The Manifold Man

How Ian Agol (BS ’92) put the finishing stroke on a three-decades-old grand vision of mathematics—for which he won the 2016 Breakthrough Prize. 

By Erica Klarreich
Illustrations by John Ritter

When Ian Agol started as an undergraduate at Caltech in the early ‘90s, he had his physics major all mapped out, down to the string theory course he was going to take in his senior year. Experiments were not his forte, however, and as his studies at Caltech progressed, he realized that many of his peers possessed a dazzling practical ingenuity that he was unlikely to attain. “There were a lot of people who were more talented at these hands-on things than I was,” he said. Instead, Agol came to understand that he had a natural affinity for abstraction. “I preferred the elegance and simplicity of mathematics.”

That self-knowledge has paid off. Now a 46-year-old professor at UC Berkeley, Agol has risen to the top of his field of mathematics, a branch that focuses on the structure of “three-manifolds.” These are shapes made by gluing ordinary pieces of three-dimensional space together in interesting ways—similar to the way that you could stitch together patches of a two-dimensional plane to form the surface of a ball, or a doughnut, or a chair, provided your patches have some stretch to them. The study of three-manifolds brings together many areas of mathematics—such as topology, geometry, and dynamics—and even physics.

In the three-manifold community, “there’s no question he’s the guy,” says mathematician Nathan Dunfield of the University of Illinois at Urbana-Champaign. “He’s the leader in the field.”

Agol was awarded the $3 million 2016 Breakthrough Prize in Mathematics, “for spectacular contributions to low dimensional topology and geometric group theory.” Earlier this year, he was elected to the National Academy of Sciences.

His colleagues say that Agol has the capacity to master huge amounts of technical material in widely different fields of mathematics and then prove deep theorems in these fields. “He gets results that otherwise might have come from someone who had spent their entire life studying that area,” Dunfield says. “The rest of us probably just aspire to one result like that.”

Agol’s most celebrated work has put the finishing touches on a grand unifying vision of the geometry and topology of three-manifolds, proposed by mathematician William Thurston three decades ago. 

Geometry deals with rigid measurements of things like angles and lengths, while topology focuses on the traits of a shape that are not rigid—features that stay the same even if a shape is stretched or bent (such as the number of holes in a shape). Thurston postulated that when it comes to three-manifolds, these two ways of studying a shape are intimately connected. In 1982, Thurston published a list of 24 questions and conjectures that laid out a path to understanding any three-manifold’s geometry in terms of its topology, and vice versa (as long as the manifold is “compact,” meaning it doesn’t stretch out to infinity).

Thurston’s list set the course for the field, and over the decades, mathematicians chipped away at his questions. In 2002, Russian mathematician Grigori Perelman proved the most famous question on the list, the geometrization conjecture, which says that any compact three-manifold can be broken into a finite number of chunks that each possesses one of eight special, highly uniform types of geometry.

Of the eight special geometries, seven were well understood; but the eighth, “hyperbolic” geometry—in which every point has the geometry of a saddle—was both the most ubiquitous and the most mysterious. By the time Thurston’s paper was about 30 years old, pretty much all of his questions had been answered except for four conjectures about these hyperbolic three-manifolds.

Then in 2012, Agol finally set these remaining conjectures to rest. He proved the virtual Haken conjecture, which says that after unrolling a hyperbolic three-manifold a few times, it’s possible to find an especially nice kind of surface inside it, called a Haken surface. Manifolds containing a Haken surface can be simplified by cutting the manifold open along the surface, and mathematicians (including Agol) had previously shown that a solution to the virtual Haken conjecture would resolve all the remaining problems on Thurston’s list.

Agol’s work implies that every hyperbolic three-manifold, no matter how complicated, can be built up by following a simple recipe. “It was fantastic work,” says Danny Calegari, a mathematician at the University of Chicago.

Agol has received numerous awards for his work on the virtual Haken conjecture—too many, in his opinion. “I feel a little embarrassed,” he says. He plans to give away $1 million of the Breakthrough Prize, in part to support mathematical endeavors.

In a sense, Agol’s work closes the book on Thurston’s vision of three-manifold geometry and topology. Researchers will have to grope around to figure out what are the next important problems to solve, but the field is anything but dead. Once mathematicians digest a big result like Agol’s, it becomes the foundation on which to erect an even more powerful theory, Dunfield says. “You take it as a known, and then ask, what are the next big questions?” 

Despite his brilliance, Agol is laid-back, with a dry, understated sense of humor, Calegari says. “It’s possible for him to be quite witty, in such a way that people standing next to him might not notice that he’s joking around.” 

Agol is a relatively unintimidating person with whom to collaborate, Dunfield says, because his genius is of the slow-burning variety. “You can talk to him and feel that you are contributing to the conversation. Then he goes away and comes back a week later with great insights.”

Agol enjoys discussing mathematics with other researchers, but he most often works solo, which allows him the freedom to stumble down narrow alleyways and dead ends without feeling pressure to produce a result quickly. “When you’re working on your own, you’re more willing to throw away a big chunk of work you’ve done,” he says. “I can create new things that might sound stupid, but I need to try them for myself. That kind of wasting time is hard to do in a collaboration.”

Agol “doesn’t like to publish piecemeal results—he likes to solve a problem,” Calegari says. “He’s comfortable going a long time without writing a paper or necessarily having anything to show for it. Eventually, he finds an approach that works, and it ends up looking very dramatic.”

Agol likes to say that he forms not just an intellectual connection to the problems he works on but also an emotional one, viewing them as his friends. This affectionate relationship can be a source of strength, he has said, since our brains are driven by emotion as much as by rationality. But “the flip side of being emotionally involved in mathematics is that it can be depressing when you don’t make progress, which is most of the time,” he wrote in an email. “One needs the mental fortitude to deal with this depression.”

That doesn’t stop Agol from diving into high-risk projects. “I prefer to work on problems where I think they’re completely out of reach, and there seems like a low probability of success, but where I might have some tools that would tell me a little bit about the problem,” he says.

In typical fashion, Agol has now turned his sights on one of the most famously difficult problems in mathematics: the four-color theorem, which says that it’s possible to color any map in a plane using only four colors in such a way that no two adjacent countries have the same color. The four-color theorem is known to be true: Kenneth Appel and Wolfgang Haken (of Haken surfaces) proved it in 1976 by means of an enormous computer computation. Since that time, mathematicians have striven to come up with a “human readable” proof—one that would actually give people some understanding of why the theorem is true.

The four-color theorem seems a far cry from the kinds of three-manifold problems Agol usually works on, but he has been interested in the problem since his undergraduate days at Caltech, when he studied it during a summer research fellowship (SURF). “It’s a problem you can describe to anyone, even in elementary school, so it’s appealing in that way,” he says.

Now, his interest has been rekindled by a novel approach to the four-color problem—proposed by Peter Kronheimer of Harvard University and Tomasz Mrowka of the Massachusetts Institute of Technology—that uses gauge theory, a type of quantum physical theory connected to three-manifolds. As is his wont, Agol has plunged into a deep study of gauge theory, even though it’s far from his areas of expertise. “I feel out of my comfort zone,” Agol says. “I have very little chance of succeeding.”

But out of his comfort zone is one of Agol’s favorite places to be. And even if he doesn’t succeed in proving the four-color theorem, he hopes that his explorations will lead back around to new insights about three-manifolds. “I think there will be some interesting mathematics that comes out of it.”