Mathematician wins groundbreaking $3 million prize for ‘magic wand theorem’

Alex Eskin, a mathematician at the University of Chicago, has won the 2019 $3 million breakthrough prize in mathematics.

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

Eskin, a 54-year-old Moscow-born American mathematician, received the prize for what the prize committee described as “groundbreaking discoveries in the dynamics and geometry of moduli spaces of abelian differentials”, specifically naming his 2013 paper with mathematician Maryam Mirzakhani who 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 a field known as moduli spaces. She collaborated with Eskin on several important pieces of this work. On August 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 40). She was the first woman to win the award, and no woman has won it since. She died of breast cancer on July 14, 2017at 40 years old.

So what does the magic wand theorem do?

“It’s useful in several different areas of math,” Eskin told Live Sciencet, noting that the idea of ​​the wand is a metaphor for the usefulness of the theorem, not an object or a physical form. “There is no wand.”

“The theorem itself that we proved belongs to an area of ​​mathematics that is not easy to explain,” he said. “It takes me hours and hours to explain to PhDs in mathematics who work in different subfields.”

However, he added, “there is a consequence [of proving it] everyone can understand.”

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

Now place a candle in the middle of the room, a candle that shines in all directions. As the light bounces around the different corners, will it still light up the whole room? Or will it miss a few spots? A side effect of proving the Magic Wand Theorem, Eskin said, is that it conclusively answers this age-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. (Ratnera former University of California, Berkeley mathematician, died a week before Mirzakhani, on July 7, 2017, at the age of 78.)

Ratner’s theorems dealt with homogeneous spaces, “where every point is like every other point, like the surface of a sphere,” Eskin said. Eskin wondered if Ratner’s ideas could be transposed to moduli spaces, where not all points are the same.

“I actually became 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 still thinking about this problem.”

Yet years passed before he could make any significant progress.

“Finally I met Maryam Mirzakhani,” Eskin said. “She’s much younger than me – I met her when she was [research fellow at Princeton University] – and we had similar research interests, and we started collaborating for a while. And she really isn’t interested in attacking the fruits at hand. She wanted to work on the difficult problems. So our plans became more and more ambitious.”

Still, they didn’t immediately begin fixing the issue that would lead to Mirzakhani’s Fields Medal and Eskin’s Breakthrough Award.

“It was kind of the biggest problem in our whole region,” he said. “She knew I was thinking about it, and I knew she was thinking about it. But we never talked about it. And it went on for a few years, and then we decided to join forces.”

Eskin compared what happened over the next five years to a mountaineering expedition, noting that he is not the first mathematician to describe a theoretical research project in this way.

An important first milestone, he said, was a January 2009 article by French mathematicians Yves Benoist and Jean-François Quint in the journal Mathematical Reports. It was in a different area of ​​math, but it turned out to be relevant in some important ways. This document led Eskin and Mirzakhani to the first route to the mountain.

“For two years we climbed it, making steady progress,” Eskin said. “And eventually 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 stuck for a year and a half,” he said. “We tried all sorts of ways to achieve this and made almost no progress.”

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

“We found a way to climb up the other side of the mountain,” he said.

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

“Using that other work, going around the back, we couldn’t get to the top either,” Eskin said. “But we somehow found enough material to build a bridge over the ravine.”

This “hardware” was a series of smaller trials, made during the ascent of this secondary road, which allowed the original road to become passable.

“From there, it took us another two years to write it and make sure everything worked,” Eskin said.

As for what he intends to do with the prize money, Eskin said, “You know, it’s pretty mind-blowing. I haven’t decided yet.”

Like past laureates, he intends to donate a large sum to an International Mathematical Union camaraderie for PhD graduate students in developing countries. As for the rest, he said, “I have no idea.”

“One of the things about math work 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 can hardly move forward. At one point, you spent five years working on a project, and you never know if it’s going to work or not… C “It’s a big part of your life invested in it. There’s always a big possibility that you’ll walk away with nothing… You need a lot of emotional stability to keep going.”

Originally published on Live Science.

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