex {GL(2,\mathbb{R})}&fg=000000$ 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 $latex {T}&fg=000000$ form topological spaces $latex {T'}&fg=000000$ in their own right. Subgroups can be defined by parameters that act as coordinates for $latex {T'}&fg=000000$. So what happens when $latex {T}&fg=000000$ 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 $latex {GL(2,\mathbb{R})}&fg=000000$ 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. The difficulty seems inherent in showing that ``bad'' mathematical objects (here, cases of the orbit closure) cannot arise: if they did then they would have certain approximations that are concrete enough to show a contradiction. The proof is a real tour de force requiring much force. But the simple manifolds it provides are nice sitting targets for analysis.
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 ``$latex {X}&fg=000000$ happens in almost all cases (but we don't know specifically which)'' into ``$latex {X}&fg=000000$ 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 $latex {L}&fg=000000$. The count can jump---e.g. when $latex {L}&fg=000000$ 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 $latex {G}&fg=000000$ be a Lie group and $latex {\Gamma}&fg=000000$ a finitely periodic structure within $latex {G}&fg=000000$---that is, a lattice. Let $latex {H}&fg=000000$ be a subgroup of $latex {G}&fg=000000$ definable by real matrices $latex {A}&fg=000000$ such that some power of $latex {A - I}&fg=000000$ is zero and the entries of $latex {A}&fg=000000$ are functions of one real parameter $latex {t}&fg=000000$. Then for every point in $latex {G/\Gamma}&fg=000000$, the closure of its orbit under $latex {H}&fg=000000$ 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.
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 sorrow and appreciation.