Alessio Figalli, a Mathematician on the Move, Wins Fields Medal

By Kevin Hartnett

In the company of Alessio Figalli, you get a sense that everything is taken care of. The young Italian mathematician is tall, fit and stylish, his R’s rolling off the tongue with an intoxicating Roman richness. He likes to be around other people, and other people like to be around him. Which is why the secret he has had to keep has been so hard for him.

Over the past few months, he’s visited friends in England, where his wife lives, and in Rome, where he grew up. He’s given lectures before friends and colleagues in Texas, North Carolina, Paris, and the Canadian mountain town of Banff. At each stop he has dined with friends and caught up with longtime collaborators. And under no circumstances has he been allowed to reveal that in February of this year, he received a call informing him that he would be receiving the Fields Medal, the most illustrious honor in mathematics.

“I’m not telling this to anyone, which means as a result you don’t really realize it,” said Figalli, a professor at the Swiss Federal Institute of Technology Zurich, back in April. “It’s a weird feeling, I’ve never experienced it. You don’t have feedback, not even from your friends.”

Now Figalli’s wait is over. This morning, at the 28th meeting of the International Congress of Mathematicians in Rio de Janeiro, he was officially named one of the four winners of the Fields Medal. The prize is conferred every four years by the International Mathematical Union to the most accomplished mathematicians in the world under the age of 40. At age 34, Figalli won the award with time to spare.

For Figalli, the Fields Medal caps a string of honors he has received during his ascent to the top of the math world. Once a classics student with no particular affinity for mathematics, he has gone on to shake the venerable mathematical discipline of analysis, which concerns the properties of certain types of equations. Figalli’s results have provided a refined mathematical understanding of everything from the shape of crystals to weather patterns, to the way ice melts in water.

“He has a wild spectrum of different contributions,” said Luis Caffarelli, a mathematician at the University of Texas, Austin, who will introduce Figalli at the Fields Medal ceremony in Rio.

While Figalli’s mathematical results are diverse, many of them turn on the innovative use of a concept called optimal transport. The idea originated in the 18th century, when a mathematician working for Napoleon Bonaparte tried to find the most efficient way to build a network of earthen fortifications. More than two centuries later, Figalli leads a community of mathematicians who have recognized that disparate mathematical problems will yield if you view them as the best way of moving a pile of dirt from one place to another.

Quick Study

Figalli was born in Rome in 1984. His father was a professor of engineering and his mother was a high school classics teacher. The Figalli home was full of books on Greek history and mythology. As a kid Figalli liked to play soccer, watch cartoons, and hang out with his friends — and, he recalls, he always made the rational decision to get his homework done first, so that he could fully enjoy himself.

“For me it was always a balance between how good a grade I could get and how much time I had to spend to get such a grade,” he said. “I was always an optimizer, I wanted the best for the least effort.”

Figalli liked math from an early age. He regarded it as an easy subject, something he was good at without having to work hard, and it took him a while to pursue the subject with any zeal. In Italy, students can enroll in either a classics or a scientific high school. Figalli had a taste for science, but his parents wanted him to study the classics, and he went along with this willingly.

“I said: ‘Why not?’ Usually there are more girls in a classics high school than a scientific high school, so that was another selling point,” he said.

Figalli turned toward serious math in his third year of high school. A mathematician colleague of his father’s encouraged Figalli to participate in the International Mathematical Olympiad, an open-ended problem-solving contest that draws the best young math minds in the world. Figalli was fascinated to discover that there were math problems whose solutions were not straightforward — you had to invent them yourself. “I loved that. It was a revelation,” he said.

Emboldened, Figalli tested into the Scuola Normale Superiore of Pisa, a university for mathematically and scientifically gifted students. There Figalli quickly confronted the limits of his education: At 18 he was sitting in math classes with the top students in Italy and he didn’t even know how to take a derivative.

“He didn’t [stand out] because he had a gap to recover compared to these highly trained colleagues,” said Luigi Ambrosio, a mathematician at the Scuola Normale and Figalli’s graduate school adviser.

But to anyone who looked closely, Figalli’s promise was apparent. He learned quickly and caught up to his peers within a year. At the start of his second year he began to work on a highly technical paper that Ambrosio had recently written. Ambrosio expected that the novice student would struggle to get anywhere with it. “Alessio came to me less than one week after and I realized he understood everything,” Ambrosio said. Figalli completed his undergraduate degree in two years.

In 2004 Ambrosio took him on as a graduate student and also arranged for him to study under Cédric Villani, a talented mathematician in Lyon, France, who would go on to win the Fields Medal himself a few years later. At the time Villani was working on a book about an intuitive idea that was experiencing a mathematical renaissance — an idea whose origins stretched back to the French Revolution.

Dirty Math

In the 1790s Gaspard Monge had a problem. He was a mathematician tasked by Napoleon with figuring out how to transport soil to the front for building fortifications. Monge wanted to find the optimal way to complete this transport — that is, he wanted to know which wagonload of dirt should end up where, so as to minimize the labor required to complete the task.

Monge made some headway on the problem, but then it languished for more than a century. It resurfaced in the 1940s when the economist Leonid Kantorovich produced the first rigorous mathematical description of optimal transport. But for decades afterward it remained of interest mainly to economists. It wasn’t until the 1980s and 1990s that mathematicians began to recognize that optimal transport was a mathematically deep question in its own right and also a tool that they might use to solve other kinds of problems.

Optimal transport is so intuitive that it’s easy to overlook the mathematical complexity it contains. The complexity comes from the enormous range of possibilities for how to move a pile of material from one place to another (or how to move many piles from many starting locations to many different destinations).

For instance, you’re not limited to transporting soil by the wagonload. Maybe it makes sense to split one wagonload between two destinations. Or to divide your pile into wheelbarrow loads, or to break it down by the shovelful, sending each one to a unique, carefully chosen destination. You can make your unit of transport infinitely small in your quest to find the optimal way to move material around — and this is where analysis comes in, as an extended form of calculus, the study of change on infinitely small or large scales.

“You can decide this piece goes here, that piece goes there. You have infinite degrees of freedom, and it’s only with advanced mathematical tools that you can really get rid of this infinite dimensionality and find in some sense your solution,” Ambrosio said.

By the time Figalli began to study at the Scuola Normale, mathematicians had recognized that the mathematics behind optimal transport was useful for much more than moving dirt. They realized that whenever you want to compare two shapes — which is something mathematicians often want to do — you can learn something by thinking about the most efficient, or optimal, way of converting one into the other.

Essential Traits

Figalli has been on the faculty at ETH Zurich since 2016, when he moved from UT Austin. In Zurich he lives in a furnished apartment on a hillside above the campus. He is rarely there for more than two weeks at a time. Instead, he’s often in England, where his wife, Mikaela Iacobelli, is a mathematician at Durham University. They met in 2013 when Figalli gave a talk at the University of Rome, where Iacobelli was a doctoral student. Iacobelli recalls being struck by the way Figalli reacted when the faculty member introducing him recited a long list of his accomplishments.

“Alessio looked a bit embarrassed, and I found this very nice because he’s really humble. I always forget about how good he is at math in our normal life,” she said.