Stanford mathematician Maryam Marzakhani has just won the Fields Prize, often called the Nobel Prize in mathematics. The first woman to win the prize, Marzakhani is best known for her work on curved surfaces, including on spheres and hyperbolic objects.
Marzakhani's work has been described as esoteric and elegant, but with implications for everything from cosmology to materials science.
According to a release from Stanford:
The award recognizes Mirzakhani's sophisticated and highly original contributions to the fields of geometry and dynamical systems, particularly in understanding the symmetry of curved surfaces, such as spheres, the surfaces of doughnuts and of hyperbolic objects. Although her work is considered "pure mathematics" and is mostly theoretical, it has implications for physics and quantum field theory ...
Mirzakhani's recent research further investigates the symmetry of surface geometry, particularly within theories regarding Teichmüller dynamics. In general, her work can best be described as pure mathematics – research that investigates entirely abstract concepts of nature that might not have an immediately obvious application.
"Oftentimes, research into these areas does have unexpected applications, but that isn't what motivates mathematicians like Maryam to pursue it. Rather, the motivation is to understand, as deeply as possible, these basic mathematical structures," said Ralph Cohen, a professor of mathematics and the senior associate dean for the natural sciences in Stanford's School of Humanities and Sciences. "Maryam's work really is an outstanding example of curiosity-driven research."
The work, however, could have impacts concerning the theoretical physics of how the universe came to exist and, because it could inform quantum field theory, secondary applications to engineering and material science. Within mathematics, it has implications for the study of prime numbers and cryptography. Despite the breadth of applications of her work, Mirzakhani said she enjoys pure mathematics because of the elegance and longevity of the questions she studies.
"I don't have any particular recipe," Mirzakhani said of her approach to developing new proofs. "It is the reason why doing research is challenging as well as attractive. It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out."
Hyperbolic objects are curly, abstract shapes that are often represented using crochet, like this one created by mathematician Daina Taimina. "The beauty of mathematics only shows itself to more patient followers," Mirzakhani told the Clay Mathematics Institute. In other interviews, she has said that when she was growing up in Iran, her greatest wish was to become a novelist. But then she became fascinated by the elegance of mathematics, which set her on the career path she's still on today.
You can learn more about Mirzakhani's work and life in this article from Quanta Magazine.