Maryam Mirzakhani
The Iranian mathematician who unlocked deep structural truths about the geometry of curved surfaces, becoming the first woman to receive mathematics' highest honour.
Maryam Mirzakhani grew up in Tehran, won gold medals at the International Mathematical Olympiad two years running, and went on to produce research so original that her peers described it as opening entire new continents in mathematics. In 2014 she became the first woman and the first Iranian to receive the Fields Medal — the discipline's most prestigious prize — for her work on the dynamics and geometry of Riemann surfaces and their moduli spaces.
From Tehran to the olympiad stage
Mirzakhani was born in Tehran in 1977 and showed an early passion for reading and storytelling before mathematics captured her fully. As a teenager she attended the Farzanegan school for gifted students and qualified for the Iranian national olympiad team. At the 1994 International Mathematical Olympiad she achieved a perfect score and took gold; she repeated the feat the following year. Those back-to-back perfect scores announced a mathematician of exceptional reach.
She completed her undergraduate degree at Sharif University of Technology in Tehran, then moved to the United States for doctoral study at Harvard University. There she worked under Fields Medalist Curtis McMullen, completing her PhD in 2004 with a thesis that solved two long-open problems simultaneously — on the count of simple closed geodesics on hyperbolic surfaces and on the Weil–Petersson volumes of moduli spaces of curves. The thesis was so substantial it produced three separate papers, each published independently.
Research: geometry in motion
Mirzakhani's mathematical world was one of curved surfaces — the hyperbolic plane and the Riemann surfaces that generalise it. A central object of study was the moduli space: the space of all possible geometric structures a surface of given topology can carry. These moduli spaces are themselves enormously complex objects, and understanding how geodesics — the shortest paths on a curved surface — wind and close upon themselves inside them was the core problem she attacked.
Her doctoral work produced a formula, now called the Mirzakhani recursion, that computes Weil–Petersson volumes of moduli spaces. The formula had immediate applications in string theory and combinatorics. Subsequent joint work with Alex Eskin and Amir Mohammadi — described by fellow mathematicians as the "magic wand theorem" — proved a profound rigidity result for the action of certain groups on moduli space, establishing that every orbit closure is an algebraic submanifold. The proof ran to over two hundred pages and settled questions that had been open for decades.
Her approach was characteristically visual and geometric. Colleagues recalled her filling large sheets of paper with diagrams, working through spatial intuition before turning to formalism.
Stanford and the Fields Medal
After a postdoctoral position at Princeton, Mirzakhani joined the faculty at Stanford University in 2008, where she remained as professor of mathematics until her death. In August 2014 she received the Fields Medal at the International Congress of Mathematicians in Seoul — the first woman in the prize's seventy-eight-year history to do so, and the first Iranian. The citation was for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
The announcement drew global attention not only for its mathematical content but as a moment of symbolic importance. Mirzakhani was characteristically measured about it, expressing hope that younger girls in Iran and elsewhere would see mathematics as a domain open to them.
The beauty of mathematics only shows itself to more patient followers.
