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An expert on curved spaces

Iranian mathematician Maryam Mirzakhani, the first woman to win the Fields Medal, dies at age 40 and leaves behind a unique legacy

Stanford University Maryam Mirzakhani made contributions to the dynamics and geometry of complex surfacesStanford University

Awarded every four years, the Fields Medal recognizes up to four mathematicians under 40 who have made exceptional contributions to the discipline. In the first 17 prize cycles between 1936 and 2010, the International Mathematical Union (IMU) awarded 51 medals, all of which went to men, evidence that male dominance in this area of ​​study can be overwhelming in more competitive strata of the career. This trend was broken in 2014, when US-based Iranian Maryam Mirzakhani, a professor at Stanford University, became the first woman to be honored.

Responsible for original contributions on the dynamics and geometry of Riemann surfaces and described as a serene figure with tremendous intellectual ambition, Mirzakhani was nevertheless surprised by the award. She ignored the IMU’s first notice that she would receive the medal—she thought the e-mail was a joke. When she was officially notified by telephone, she had to reveal a personal problem. A victim of breast cancer, she was recovering from a cycle of chemotherapy, and she was not sure she would be able to attend the ceremony or whether she would be prepared to face the attention of the press. In the end, she attended the event. A human shield formed by friends spared her harassment at the ceremony held in Seoul, South Korea, on August 13, 2014. Thirty-five months later, Mirzakhani died in a hospital in the United States on July 14 as a result of her cancer, which had spread to her bone marrow. She was 40 years old.

Mirzakhani was defined as a “master of curved spaces” by Canadian mathematician Manjul Bhargava of Princeton University, who also won the Fields Medal in 2014. “Everyone knows that the shortest distance between two points on a flat surface is a straight line. But if the surface is curved—for example, the surface of a ball or a donut—then the shortest distance…will also be along a curved path, and can thus be more complicated. Maryam proved many amazing theorems about such shortest paths—called ‘geodesics,’” he told The New Yorker.

Bhargava and Mirzakhani were never research partners, but in a funny situation, they solved a simple combinatorial problem together, The New Yorker article reports. After receiving the medals in Seoul, they realized that the organization’s presenters had not noticed that the name of each winner was engraved on their medals. Bhargava had been handed the medal that belonged to British mathematician Martin Hairer, who, in turn, had received Mirzakhani’s, who herself had received the award meant for Artur Ávila, the first Brazilian to win the honor (see Pesquisa FAPESP, issue No. 223). Ávila, meanwhile, had received Bhargava’s medal. The festive atmosphere made it difficult for the four recipients to gather together in one place. After sharing a laugh over the situation, Bhargava and Maryam discussed what would be the fastest way for each pair to meet up and exchange medals.

With a degree in mathematics from Sharif University of Technology in Tehran, Mirzakhani moved to the United States to pursue a PhD at Harvard and began working at Stanford in 2009. In the house where she lived with her husband, Czech computer scientist Jan Vondrák, and her daughter, Anahita, now 6 years old, she used the floor to write on huge sheets of paper, sketching ideas, formulas, and diagrams of hyperbolic surfaces—abstract surfaces with donut-like shapes with one or more holes in which distances and angles are measured according to a certain set of equations. On some hyperbolic surfaces, the shortest path between two points may be a long geodesic, while on others, it may be a short loop, like the circle of a sphere.

Her doctoral dissertation, which was precisely on these geodesics on surfaces of hyperbolic space, was described as a highly original contribution. “It’s the kind of mathematics you immediately recognize belongs in a textbook,” Alex Eskin, a mathematician at the University of Chicago, told Quanta magazine. Eskin, Mirzakhani, and mathematician Amir Mohammadi of the University of California, San Diego worked together on a project on the dynamics of abstract surfaces connected to billiard tables that culminated in the solution of the so-called “magic wand theorem,” which relates to entire spaces composed of hyperbolic surfaces.

The researcher’s notes

Exceptional Talents
Mirzakhani was born and raised in Iran. In middle school, one of the teachers discouraged her from becoming a mathematician, saying she did not have any particular talent for the subject. But she went to an all-girls high school in Tehran run by the country’s National Organization for Exceptional Talent Development. In 1995, at the age of 18, she won a gold medal at the 36th International Mathematical Olympiad in Toronto, Canada. Three days after her death, Maryam Mirzakhani was honored at the opening of the 58th year of the Olympiad held in Rio de Janeiro. Gender disparity in mathematics, a problem that Mirzakhani’s example sought to combat, was also a focus of the most recent event, which, for the first time, awarded a trophy to the five girls who contributed the most to the success of their teams. Of the 623 high school students from the 112 countries that participated in the Olympiad, only 65 were girls. Among the representatives of Brazil, there were only boys.

According to Marcelo Viana, director of the Brazilian Institute for Pure and Applied Mathematics (IMPA), the cultural pressure against women in mathematics is persistent and feeds on the false idea that women have a natural propensity for other fields of study, such as the humanities, but not for harder or abstract areas of knowledge. “This is nonsense that perpetuates itself. We have to believe that it is possible to reverse the situation in the medium and long term and act accordingly,” he says. “If there are few examples to provide inspiration, girls end up thinking math is not for them, which feeds a vicious circle.”