With Snowflakes and Unicorns, Marina Ratner and Maryam Mirzakhani Explored a Universe in Motion

Amie Wilkinson in The New York Times:

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.

…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 here.