Maryam Mirzakhani was an Iranian mathematician who made several important contributions to geometry and was the first woman to win a Fields Medal in 2014. Known for her creativity and courage in tackling challenging problems, her work connected the fields of hyperbolic geometry, complex analysis, topology, and dynamics. (Image by Maryeraud9 shared under a CC BY-SA 4.0 license)
The International Women in Mathematics Day on 12 May commemorates the birthday of Maryam Mirzakhani, who made significant contributions to the geometry of curved surfaces. This year, over 200 events were held across the world to honour her legacy and celebrate the work of women mathematicians. The Institute of Mathematical Sciences (IMSc) celebrated the day by hosting research talks by women scientists from various institutions.
The event hosted at IMSc opened with a talk about ‘Adjoint of ideals’ by Clare D’Cruz from the Chennai Mathematical Institute. Jaya N Iyer from IMSc presented a talk on ‘Rational connectedness of algebraic varieties’, followed by a talk on ‘Arithmetic of moments of elliptic curves’ by V Uma from Indian Institute of Technology (IIT), Madras. The final two talks were on ‘The art of colouring: Fast and exact with subset convolution’ by IMSc’s Sushmita Gupta and ‘Arithmetic of moments of elliptic curves’ by Bidisha Roy from the IIT Tirupati.
Participants at the International Women in Mathematics Day celebrations at IMSc. (Image: Pritesh Kumar)Maryam Mirzakhani: A spirited surfer of surfaces
Maryam Mirzakhani was an Iranian mathematician who did pathbreaking research on the geometry of surfaces. She was a Professor at Stanford University who became the first woman to win the Fields Medal for her contributions to mathematics.
Her interest in mathematics began when she was a school student in Tehran. Mirzakhani and her constant friend, Roya Beheshti Zavareh, were among the first girls in Iran to participate in national-level mathematics competitions. Initially driven by the challenge of these competitions, Mirzakhani soon realised that she really enjoyed doing mathematics. She co-wrote her first paper while still in school, a proof of an open problem introduced during an Olympiad workshop. In 2014, Mirzakhani and Beheshti earned a place in the Iranian team representing the International Mathematical Olympiad, where Mirzakhani won gold and Beheshti won silver. Mirzakhani struck gold again the following year with a perfect score.
Mirzakhani finished her undergraduate studies in Sharif University of Technology, Tehran when she co-wrote an introductory book on number theory in Farsi, along with Beheshti. While aimed at school students preparing for the mathematics Olympiad, it also carries deeper insights into the subject. Mirzakhani went on to do her PhD with the mathematician Curtis T McMullen, a Fields Medalist himself, from Harvard University.
For her doctoral thesis, Mirzakhani counted paths on the surface of a donut-shaped object called a torus. Starting from a point and walking in a straight line on such a surface, it is possible to take many different kinds of paths. Imagine a tiny ant on top of a donut laid flat. If the ant walked on its surface in a straight line taking the shortest path to the bottom of the donut and walked further, it would simply end up where it started. This path would fall on the edge of a vertical cross-section of the donut. Apart from such paths that close in on themselves, it is also possible to take other straight paths where the ant would keep walking forever. For her PhD, Mirzakhani studied the number of closed, non-crossing paths on a torus with two holes (imagine a pair of donuts stuck together in a figure ‘8’). To do this, she used a mathematical concept called “moduli space” that maps the different geometric structures possible on a surface.
“She would start describing sort of elaborate stories or mathematical narratives. And these narratives were very ambitious and speculative, they were almost like science fiction,” says McMullen in a 2020 documentary. “She was very adept at finding the right question.”
In her later work with Alex Eskin, when she was a faculty member at Stanford University, Mirzakhani worked on studying all the possible paths a billiards ball can take on a pool table. But not just a regular rectangular pool table, but tables with diverse shapes differing in their number of edges.
“Trying to prove a theorem is like trying to climb a mountain which nobody has ever climbed,” says Eskin in the film. “And at some point we kind of felt like we could see the top, but then there is this ravine in front of us,” he says, about facing formidable challenges just when they thought the solution was in reach. “It was an incredibly intense experience. Maryam was just absolutely amazing, both in terms of the mathematics, but also the mental stability aspect, which is necessary for this kind of work.”
Mirzakhani was diagnosed with breast cancer in 2013, a year before she won the Fields Medal, and passed away in 2017, a tremendous loss to the mathematical community.
“Her work on Riemann surfaces and their moduli spaces bridges several mathematical disciplines – hyperbolic geometry, complex analysis, topology, and dynamics – and influences them all in return… Fluent in a remarkably diverse range of mathematical techniques and disparate mathematical cultures, she embodies a rare combination of superb technical ability, bold ambition, far-reaching vision and deep curiosity,” reads her Fields Medal citation.
