Marina
Ratner and Maryam Mirzakhani could not have been more different, in
personality and in background. Dr. Ratner was a Soviet Union-born Jew
who ended up at the University of California,
Berkeley, by way of
Israel. She had a heart attack at 78 at her home in early July.
Success
came relatively late in her career, in her 50s, when she produced her
most famous results, known as Ratner’s Theorems. They turned out to be
surprisingly and broadly applicable, with many elegant uses.
In
the early 1990s, when I was a graduate student at Berkeley, a professor
tried to persuade Dr. Ratner to be my thesis adviser. She wouldn’t
consider it: She believed that, years earlier, she had failed her first
and only doctoral student and didn’t want another.
Dr. Mirzakhani was a young superstar from Iran who worked nearby at Stanford University. Just 40 when she died of cancer in July, she was the first woman to receive the prestigious Fields Medal.
I
first heard about Dr. Mirzakhani when, as a graduate student, she
proved a new formula describing the curves on certain abstract surfaces,
an insight that turned out to have profound consequences — offering,
for example, a new proof of a famous conjecture in physics about quantum
gravity.
I
was inspired by both women and their patient assaults on deeply
difficult problems. Their work was closely related and is connected to
some of the oldest questions in mathematics.
The
ancient Greeks were fascinated by the Platonic solid — a
three-dimensional shape that can be constructed by gluing together
identical flat pieces in a uniform fashion. The pieces must be regular
polygons, with all sides the same length and all angles equal. For
example, a cube is a Platonic solid made of six squares.
Early
philosophers wondered how many Platonic solids there were. The
definition appears to allow for infinite possibilities, yet, remarkably,
there are only five such solids, a fact whose proof is credited to the
early Greek mathematician Theaetetus. The paring of the seemingly
limitless to a finite number is a case of what mathematicians call
rigidity.
Something
that is rigid cannot be deformed or bent without destroying its
essential nature. Like Platonic solids, rigid objects are typically
rare, and sometimes theoretical objects can be so rigid they don’t exist
— mathematical unicorns.
In
common usage, rigidity connotes inflexibility, usually negatively.
Diamonds, however, owe their strength to the rigidity of their molecular
structure. Controlled rigidity — that is, flexing only along certain
directions — allows suspension bridges to survive high winds.
Dr.
Ratner and Dr. Mirzakhani were experts in this more subtle form of
rigidity. They worked to characterize shapes preserved by motions of
space.
One
example is a mathematical model called the Koch snowflake, which
displays a repeating pattern of triangles along its edges. The edge of
this snowflake will look the same at whatever scale it is viewed.
The
snowflake is fundamentally unchanged by rescaling; other mathematical
objects remain the same under different types of motions. The shape of a
ball, for example, is not changed when it is spun.
Dr.
Ratner and Dr. Mirzakhani studied shapes that are preserved under more
sophisticated types of motions, and in higher dimensional spaces.
In
Dr. Ratner’s case, that motion was of a shearing type, similar to a
strong wind high in the atmosphere. Dr. Mirzakhani, with my colleague
Alex Eskin, focused on shearing, stretching and compressing.
These
mathematicians proved that the only possible preserved shapes in this
case are, unlike the snowflake, very regular and smooth, like the
surface of a ball.
The
consequences are far-reaching: Dr. Ratner’s results yielded a tool that
researchers have turned to a wide variety of uses, like illumining
properties in sequences of numbers and describing the essential building
blocks in algebraic geometry.
The
work of Dr. Mirzakhani and Dr. Eskin has similarly been called the
“magic wand theorem” for its multitude of uses, including an application
to something called the wind-tree model.
More
than a century ago, physicists attempting to describe the process of
diffusion imagined an infinite forest of regularly spaced identical and
rectangular trees. The wind blows through this bizarre forest, bouncing
off the trees as light reflects off a mirror.
Dr.
Mirzakhani and Dr. Eskin did not themselves explore the wind-tree
model, but other mathematicians used their magic wand theorem to prove
that a broad universality exists in these forests: Once the number of
sides to each tree is fixed, the wind will explore the forest at the
same fundamental rate, regardless of the actual shape of the tree.
There
are other talented women exploring fundamental questions like these,
but why are there not more? In 2015, women accounted for only 14 percent
of the tenured positions in Ph.D.-granting math departments in the
United States. That is up from 9 percent two decades earlier.
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Dr. Ratner’s theorems are some of the most important in the past half-century, but she never quite received the recognition she deserved. That is partly because her best work came late in her career, and partly because of how she worked — always alone, without collaborators or graduate students to spread her reputation.
Berkeley did not even put out a news release when she died.
By contrast, Dr. Mirzakhani’s work, two decades later, was immediately recognized and acclaimed. Word of her death spread quickly — it was front-page news in Iran. Perhaps that is a sign of progress.
I
first met Dr. Mirzakhani in 2004. She was finishing her Ph.D. at
Harvard. I was a professor at Northwestern, pregnant with my second
child.
Given her reputation, I expected to meet a fearless warrior with a single-minded focus. I was quite disarmed when the conversation turned to being a mathematician and a mother.
“How do you do it?” she asked. That such a mind could be preoccupied with such a question points, I think, to the obstacles women still face in climbing to math’s upper echelons.
At Harvard, the number of tenured women research mathematicians is currently zero. At my institution, the University of Chicago, until 2011 only one woman had ever held such a position.
We are only gradually joining the ranks, in what might be called a “trickle up” fashion.
Students often tell me that my presence on the faculty convinces them that women belong in mathematics. Though she would have shrugged it off, I was similarly inspired by Dr. Ratner.
I hope I played this role for Dr. Mirzakhani. And for all of her reticence about being famous, Dr. Mirzakhani has inspired an entire generation of younger women.
There are a surprising number of social pressures against becoming a mathematician. When you’re in the minority, it takes extra strength and toughness to persist. Dr. Ratner and Dr. Mirzakhani had both.
For the inspiration they provide, but above all for the beauty of their mathematics, we celebrate their lives.
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