Maryam Mirzakhani: Remembering a Mathematical Pioneer

Hello everybody! Today, we're going to be talking about one of the great female mathematicians, Maryam Mirzakhani.

Mirzakhani was the first woman to win the Fields Medal, the equivalent of a Nobel Prize in math, which she won just over five years ago in 2014. Mirzakhani was investigating new and different types of geometry, such as hyperbolic geometry (when space curves inward) and those originally discovered by the great mathematician Bernhard Riemann over a century before her. Officially, she won the Fields Medal for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."

She made dramatic and spectacular advancements to mathematics as a field, and she was invaluable to the community. She taught at Stanford University and fostered a legacy of inspiring young girls to pursue math.

In 2013, Mirzakhani was diagnosed with breast cancer. In 2016, the cancer spread to her bones and liver, and she died on July 14, 2017 at Stanford Hospital when she was only 40 years old. This tragedy was not only a blow to her family and everyone who knew her but also to the entire math community.

Before I continue, watch this video! This is beautifully made from the Simons Foundation and it'll introduce this perfectly:

In remembering her life, I'll start with Mirzakhani's childhood. She grew up in Tehran, Iran, and she originally wanted to be a writer. The Iran-Iraq war was raging for eight years throughout her childhood, and though she grew up in a supportive, encouraging and STEM-oriented family (her father was an electrical engineer), there were many hard times throughout her childhood and the war.

As a child, Mirzakhani succeeded in almost every class, except math. The way math was taught at her school simply didn't agree with her, and she was angry and frustrated with the subject.

Mirzakhani as a child. Photo source unknown.

Naturally resilient and determined to do better, Mirzakhani eventually discovered the more passionate, poetic side of math and fell in love with it. She and her best friend, Roya Beheshti, became inspired to explore high-level challenging math problems in old International Mathematical Olympiad competitions, the world championship for math for high school students.

The two petitioned for a higher-level math class at their all-girls high school, and pretty soon, Mirzakhani and Beheshti became the first Iranian women to qualify for the International Mathematical Olympiad. Two years in a row, Mirzakhani took home a gold medal, and in 1995, she gained even more recognition. She had scored a perfect score.

Mild digression here: I think Mirzakhani's dramatic change in thought in rediscovering math is a perfect illustration of the kinds of ideas I've talked about in last week's article, "Stop Teaching Our Students to Fear Math" and my first article, "Math: The Poetry of Ideas," so I'd love it if you'd read those after this!

After her impressive launch to the top of the IMO, Mirzakhani stayed in Iran for her undergrad degree. She attended the Sharif University of Technology in Tehran and even published three papers throughout her time there.

On a field trip to a university technology competition, Mirzakhani's bus skidded into a ravine, a car crash that killed seven of her peers and two bus drivers. She survived and went on to graduate from Sharif University.

In 1999, Mirzakhani decided to move to the United States for graduate school at Harvard where she also got her PhD. There, she began to explore the subjects that would eventually become the substance for her Fields Medal.

One was hyperbolic geometry, where space is shaped like a Pringles chip or a coral reef. In hyperbolic geometry, space always curves inward, the opposite of a sphere which has outward or positive curvature. In typical geometry, if you draw a line and a point, there's only one line parallel to your line through your point, but in hyperbolic geometry, there's an infinite number of parallel lines! (I'd love to write an article about these geometries, called non-Euclidean geometries, so just let me know if that's something you'd like to see!)

A group of women that figured out how to crochet hyperbolic space were some of the first to show mathematicians how to visualize it. Photo courtesy of Daina Taimina.

Another was moduli spaces, where somehow every point represents a surface and the number of dimensions vastly exceeds our small 3D world.

Not only revolutionizing her subject but also the way in which she worked, Mirzakhani wrote on giant white sheets of paper, doodling, trying out ideas, and searching, and only later putting it all into words and mathematical symbols. Her daughter, Anahita, always called it "painting."

