“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.” – Maryam Mirzakhani
Is the Earth flat? Hopefully by now, it is common knowledge that the Earth is indeed not flat, but rather holds a spherical shape. Yet decades ago, there was a point in which our ancestors believed that the Earth was flat. And this wasn’t without reason, as any patch of Earth that you stand on feels very much like a two-dimensional plane. This is because the Earth is an example of a mathematical surface – a shape that can be covered in overlapping pieces, all of which can be mapped on a plane, making it homeomorphic to the plane. And there are many different types of mathematical surfaces, and they all depend on the number of holes in it. For example, a donut shaped figure contains one hole, and is therefore different from a spherical surface with no holes or a pretzel shaped surface with three holes. To put a name to this concept, a donut shaped surface with two or more holes that have a non-standard geometry is called a hyperbolic surface. And geometric objects whose points each represent a different hyperbolic surface are known as Riemann surfaces. This way of looking at mathematical surfaces is very abstract, but necessary in order to understand the work of Maryam Mirzakhani, who is the subject of this week’s Wonder Woman Wednesday post!
Maryam Mirzakhani was born and raised in Tehran, Iran where she grew up in the midst of the war-torn country. Despite early challenges, she considers herself lucky in that she was able to attend a good high school and further her education by obtaining a Ph.D in mathematics at Harvard University. Mirzakhani’s interest in mathematics began in high school when she discovered her fascination with solving mathematical problems and treating them like puzzles. She is often complimented for her strong geometric intuition which allows her to grapple directly with difficult subjects like the geometry of moduli spaces. Her rare combination of superb technical ability, bold ambition, far reaching vision, and deep curiosity has led her to the success and accomplishments she has accrued over the course of her life so far. Now a mathematics professor at Stanford University, Mirzakhani was the first woman to win the prestigious Field’s Medal in 2014.
The Field’s Medal is dubbed the Nobel Prize of Mathematics and is considered the most prestigious honor a mathematician can receive. It is officially titled the International Medal for outstanding Discoveries in Mathematics and was started in 1936. Mirzakhani’s winning of the prize in 2014 marks the first time a woman has ever received this honor and it shatters another barrier for women in STEM fields worldwide. In a way, the Field’s Medal is stacked against women in that it is restricted to mathematicians younger than 40, which are the years in which many women dial back their careers to raise children. Furthermore, mathematics itself is a field that is still highly dominated by men, and according to The Washington Post, only 9 percent of tenure-track positions in math are held by women. Despite the odds being clearly against women in this field, Mirzakhani’s win is a testament to the fact that women can in fact become successful in mathematics, and is an accomplishment to be celebrated by all women in STEM. Mirzakhani herself has said “This is a great honor. I will be happy if it encourages young female scientists and mathematicians,” and “I am sure there will be many more women winning this kind of award in coming years.”
So what led to her winning of this award? Mirzakhani’s earliest work involved solving the problem of calculating the volumes of moduli spaces of curves on Riemann surfaces. She solved this by drawing a series of loops across their surfaces and calculating their lengths. She worked with geometry and dynamical systems and won the Field’s Medal for her sophisticated and highly original contributions to the fields of geometry and dynamical systems, particularly in understanding the symmetry of curved surfaces. For instance, geodesics are straight lines on a hyperbolic surface, and the number of closed geodesics of a given length on a hyperbolic surface has long been known to grow exponentially as the length of the geodesics grows according to this expression:
Yet for the longest time, mathematicians couldn’t figure out just how many simple closed geodesics of a given length a hyperbolic surface can have. Mirzakhani answered this problem in her doctoral dissertation in 2004. She developed a formula for how the number of simple geodesics of length L grows as L gets larger. Turns out that, the number whose length is less than or equal to L grows much more slowly, according to the expression below, where g is the number of holes of the surface.
All of this work is extremely abstract and hard to conceptualize. The bottom line is that moduli space is a world in which many new discoveries await, and it is up to people like Mirzakhani to lead the explorations for more mathematical discoveries. She summarizes her passion for pure mathematics by stating that “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.” Hopefully she can indeed “find a way out” as these mathematical discoveries have great significance in many fields including physics and quantum field theory.
For more information about Maryam Mirzakhani and her research interests, check out the following video below by WIRED Science!
Article Written By: Lisa He