Marina Ratner, an influential mathematician and Russian-Jewish émigré who defied the notion that the best and the brightest in her field do their finest work when they are young, died on July 7 at her home in El Cerrito, Calif. She was 78.
The cause was cardiac arrest, said her daughter, Anna Ratner.
Friends and colleagues have said that Dr. Ratner started as a good but unexceptional mathematician. “The beginning of her career was not particularly promising,” said Anatole Katok, of Pennsylvania State University, who met her in the early 1960s when both lived in the Soviet Union.
A common belief is that a mathematician who does not do great work by age 40 never will. But Dr. Ratner was about that age when she set off on an ambitious effort to connect the physics of the motion of objects with more abstract ideas of number theory.
She proved her most influential theorem after she turned 50.
“She struggled and went unrecognized for a long time,” said Artur Avila, a mathematician at the French National Center for Scientific Research in Paris and the Institute for Pure and Applied Mathematics in Rio de Janeiro. “She is also one of the main examples to counter the myth that mathematics is a young person’s game.”
Amie Wilkinson, of the University of Chicago, pointed to the classic video game “Asteroids” to explain Dr. Ratner’s work.
“When an object crosses the right edge of the screen, it instantaneously appears on the left edge of the screen moving in the same direction,” Dr. Wilkinson said in an email. “The simplest case of Ratner’s Theorem applies to the motion of objects in the ‘Asteroids’ game. If an object moves in a straight line indefinitely without turning, there are only two possibilities: Either it will return to its initial position and repeat its path indefinitely, or it will never return to its initial position and instead will visit every possible region of the space.”
Dr. Ratner showed that for a higher dimensional, more complicated video game, an equivalent statement was true.
Her dynamics research helped unravel mathematical problems that had resisted more direct, traditional approaches of attack.
Dr. Avila said Dr. Ratner’s work had been the basis for that of younger mathematicians like Elon Lindenstrauss and Maryam Mirzakhani, two winners of the Fields Medal, the most prestigious honor in mathematics. Dr. Mirzakhani, the first woman to win a Fields, also died this month.
“What is remarkable about these results of Ratner is how many unexpected applications they had,” Dr. Lindenstrauss said in an email. “It is almost as if this dynamical fact was a philosopher’s stone that allowed many mathematicians to show quite remarkable things, in remarkably diverse situations.”
Marina Ratner was born on Oct. 30, 1938, in Moscow, the daughter of scientists. She graduated from Moscow State University in 1961. She then worked for four years as an assistant for Andrey Kolmogorov, a prominent Russian mathematician, before attending graduate school at Moscow State. She completed her doctorate in 1969.
When she applied for a visa to leave the Soviet Union for Israel, Dr. Ratner was pressured to leave a teaching job at the High Technical Engineering School in Moscow, Anna Ratner said.
“She had a very hard time in Russia,” said Alexandre Chorin, a colleague at the University of California, Berkeley. “The Russians took a variety of steps to penalize her.”
Dr. Ratner and her daughter immigrated to Israel in 1971, where she was a lecturer at the Hebrew University of Jerusalem. She was able to pursue her mathematical research but was unable to find a permanent position.
Her work caught the attention of Rufus Bowen, however, at Berkeley, and he lobbied the university to hire her. It did, in 1975, initially for a temporary position, and even that, given her relatively meager record, was controversial in the department. She eventually became a tenured professor.
Dr. Ratner was a member of the American Academy of Arts and Sciences and the National Academy of Sciences.
In addition to her daughter, she is survived by two grandchildren.
A marriage to Alexander Samoilov ended in divorce.
Dr. Ratner’s style of working may have contributed to her not receiving as much acclaim as some thought she deserved. She always worked alone. At Berkeley, she earned high marks as a teacher of undergraduates but taught only one graduate student.
She also did not always do the best job of promoting her achievements.
Étienne Ghys, a mathematician at École Normale Supérieure de Lyon in France, recalled spending six months trying to understand the results of her dynamics research to present them at a seminar. When he discussed the papers with her, he told her that he had the feeling that she had written the papers not for other mathematicians to understand but mainly to convince herself that the theorems were correct.
Dr. Ghys said Dr. Ratner replied: “Yes! Exactly! You understood why and how I write mathematics.”
The mathematics section of the National Academy of Sciences lists 104 members. Just four are women. As recently as June, that number was six.
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.
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.