Mathematicians solve one of the mysteries of two-dimensional shapes

IAS - October 2020

Pure mathematicians like Daniel Cristofaro-Gardiner thrive on the creative challenge of imagining connections between things that seem unconnected and then using these connections to puzzle out old problems. Take the simplicity conjecture, which Cristofaro-Gardiner solved earlier this year with two of his colleagues.

The conjecture emerged from the work of mathematicians who, during the 1960s-70s, became fascinated with questions about the structure of so-called transformation groups of manifolds. Such transformation groups measure the different ways a given shape can be squeezed and molded, and mathematicians sought to ask if these  transformation groups were “simple”, meaning that they cannot be broken usefully into smaller components. They determined that the transformation group consisting of compactly supported volume-preserving homeomorphisms of three-dimensional balls (or even balls in more dimensions) were indeed all simple.

Proving the simplicity of the transformation groups associated to these higher-dimensional balls sounds more difficult than proving their simplicity in just two dimensions. But, in fact, this question about the transformation group of the two-dimensional ball (i.e. the disk) remained mysterious and unsolved for decades despite many attempts. The question came to be known as the simplicity conjecture.

Finally, earlier this year, a trio of mathematicians—Cristofaro-Gardiner, Vincent Humilière, and former IAS member Sobhan Seyfaddini—at last succeeded. “We proved that the group of compactly supported area-preserving homeomorphisms of the two-disc is, in fact, not simple,” says Cristofaro-Gardiner, von Neumann Fellow in the School of Mathematics at the Institute for Advanced Study (IAS) in Princeton, New Jersey. Although Cristofaro-Gardiner can summarize the answer to the simplicity conjecture in a single sentence, solving it was, unsurprisingly, far from straightforward. The paper that shows just how the proof was solved runs 111 pages.

How did these mathematicians succeed where so many before them had failed? “When you have an old problem that’s attracted a lot of interest, like this one, you want to bring new ideas to the table. We had a really good mix of expertise for attacking this problem and some tools that others had not tried before,” says Cristofaro-Gardiner. Specifically, they applied tools from a quickly evolving area of mathematics called symplectic geometry. Symplectic geometry uses insights from geometry to study the dynamics of systems. In doing so, it elucidates the deep mathematical symmetries between an object’s position in space and its momentum.

Consider the rotations of the planets around the sun. Symplectic geometry examines how the solar system is evolving over time by visualizing “the entire system as a very high-dimensional land,” says Cristofaro-Gardiner. Imagine the solar system as moving around in a 54-dimensional space, he explains, with each planet plus the sun having six dimensions—“three dimensions because we live in three-dimensional space plus three additional dimensions to keep track of how fast each object is moving.”

This same mathematical approach can also be used to analyze the behavior of water waves, falling apples, and spinning tops. And in the case of the simplicity conjecture, Cristofaro-Gardiner and his colleagues took advantage of the inherently symplectic structure of two-dimensional discs, which is at the heart of the conjecture. Using a tool from four-dimensional geometry called the Seiberg-Witten theory, “we turned this problem about two-dimensional geometry into a problem about four-dimensional geometry, where we could then exploit this symplectic structure in the two-dimensional case,” he says.

If this approach doesn’t sound entirely comprehensible, don’t be alarmed. Pure mathematics, like symplectic geometry, consists of abstract problems based on complex, imaginary numbers, and it can be challenging even for mathematicians working in other areas. Ultimately, though, the intellectual puzzles of such pure mathematics can lead to a deeper understanding of mathematics itself.

The solution to simplicity conjecture did just that, according to Helmut Hofer, Hermann Weyl Professor at IAS. “It is always wonderful to see how a very resistant problem is ultimately resolved. In this case, the resolution is unexpected, and there is a lot to be learned,” Hofer says.

In particular, the breakthrough adds “another argument to an already impressive list about the future importance of the emerging field of continuous symplectic geometry,” says Hofer, a leader in the foundations of symplectic geometry.

In the lead-up to solving the simplicity conjecture, Cristofaro-Gardiner says he benefited from being immersed in the friendly, informal atmosphere of IAS. “Sometimes you find new research opportunities that you hadn’t previously considered,” he says, plus “in my particular area, there’s a really strong group at IAS, and it’s just so enjoyable having people around you can tell about what you’re working on.”

Although he, Humilière, and Seyfaddini started working on the problem in earnest in summer 2018, the 2019-2020 von Neumann fellowship at the Institute gave him the unimpeded time needed to complete the concrete calculations required for solving the problem. “If you have a really challenging project, IAS is a great place to be,” says Cristofaro-Gardiner. He happily accepted when offered the chance to extend the fellowship for an additional year, after which he plans to join the faculty at University of Maryland.

He noted that people often imagine that advances in mathematics come about in magical moments of sudden inspiration or insight, but such Eureka moments are rare. The simplicity conjecture necessitated a persistent, laborious dedication to solving the problem—and some good fortune.

“You have this super complicated system of equations, and you have to make sure that you’re doing it all correctly. And then, in a way, it’s sort of just like blind luck at that point, because either the count comes out to be what you want… or it doesn’t, and the whole project is a total waste of time,” says Cristofaro-Gardiner. Among the approaches that he and his colleagues used was a relatively obscure algebraic approach called periodic Floer homology that allowed them to make sure that what they were claiming was consistent with the principle of invariance, a powerful concept in mathematics.

Moving forward, they plan to extend their simplicity conjecture work beyond the case of the disc to all surfaces (mathematicians call all two-dimensional shapes surfaces). As well, he aims to explore how some of the tools that he developed and applied to the problem, particularly periodic Floer homology, can be used to investigate other questions in symplectic geometry.

“It’s fun to do research and try to learn and to understand,” he says, “and that certainly will keep us busy for a while.”

—Jennifer Fisher Wilson