The universe in motion
The mathematics section of the National Academy of Sciences lists 104 members. Just four are women. As June, that number was six.
Marina Ratner and Mary am Mirzakhani could not have been more personality and in background. 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 Rat ne r’ 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 Ra tn er 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.
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 Mirzakhani when, as a she proved a new formula describing the curve son certain abstract surfaces, an insight that turned out to have profound consequences—offering, for example, a new proof of a famous conjecture in quantum gravity.
I was inspired by both women and their patient assault son deeply difficult problems. Their work was closely related an disconnected 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 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 manythere 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 limit less to a finite number is a case of what mathematicians call rigidity.
Something that is rigid can not 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 in flexibility, 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.
Rat ne rand Mir zak ha ni were experts in this more subtle form of rigidity. They worked to characterize shapes preserved by motions of space.
One example is a mathematicalmodel 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 fundamentallyunchanged by re scaling; other mathematicalobjects remain the same under different types of motions. The shape of a ball, for example, is not changed when it is spun.
Rat ne rand Mir zak ha ni studied shapes that are preserved under more sophisticated types of motions, and in higher dimensional spaces.
In Rat ne r’ s case, that motionwas of as hearing type, similar to a strong wind high in the atmosphere. Mir zak ha ni, with my colleague Alex E skin, focused ons hearing, stretching and compressing.
These mathematicians proved that the only possiblepreserved shape sin this case are, unlike the snowflake, very regular and smooth, like the surface of aball.
The consequences are far-reaching: 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 Mir zak ha ni and E skin has similarly been called the“magic wand theorem” for its multitude of uses, including an applicationto something called the wind-tree model.
More than a century ago, physicists attempting to describe the process of diffusionimagined 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.
Mirzakhani and E skin did not themselves explore the wind-tree-model, but other mathematicians used their magic wand theorem to prove that abroad 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 fundamentalquestions 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.
Ratner’s theorems are some of the most important in the past half-century, but she never quite received the recognition she de served. That is partly because her best work came late in her career and partly because of how she worked — always alone, without collaboratorsor graduate students to spread her reputation.
Berkeley did not even put out a news release when she died.
By contrast, 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 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 fear less warrior with a single-minded focus. I was quited is armed 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.
There area surprising number of social pressures against becoming a mathematician.When you’ re in the minority, it takes extra strength and toughnessto persist. Rat ne rand Mir zak ha ni had both.
For the inspiration they provide, but above all for the beauty of their mathematics, we celebrate their lives.