Zimmer's conjecture

(Illustration by Michael Vendiola)

Burden of proof

Three mathematicians team up to advance Zimmer’s conjecture. Yes, that Zimmer. 

When president Robert J. Zimmer first arrived at UChicago in 1977, as a Dickson Instructor of Mathematics, he continued the work he had started as a graduate student at Harvard and prolifically published on while a faculty member at the US Naval Academy. This research included the relationship between dynamics, which is the study of repeated transformations, and Lie theory, an area of math that studies symmetries.

As a UChicago mathematics professor, Zimmer refined his research, which also included ergodic theory and differential geometry, eventually developing what is now known as the Zimmer program, a body of work first outlined in the early 1980s that includes Zimmer’s conjecture.

Mathematics professor Benson Farb, whose research relates to the Zimmer program, explains that the study of symmetry is fundamental to our understanding of the world. “The Zimmer program posits an essentially complete description of a wide swath of symmetries,” says Farb. “More precisely, Zimmer’s vision was that spaces that admit enough of a certain kind of symmetry must satisfy so many constraints that we should in fact be able to list all the possibilities, even though there are infinitely many of them.”

Zimmer’s work was influential because, building on the Fields Medal work of mathematician Gregory Margulis, he introduced new connections between disparate subfields, spurring research by several generations of mathematicians.

While work on the Zimmer program was active for around 25 years, the field has been quiet in the past decade. “The possible progress on much easier special cases of the Zimmer program was tapped out,” says Farb, “and everyone was simply completely stuck.”

Around 10 years ago scholars drifted into other areas. But work on the Zimmer program sprang back to life in fall 2016 with a collaboration between Aaron Brown and Sebastian Hurtado-Salazar, who hold the same Dickson Instructor positions Zimmer held, and Indiana University professor David Fisher, SM’94, PhD’99, a former doctoral student of Zimmer’s. They presented “what may be the biggest breakthrough ever on the Zimmer program,” says Farb, “verifying a large chunk of Zimmer’s conjecture by using a host of new ideas.”

Inquiry asked Brown and Hurtado-Salazar about the conjecture, their process, and the proof.

How did this collaboration come together?

Aaron Brown: We were in the right place at the right time. It was Sebastian’s idea for the three of us to talk when David was visiting. He had some tools he thought would be useful to apply to Zimmer’s conjecture, and the work I’ve been doing the past few years with another group of coauthors—Federico Rodriguez Hertz and Zhiren Wang at Penn State—ended up being a new idea behind how to put everything together. Our proof uses a piece of relatively recent machinery that wasn’t available to other people who have worked on Zimmer’s conjecture.

Sebastian Hurtado-Salazar: One of the other tools we used was developed in a very different area of mathematics. Sometimes you can just google if someone has already done the work. We found something called strong property (T) that was developed in relation to the Baum-Connes conjecture, part of operator K-theory, if I’m not mistaken, which is unrelated to Zimmer’s conjecture, and which I don't understand at all.

How long did the proof take?

Brown: It took us about 10 hours to create a strategy and plausible outline. It took another few weeks to flesh out the outline details and another four months to check all those details. Normally it can take anywhere from a few months to years.

What does Zimmer’s conjecture say in simple terms?

Brown: The conjecture roughly states that certain groups of symmetries of some objects don’t act on certain spaces: they can’t be part of a group of transformations. The conjecture and our proof are interesting in that they combine topics and tools from several areas of mathematics—group actions, geometry, dynamics, smooth ergodic theory, and functional analysis.

Hurtado-Salazar: When you have some set of equations—let’s say just with numbers—if you have too many equations and too few variables, usually there’s no solution, or the solutions are in some sense obvious. The Zimmer program is this kind of problem, but the variables are not just numbers; they are complicated geometric objects with many parameters. So we proved that that was the case—that in fact for this type of equation and these variables, there is no nontrivial solution under some specific conditions.

These variables and equations have a very precise geometric meaning. One could think of it like this: when certain objects, called higher rank lattices, appear in a higher dimensional space, they appear in a very geometric way.

How does the way an object appears relate to dimensions?

Hurtado-Salazar: An example of a low-dimensional object is a circle, like a circular string in three-dimensional space. You can cut the circular string, tangle the string in some weird way, and glue it back together, and now you have something in three-dimensional space that looks different. After this process the string is what is called in mathematics a knot. This kind of object is not very rigid, in the sense that you could put it into a higher dimensional space in many different ways. But there are these other higher dimensional objects, higher rank lattices, that, when put in higher dimensional spaces, sit very naturally. There’s no way of cutting them apart and putting them back together in whatever way you choose. There are only very specific ways they fit back together.

The Zimmer program is a sort of formalization of this concept. It’s about how far you can push the geometric rigidity, to understand how rigid these higher rank lattices are. The philosophy is that they always appear in a specified natural way.

What does your proof mean?

Hurtado-Salazar: It gives certainty to the rigidity properties of these objects. It was already expected by the conjectures that this was the case, but in math, if there’s no proof, there’s nothing. Zimmer had intuition and a lot of evidence. Now there’s complete certainty—if there’s no mistake!

Brown: Also, we didn’t prove the full conjecture. We proved it for one case of objects. We’re now working on the other case.

Have you spoken with President Zimmer about it?

Hurtado-Salazar: We all met in the Quadrangle Club, but we didn’t talk much about math. But he did mention that he thought about this conjecture every day for like 10 years.

Zimmer no longer has time for serious original thought in mathematics, but he follows what others are doing in the area and occasionally has exchanges with colleagues. “I remain totally fascinated by these problems,” he says, “although I seem to have escaped my hourly obsession with them.”

He was confident that someday there would be fundamental new insights brought to bear. “It is particularly gratifying that this was achieved by two young mathematicians at Chicago and a former doctoral student of mine,” he says. “Their arguments are original, powerful, and beautiful, and I was surprised by some of the particular techniques that were involved.”

Now that one case has been proven, Zimmer has hopes that the other will follow and that the new techniques, when fully worked out, will yield more results. After all, he says, “One only makes conjectures in the hope they will be proven.”