Mathematician Wins $3 Million Breakthrough Prize for 'Magic Wand Theorem'

Alex Eskin, a mathematician at the University of Chicago, has won the $3 million 2019 Breakthrough Prize in Mathematics.

The Breakthrough Prizes were founded in 2013 by a group of tech billionaires (as well as multihundred millionaire Anne Wojcicki, co-founder and CEO of genomics and biotech company 23andMe). The prizes are awarded each year to researchers in mathematics, fundamental physics and the life sciences. Past winners decide who will win in each category.

Eskin, a 54-year-old American mathematician born in Moscow, received the award for what the prize committee described as "revolutionary discoveries in the dynamics and geometry of moduli spaces of Abelian differentials," specifically calling out his 2013 paper with mathematician Maryam Mirzakhani that proved their "magic wand theorem."

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Mirzakhani, a former Stanford University professor born in Tehran, Iran, was also famous in the world of mathematics for her work in an area known as moduli spaces. She collaborated with Eskin on several important pieces of this work. On Aug. 13, 2014, she won the Fields Medal (the most prestigious prize in mathematics, awarded once every four years to two, three or four mathematicians under age 40). She was the first woman to win the prize, and no woman has won it since. She died of breast cancer on July 14, 2017, at age 40.

So, what does the magic wand theorem do?

"It's useful in several different areas of mathematics," Eskin told Live Sciencet, noting that the idea of the wand is a metaphor for how useful the theorem is, not a physical object or shape. "There is no wand."

"The theorem itself which we proved is in an area of mathematics that is not easy to explain," he said. "It takes me hours and hours to explain to math Ph.D.s that work in different subfields."

However, he added, "There is a consequence [of proving it] which anyone can understand."

Imagine a room made out of perfect mirrors, Eskin said. It doesn't have to be a rectangle; any weird polygon will do. (Just make sure the angles of the different walls can be expressed as ratios of whole numbers. For example, 95 degrees or two-thirds of a degree would work, but pi degrees would not.)

Now place a candle in the middle of the room, one that shines light in every direction. As the light bounces around the different corners, will it always illuminate the whole room? Or will it miss some spots? A side effect of proving the magic wand theorem, Eskin said, is that it conclusively answers this old question.

"There are no dark spots," he said. "Every point in the room is illuminated."

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Eskin said he first became interested in the ideas behind the magic wand theorem as a graduate student doing research related to a series of proofs known as Ratner's theorems, which mathematician Marina Ratner proved in the early 1990s. (Ratner, a former University of California, Berkeley mathematician, died one week before Mirzakhani, on July 7, 2017, at age 78.)

Ratner's theorems dealt with homogeneous spaces, "where every point is like every other point, such as the surface of a sphere," Eskin said. Eskin wondered whether Ratner's ideas could be carried forward into moduli spaces, where not all the points are the same.

"I actually got obsessed with this problem," Eskin said. "I had to work on other things because I was young, and you have to publish [research] to get hired. But I was always thinking about this problem."

Still, years passed before he was able to make significant progress.

"Eventually, I met Maryam Mirzakhani," Eskin said. "She's a lot younger than I am — I met her when she was a [research fellow at Princeton University] — and we had similar research interests, and we started collaborating for a while. And she's very much not interested in going after the low-hanging fruit. She wanted to work on the difficult problems. So, our projects got more and more ambitious." 

Still, they didn't immediately start plugging away at the problem that would help lead to Mirzakhani's Fields Medal and Eskin's Breakthrough Prize.

"This was kind of the biggest problem in our whole area," he said. "She knew I was thinking about it, and I knew she was thinking about it. But we never talked about it. And this went on for a couple of years, and then we just decided to join forces."

Eskin compared what took place over the next five years to a mountain-climbing expedition, noting that he's not the first mathematician to describe a theoretical research project this way.

An important early milestone, he said, was a January 2009 paper by French mathematicians Yves Benoist and Jean-François Quint in the journal Comptes Rendus Mathématique. It was in a different area of mathematics, but it turned out to be relevant in some important ways. That paper led Eskin and Mirzakhani to the first route up the mountain.

"For two years then, we were climbing it, making steady progress," Eskin said. "And finally, we got to a place where we could see the top. But we hit a ravine, and we couldn't cross that ravine."

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"We were basically stuck for a year and a half," he said. "We were trying all kinds of ways to go at this and basically made absolutely no progress." 

At some point, though, they decided to stop trying to cross the ravine.

"We found a way to climb the other side of the mountain," he said.

Their new approach no longer started from the 2009 French paper but instead leaned heavily on earlier work by Israeli mathematician and 2010 Fields Medal winner Elon Lindenstrauss.

"Using this other work, going around the back, we couldn't reach the top either," Eskin said. "But we kind of found enough material that we could build a bridge over the ravine." 

That "material" was a series of smaller proofs, made while climbing that back route, that allowed the original route to become passable.

"From there, it took us another two years to write it down and make sure it all worked," Eskin said.

As for what he intends to do with the prize money, Eskin said, "You know, it's kind of stunning. I haven't decided yet."

Like past winners, he intends to donate a significant sum to an International Mathematical Union fellowship for graduate students pursuing doctorates in developing countries. As for the rest, he said, "I just have no idea." 

"One of the things about working in math is that the highs are very high and the lows are very low," Eskin said. "It's very frustrating, because for a long time, you basically can make no progress. At some point, you've spent five years working on a project, and you never know if it's going to work or not … It's a big part of your life invested in this. There's always a big possibility you'll come out of it with nothing ... You need a lot of emotional stability to keep on going."

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Originally published on Live Science.