Maryam Mirzakhani won the Fields medal yesterday.
As a child in Tehran, she didn't intend to become a mathematician - she just wanted to read every book she could find! She also watched television biographies of famous women like Marie Curie and Helen Keller. She started wanting to do something great... maybe become a writer.
She finished elementary school while the Iran-Iraq war was ending, and took a test that got her into a special middle school for girls. She did poorly in math her first year, and it undermined her confidence. “I lost my interest in math," she said.
But the next year she had a better teacher, and she fell in love with the subject. She and a friend became the first women on Iranian math Olympiad team. She won a gold medal the first year, and got a perfect score the next year.
After getting finishing her undergraduate work at Sharif University in Tehran in 1999, she went on to grad school at Harvard. There she met Curtis McMullen, a Fields medalist who works on hyperbolic geometry and related topics.
Hyperbolic geometry is about curved surfaces where the angles of a triangle add up to less than 180 degrees, like the surface of a saddle. It's more interesting than Euclidean geometry, or the geometry of a sphere. One reason is that if you have a doughnut-shaped thing with 2 or more holes, there are many ways
to give it a hyperbolic geometry where its curvature is the same at each point. These shapes stand at the meeting-point of many roads in math. They are simple enough that we can understand them in amazing detail - yet complicated enough to provoke endless study.
Maryam Mirzakhani took a course from McMullen and started asking him lots of questions. “She had a sort of daring imagination,” he later said. “She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, ‘Is it right?’ I was always very flattered that she thought I would know.”
Here's a question nobody knew the answer to. If an ant walks on a flat Euclidean plane never turning right or left, it'll move along a straight line and never get back where it started. If it does this on a sphere, it will
get back where it started: it will go around a circle. If it does this on a hyperbolic surface, it may or may not
get back where it started. If it does, the curve it moves along is called a closed geodesic
The ant can go around a closed geodesic over and over. But say we let it go around just once: then we call its path a simple
closed geodesic. We can measure the length of this curve. And we can ask: how many simple closed geodesics are there with length less than some number L?
There are always only finitely many - unlike on the sphere, where the ant can march off in any direction and get back where it started after a certain distance. But how many?
In her Ph.D. thesis, Mirzakhani figured out a formula for how many. It's not an exact formula, just an 'asymptotic' one, an approximation that becomes good when L becomes large. She showed the number of simple closed geodesics of length less than L is asymptotic to some number times L to the power 6g-6, where g is the number of holes in your doughnut.
She boiled her proof down to a 29-page argument, which was published in one of the most prestigious math journals:
• Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Mathematics 168
(2008), 97–125, http://annals.math.princeton.edu/wp-content/uploads/annals-v168-n1-p03.pdf
This is a classic piece of math: simple yet deep. The statement is simple, but the proof uses many branches of math that meet at this crossroads.
What matters is not just knowing that the statement is true: it's the new view of reality you gain by understanding why
it's true. I don't understand why this particular result is true, but I know that's how it works. For example, her ideas also gave here a new proof of a conjecture by the physicist Edward Witten, which came up in his work on string theory!
This is just one of the first things Mirzakhani did. She's now a professor at Stanford.
"I don't have any particular recipe," she said. "It is the reason why 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."
She has a lot left to think about. There are problems she has been thinking about for more than a decade. "And still there’s not much I can do about them," she said.
"I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers."
I got some of my quotes from here:http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/
and some from here:http://www.theguardian.com/science/2014/aug/13/interview-maryam-mirzakhani-fields-medal-winner-mathematician
They're both fun to read.#spnetwork
doi:10.4007/annals.2008.168.97 #geometry #mustread