Mirzakhani in 2014. Photo courtesy of Thomas Lin, Quanta Magazine

Mirzakhani's PhD thesis, focusing on the volume of moduli spaces as well as Riemann surfaces, was published in three separate papers in high-level mathematical journals. She even proved the Witten conjecture, an important problem in theoretical physics that was also intertwined with the math world.

Like many other great mathematicians, Mirzakhani's work was clearly elegant and magnificent in the connections she made. Those that knew her, including her PhD advisor and her husband, described her as daring, ambitious and modest. Her PhD work stretched throughout many fields, all the way down to the quantum level of physics.

After her PhD, Mirzakhani hopped around from job to job, publishing lots of groundbreaking papers at different institutes, before ending up at Stanford University in 2008 where her professorship and expertise was very much cherished.

She won the Ruth Lyttle Satter Prize by the American Mathematical Society in 2013, given to a woman whose work of the last six years had redefined her field, and in 2014, she won something even more prestigious: The Fields Medal.

The Fields Medal is the equivalent of a Nobel Prize in math and is the highest honor for any mathematician. Despite the immense achievement, Mirzakhani was uncomfortable with being a symbol for women, Iranians, and immigrants in math. She saw her work as important in its own right and disliked all the press that came with it, especially as she had just completed a demanding treatment for breast cancer. Though she tended to avoid the limelight, she still revolutionized how the world thinks about women in math and inspired many young girls in both the U.S. and her home country of Iran.

Still, the cancer returned and it managed to beat even her incredibly optimistic personality. Her thesis advisor and chair of the Harvard Department of Math, Curtis McMullen, described her proofs as "science-fiction stories," creative outlets into the outside world. With her death, it may take decades for us to gain the same outlook and intuition again.

Mirzakhani had been planning to explore more than algebraic geometries but to tackle other mystical fields from number theory to ergodic theory to combinatorics. We will never get to see her insight into those fields. Her death has dealt a massive blow to the entire math community with the loss of both her personality and her genius.

Mirzakhani may be gone, but we can allow her legacy to live on. Talking about her and understanding and appreciating her story go a long way towards realizing not only her achievements to the math world but her successes in changing society's perspective about women in math. As Stanford President Marc Tessier-Lavigne put it, "Maryam is gone far too soon, but her impact will live on for the thousands of women she inspired to pursue math and science."

Today, the Maryam Mirzakhani Prize in Mathematics is awarded every other year to a mid-career mathematician, and more relevantly for my members, there's a Maryam Mirzakhani AMC 10 Prize. The top-performing girls on the AMC 10A will be recognized, and one will receive $5000.

Together, these awards and many others do a lot to preserve Mirzakhani's legacy, and her name will carry on in the field for many years to come. For now, all we can do is remember her and cherish the legacy she has left in mathematics: both as a pioneer in her field and as a pioneer for women.

If you'd like to learn more about Maryam Mirzakhani, check out these links:

I borrowed heavily from the first two here to write this article and all of them are amazing:

1. (a New York Times article)

2. (more of a standard article)

3. (I really like this article!)

4. (standard information and some great quotes!)

These videos are also great to watch!

One more thing before I go:

If you've been wondering why the GLeaM logo is what it is, I'm going to paste our logo story (which is on the homepage) here again:


The great mathematician Maryam Mirzakhani's family often described the way she worked as "painting," as she crouched on the floor drawing on large sheets of paper. Mirzakhani was the first woman to win the Fields Medal, a prestigious award that is the equivalent of the Nobel Prize for math. She was a major advocate for women in mathematics, and she died in 2017 of breast cancer. The paintbrush is a symbol of her, but also of the beauty of mathematics, closely intertwined with art in the way ideas transition marvelously into one another. Around the paintbrush is a fractal, the Sierpinski triangle, one of the most beautiful aspects of math, reminding us to pursue seemingly impossible dreams in surprising fractional dimensions.

Two weeks ago, we talked about fractals and the Sierpinski triangle, and this week, we talked about Maryam Mirzakhani, so I think my logo should make sense to everybody now!

Thank you everybody, and have a great week!

Cover Image Courtesy: Newsweek