# Monday, August 18, 2014

### The Fields Medal

**by Jonathan Kujawa**

The big news in math this week was the opening of the quadrennial International Congress of Mathematicians (ICM) in Seoul. A number of prestigious awards are given at the ICM. Most famously this includes the Fields medal and the Nevanlinna prize (aka the Fields medal for computer science). Up to four winners of the Fields medal are announced along with the winner of the Nevanlinna prize. All the winners must be no older than 40.

I had the pleasure to attend the 2006 ICM in Madrid. This is the ICM famous for Grigori Perelman refusing to accept the Fields medal for his work in finishing the proof of the Poincaré conjecture. Perelman (or at least the media version of him) comes across as the stereotypical eccentric mathematician uninterested in worldly things. Fortunately for the PR folks, this year's winners all appear to be the sort you'd enjoy having over for dinner and drinks.

This year the Fields medal went to Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani. The Nevanlinna prize went to Subhash Khot. An excellent profile of each of the winners, including very nicely done videos, can be found on the Quanta website. The profiles are a bit short on the actual math of the winners. If you'd like a more meaty discussion of their work, former Fields medalist Terry Tao wrote blog posts here and here giving a more technical overview. Even better, former Fields medalist Timothy Gowers is blogging from the ICM itself! He's giving summaries of the main talks as well as his more general impressions while at the event. I can also recommend that you check out the excellent overviews of some of the winners' work on John Baez's Google+ page.

Rather than talk about the details of the winners' work [1], I wanted to point out a meta-mathematical common feature of their research. This is the idea of studying a collection of objects as a whole, rather than one by one.

Any algebra or calculus student will tell you what math is: you are given a specific problem, you solve it and find the exact solutions, and then move on to the next problem. The truth, however, is that nowadays mathematicians often study a family of problems collectively.

For example, you might want to study all the 3D shapes which have exactly one hole through them (like a coffee mug). One approach you could take is to imagine a "space" where at each location is one of these shapes. Why do this? Well, it can be that each individual problem or object is too easy or too hard to deal with on its own. In this case it often turns out to be remarkably fruitful to study them in toto.

To continue with our example, imagine further that the shapes are arranged in this space so that the closer they are the more similar their shape. This provides a huge conceptual breakthrough. Now you can talk about all the shapes in the neighborhood of a coffee mug (that is, all the shapes most similar to it), or imagine traveling a path in this space which goes between the coffee mug and the doughnut. Traveling this path can be imagined as a movie showing you the shapes you pass by on your journey [2]:

Collecting all your objects of interest and viewing them as points in a space is oftentimes called a configuration space or a moduli space [3]. The power of this point of view is that now you can use the tools and concepts of geometry to organize and study the problems you're thinking about. This turns out to be a a remarkably useful approach.

In the case of Mirzakhani it is something of an Ouroboros: each object she studies is a geometric object and so, in the end, she is using geometry to answer questions about other, different, geometries! In the same spirit, Bhargava studied elliptic curves (which we saw are used in cryptography) by studying them collectively. In the case of Bhargava, he and Shankar proved that the famous Birch and Swinnerton-Dyer conjecture holds for at least 2/3 of all elliptic curves without verifying it for any individual curve! Similar philosophies can be seen in the work of the other winners, as well.

The headline news this year, of course, is that Mirzakhani is the first female to win the Fields medal. But that's not the half of it: Avila is the first winner from South America and Bhargava is the first of Indian origin (although born in Canada). Heck, even the token white male, Hairer, is at a lesser known research university.

Before I go any further, let me emphasize in strongest terms that all of the winners are doing world class work and clearly deserve the Fields medal. There are, of course, other mathematicians who you could justly argue should have also won. Especially those who will no longer be eligible due to their age [4]. But no one can fairly argue that any of these folks shouldn't have won.

This year's winners reflect one of the things I love about mathematics as a discipline (as opposed to the human endeavor). Mathematics doesn't care a whit where you live, what you look like, how much money you have, or any of that nonsense. Anyone can do beautiful, true mathematics. The most famous example of this is Ramanujan: he was a self taught mathematician in turn-of-the-century India whose work continues to keep mathematicians awake at night.

This year's winners show this. Not only is Mirzakhani a women, she was born and raised in Iran. Not only is Avila from Brazil, but his father grew up in the rural Amazon. Great mathematicians can be found anywhere. The ICM embodies this sentiment. People attend from around the world to learn and celebrate the most recent advances in mathematics.

But the flip side is that mathematics is also a human enterprise. The media focus on the flash of the few who win prizes rather than the many who have done great work is one example. And another is the very poor job we do of identifying, encouraging, and developing scientific talent. It is little accident that both Bhargava and Hairer have parents who are mathematicians.

I have often thought of the many Einsteins, for lack of a better term, we miss out on because they are of the wrong gender, of the wrong background, were born in the wrong place, had parents or teachers who didn't know any better, or any of a myriad other ultimately idiotic reasons. Imagine where we would be today if we hadn't spent millennia recruiting essentially only from the tiny pool of white, European men.

In the Quanta profile of Mirzakhani I found the following anecdote particularly compelling:

To her dismay, Mirzakhani did poorly in her mathematics class that year. Her math teacher didn’t think she was particularly talented, which undermined her confidence. At that age, “it’s so important what others see in you,” Mirzakhani said. “I lost my interest in math.”

The following year, Mirzakhani had a more encouraging teacher, however, and her performance improved enormously. “Starting from the second year, she was a star,” Beheshti said.

Mirzakhani and mathematics would be much poorer today if not for her second year teacher!

Unfortunately I see discouraged students every day. In a recent class I taught, two women frequently commented on how mediocre they are at math. And this despite my protests that they were easily near the top of the class!

Certainly things have vastly improved. A personal hero of mine, Emmy Noether, was one of the great mathematicians of the first half of the previous century. Nevertheless, when Hilbert tried to get her a position at the University of Göttingen the faculty protested the idea of a women professor. Hilbert's rebuttal is a classic: "I do not see that the sex of the candidate is an argument against her admission as *privatdozent*. After all, we are a university, not a bath house."

Such overt prejudices have hopefully gone to the wayside. But there are still a great number of institutional, systemic, and societal discouragements. Take the Fields medal's age limit of 40 years as a somewhat silly but real example. It was set in an age when the candidates were male and their stay-at-home wives would be the one who took care of the kids. It may be inadvertent, but the message is clear when the same years which count towards the Fields medal are those when you typically have children [5].

Fortunately, as this year's Fields medalists show, things are improving. It's possible to envision a day when it will be unremarkable if all the Fields medalists in a given year are from the majority: women, Chinese, African, etc. And I'm happy to say that the field as a whole is working quite hard at addressing these issues.

[1] Even the closest winner (Bhargava) is a long ways from my areas of expertise. Try the links I provided if you prefer to get to the meat of the nut.

[2] Thanks to wikipedia for the mug to doughnut video.

[3] In case you want to impress at the next cocktail party. :-)

[4] Such as your correspondent. :-)

[5] Which makes it all the more remarkable that Mirzakhani also has a young child.

Posted by Jon Kujawa at 12:55 AM | Permalink