Looking at the universe in motion, mathematically

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. Marina was a Soviet Union-born Jew who ended up at the University of California, Berkeley, USA, 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.

Maryam was a young superstar from Iran who worked at Stanford University, USA. Just 40 when she died of cancer in July, she was the first woman to receive the prestigious Fields Medal. I first heard about Maryam 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 like 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. In common usage, rigidity connotes inflexibility, usually negatively. Diamonds, however, owe their strength to the rigidity of their molecular structure.

Marina and Maryam were experts in this more subtle form of rigidity. They worked to characterise 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. Marina and Maryam studied shapes that are preserved under more sophisticated types of motions, and in higher dimensional spaces.

In Marina’s case, that motion was of a shearing type, similar to a strong wind high in the atmosphere. Maryam, with 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.

The consequences are far-reaching: Marina’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 Maryam and Alex has similarly been called the ‘magic wand theorem’ for its multitude of uses.

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. Maryam and Alex 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.

(The author is a professor of mathematics at the University of Chicago, USA)