Tonight the Breakthrough Prize Foundation awarded the second annual Breakthrough Prize in Mathematics to Ian Agol of the University of California, Berkeley, for his work in geometric topology, which completed a revolution in the field that began more than 30 years ago. With an award of $3 million apiece in the categories of life sciences, physics and mathematics, the Breakthrough Prizes are the world’s richest science prizes.
Agol’s field, topology, is the branch of mathematics that pretends all shapes are made of putty or stretchy rubber. It studies those properties that remain the same when the space is squished or stretched, as long as there is no tearing or gluing. You can think of topological properties as the large-scale properties of a space. Geometry, on the other hand, looks at finer properties, those that depend on exactly how the space is put together. Topologists have long had a fairly complete understanding of how topology and geometry interact for two-dimensional surfaces, or 2-manifolds. Three-dimensional manifolds are a different story.
An appetizing way to understand 2-manifolds and 3-manifolds is to think of a doughnut. The glaze—the two-dimensional donut-shaped surface—is the 2-manifold. The 3-manifold is the whole doughnut, filling and all. We interact with 3-manifolds all the time in our everyday life, but mathematicians also study 3-manifolds that are more abstract and cannot be represented visually in the real world.
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Agol’s work is the culmination of a research program begun by mathematician William Thurston. In 1982 Thurston published a landmark paper laying out all the key questions and some possible answers for how to build and work with 3-manifolds. The paper served as a road map for research in the subject until 2012, when Agol provided answers to the last of Thurston’s major lingering questions about 3-manifolds.
The marquee conjecture from Thurston’s work is called the geometrization conjecture. (It is now the geometrization theorem because Russian mathematician Grigori Perelman proved it in the early 2000s. He famously refused the prestigious Fields Medal and the million-dollar Millennium Prize for his work. The theorem states that all 3-manifolds, no matter how complicated they are topologically, have only a few different geometric descriptions. Most of these descriptions allow mathematicians to understand three-dimensional geometry by understanding two-dimensional geometry, which is fundamentally simpler. But one type of geometry, called hyperbolic, resisted this simplification. Agol’s work gives researchers a way to study these hyperbolic 3-manifolds using surfaces as well.
Specifically, Agol proved the virtual Haken and virtual fibering conjectures. Topologists say a space has a property “virtually” if it can be “covered” by a space that does have the property. “Covered,” in this case, is a technical term closely related to but not exactly like the everyday act of wrapping a present.One way to understand this idea is to think about coiling up a garden hose on a circular reel. In that picture we could say the hose is a cover of the circle or the circle is a virtual hose. The power of “virtual” is that it allows you to understand the object that is covered by understanding the better-behaved cover. Returning to the garden hose, the circle and the hose are not exactly the same, but they share some similarities, and a deep, Zen-like understanding of the hose will help one understand the circle.
Haken manifolds, named after German mathematician Wolfgang Haken, can be cut into smaller pieces in an iterative process. If a manifold yields to this type of decomposition, it becomes easy to understand it by understanding the pieces left at the end. The virtual Haken conjecture states that many manifolds that are not Haken are virtually Haken‑in other words, studying the Haken cover can help researchers understand the manifold that lurks beneath.
The virtual fibering conjecture ties geometry to dynamics, the study of how spaces change over time. If you drag a circle along a line segment, you get a cylinder. Then you can glue the top circle to the bottom to get a torus—the mathematical term for shape that looks like an inner tube. You could see the torus as a diagram tracing out the circle’s movement through space over time. Jumping up a dimension, you can do something similar by dragging a surface along a line segment and gluing the top surface to the bottom to get a 3-manifold called a surface bundle. The virtual fibering conjecture states that a large set of manifolds are not quite surface bundles, but up to the wiggle room of the word “virtually,” they might as well be. “A 3-manifold has lots of different lives,” University of Chicago mathematician Danny Calegari says. It can be described geometrically, dynamically, combinatorially, and so on. “You want to reconcile the different points of view.” Agol’s work reconciling several different viewpoints is the basis for his award.
Although the Breakthrough Prize is an individual award, Agol’s success illustrates the importance of collaboration in mathematics. “I feel like I only deserve a small part of it because I’ve made so much use of other people’s work and relied a lot on collaborators and people who did work before me,” Agol says. His theorem builds most immediately on work of McGill University mathematician Daniel Wise, who shared the 2013 Oswald Veblen Prize in geometry with Agol. Agol also relied on work of Jeremy Kahn and Vlad Markovic, and part of the proof of the virtual Haken conjecture was written jointly with Daniel Groves and Jason Manning; many other people made important contributions along the way. “I find that when you’re talking to people, it puts your mind in a different reference frame where you make intuitive leaps,” Agol says. “You’re in verbal mode, not contemplative mode.”
Richard Taylor of the Institute for Advanced Study was one of the recipients of last year’s Breakthrough Prize in mathematics, and he chaired the Selection Committee this year. “Agol’s work embodied these two things we were looking for,” Taylor says. “He’s clearly at the top of his game, and it’s also more than one result. This isn’t a prize for one theorem. It’s a prize for people who have made a series of contributions.”
Agol’s proof of the virtual Haken conjecture in some ways marks the end of an era, but as Taylor says, “It’s probably not the case that 3-manifold topology has come to an end.” Agol says there are still plenty of interesting questions to ask about 3-manifolds. “For me, one of the main programs is to try to connect up what has been done in hyperbolic geometry—the geometrization conjecture and the picture we have there—with other areas of 3-manifold topology.” There is also the question of computational complexity: If someone hands you a 3-manifold, how long will it take to find the Haken manifold that covers it and then to decompose it into smaller pieces? In addition, the relatively complete picture of 3-manifolds could help researchers understand the heady world of four-dimensional spaces in much the same way surfaces helped them understand 3-manifolds.
Agol says he hopes to use his $3-million prize to give back to the mathematics community, perhaps by supporting mathematicians in developing countries as past recipients have done. He says winning the award is an honor but he did not enter math expecting to win prizes. “Finding out about the prize was never as exciting as the actual moment of thinking I had figured out the virtual Haken question.”