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# Remembering Iranian Genius Maryam Mirzakhani, The Only Woman to Win Fields Medal

Maryam Mirzakhani, the Iranian mathematics genius, was born in Tehran in 1977. She attended school there and obtained her BSc in mathematics in 1999 from Sharif University of Technology in Iran. In 1994, while still in school, Mirzakhani achieved the gold medal level in the International Mathematical Olympiad, becoming the first female Iranian student to do so. In the 1995 Olympiad, she became the first Iranian student to achieve a perfect score, and won two gold medals. In 2004, Mirzakhani earned a PhD from Harvard University where she worked under the supervision of Fields medalist Curtis Mullen. In 2008, Mirzakhani joined Stanford University as a professor.

Maryam Mirzakhani was probably gifted with a lot of patience, because despite doing poorly in mathematics for a few years in school, she was mourned as one of the best mathematical minds in the world when she died on July 14. She was 40 years. She had been battling breast cancer since 2013; the disease spread to her liver and bones in 2016.

Mirzakhani rose to international fame when she became the first woman to win the Fields Medal in 2014, the most prestigious award in the field of mathematics. The 52 previous recipients had all been men. She won the prize for her “outstanding contributions to the dynamics and geometry of Rieman surfaces and their moduli spaces.” She shared the award with three male mathematicians. “Mastering these approaches allowed Mirzakhani to pursue her fascination for describing the geometric and dynamic complexities of curved surfaces spheres, doughnut shapes and even amoebas – in as great detail as possible.”

Surfaces are basic objects in mathematics, appearing in many guises. The surface of our planet is a sphere, but from local observations alone one cannot be sure of this: the Earth could be shaped like a bagel, for example, or a bagel with a few handles attached. A bagel-like surface is known in mathematics as a torus.

To make a torus, one can take a square piece of material and glue the bottom edge to the top to form a cylinder, then bend the cylinder and glue its ends together. A less distorted view of the torus is obtained by thinking of a square video screen with a character that wanders off the top only to reappear at the bottom directly below where it exited, and then wanders off the left edge but reappears at the right, moving with the same speed and direction at the same height. This character is living on a flat torus.

One can vary the shape of the torus by making the screen rectangular, or by skewing it to be a parallelogram (identifying points on opposite sides with a suitable shift). The variety of shapes that arise is described by a moduli space – an object where each point represents a specific flat torus.

If we replace our square screen with a regular octagon, retaining the rule that when our character disappears across an edge it emerges at the opposite edge, then our character is no longer living on a torus: we are now observing it moving around a “surface of genus 2”: a sphere is a surface of genus 0, a bagel is of genus 1, and genus 2 can be drawn as a bagel with a handle, and so a second hole. If we replace the octagon by more complicated polygons, we observe our character living on higher genus surfaces: we are looking at flat models for surfaces obtained from a bagel by attaching more handles.

With a rectangular screen, the four corners of our flat model fit together so that the glued-up torus is flat everywhere. In the higher genus case, naive gluing produces a cone point, so a distorted, non-flat geometry is needed to get a smooth, homogenous glued-up surface. This is hyperbolic geometry, which lies at the heart of much of what Maryam achieved. The moduli space for tori is itself a surface, but the moduli spaces for higher genus surfaces are high-dimensional objects whose beguiling structure is enormously rich, complicated, and presenting huge challenges to our understanding.

“I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.”

“Maryam was a wonderful colleague,” said Ralph L. Cohen, the Professor of Mathematics at Stanford. “She not only was a brilliant and fearless researcher, but she was also a great teacher and terrific PhD adviser. Maryam embodied what being a mathematician or scientist is all about: the attempt to solve a problem that hadn’t been solved before, or to understand something that hadn’t been understood before. This is driven by a deep intellectual curiosity, and there is great joy and satisfaction with every bit of success.”

A calm, modest and friendly person, with immense intellectual ambition, Maryam spoke eloquently about the fun that she had unravelling the intricate mysteries that she spent her life exploring, and the joy of getting to know the key characters that emerged and evolved in the unfolding of her mathematical plots – a joy that resonated with her childhood dream of becoming a writer. Her work had implications in fields ranging from cryptography to “the theoretical physics of how the universe came to exist”.

She will surely be remembered for good!