The much-vaunted Ricci flow equation was introduced by the mathematician Richard Hamilton in 1982. The equation is a tool: When applied to a manifold, a curved space in higher dimensions, the equation evolves the geometry of the space, making it smoother, more like a sphere. The Ricci flow is often compared to the heat equation, which describes how heat flows and distributes through space more evenly over time.

 

Dr. Hamilton applied his Ricci flow tool in pursuing a solution to the Poincaré conjecture, a long-elusive problem that states: Any finite and closed three-dimensional shape, without any holes, is “homeomorphic” to the sphere—that is, it has the same loose topological make-up. Using the Ricci flow, Dr. Hamilton’s strategy was to deform this shape, a.k.a. the Riemannian manifold, so that it was perfectly round, or spherical. During this process, however, the geometry occasionally blew up to infinity, forming a “singularity” with unwieldy high curvature. By performing “surgery” and lopping out such problematic spots, Dr. Hamilton carried on toward the larger problem, but he couldn’t finish it off.

 

It was the Russian mathematician Gregori Perelman who realized Dr. Hamilton’s vision. By finding a crucial characteristic of singularities—ruling out all shapes except extant cylindrical or spherical singularities—he solved the Poincaré conjecture in 2002.

 

During the Fall 2024 program on New Frontiers in Curvature at the Simons Laufer Mathematical Sciences Institute, Richard Bamler, of UC Berkeley, gave an introductory talk illuminating new research (some with collaborators) in the 3-dimensional realm of Ricci flow—namely his own recent contributions that moved the ball forward and promised to keep it rolling. 

 

These results confirmed outstanding conjectures suggested by Dr. Perelman, providing proof that: 1) There are only finitely many singularities and thus finitely many necessary surgeries in the Ricci flow; and 2) The flow is capable of resolving singularities on its own, thus not relying on the somewhat cumbersome manual surgery process (this is joint work with Bruce Kleiner, at NYU’s Courant Institute, and John Lott, at UC Berkeley).

 

Overall, Dr. Bamler’s work provided a subtle new framework, known as structure theory, for analyzing singularities and extending the flow past them. While to date all results have been in dimension three, the new goal is to study the flow in higher dimensions.

 

“I wanted to advertise this new viewpoint and this new way of thinking,” Dr. Bamler said of his time at SLMath. He was positively surprised with the response; many mathematicians were keen to learn more.

 

Eleonora Di Nezza, at the Mathematics Institute of Jussieu-Paris Rive Gauche, welcomed the opportunity to talk to Dr. Bamler in person (both had participated in a related program at SLMath/MSRI in 2016, around the time Dr. Bamler had his first breakthrough). “His work is a masterpiece,” Dr. Di Nezza said. But it is long, difficult to digest. During the program, she observed, “everyone wanted to talk to him. He was overwhelmed.” 

 

Dr. Di Nezza is intrigued by Ricci flow for Kähler manifolds—special Riemannian manifolds with an extra, complex structure. The Kähler structure is preserved by the Ricci flow, motivating the name Kähler-Ricci flow.  

 

The significance of Bamler’s theory in the Kähler setting is yet to be determined. “No one has looked at it yet, we have to try,” said Dr. Di Nezza, a co-organizer of the concurrent Fall 2024 program at SLMath on Special Geometric Structures. “The Kähler structure gives more rigidity, in a sense, so sometimes better things can happen.”

 

One challenge is simply getting the two distinct research communities into closer dialogue and finding time to parse the nuances. “Sometimes people don’t use the same language, but they’re saying the same thing,” said Dr. Di Nezza. Discussions in these directions began between Dr. Bamler, Dr. Di Nezza and other members of the Special Geometric Structures program, including Henri Guenancia of the Mathematics Institute of Bordeaux, Vincent Guedj of the Toulouse Mathematics Institute. To continue the Kähler line of investigation, Dr. Di Nezza invited Dr. Bamler to give a week-long mini-course on the subject, in Paris this summer. The aim is to build a solid foundation for the application of Dr. Bamler’s techniques to Kähler-Ricci flow.

 

For his part, Dr. Bamler is focused on applying the Ricci flow to manifolds in four dimensions and proving more topological conjectures.

 

“I have a plan laid out,” he said.

 

He is collaborating with his former post-doctoral student, Eric Chen, at the University of Illinois Urbana-Champaign, and his former Ph.D. student, Yi Lai, at the University of California Irvine. The first step is to understand what singularities look like, how they behave, and perform surgery accordingly.

 

But in the 4-dimensional landscape, things are more complicated. “In dimension three, there are three different curvatures,” Dr. Bamler said. “In dimension four, depending on how you count, there are 14 different curvatures.” 

 

With Dr. Chen, the focus is on conical singularities that emerge. “We made a fair amount of progress,” said Dr. Chen of their work during the term. 

 

With Dr. Lai, the focus is on soliton singularity models. Specifically, her area of expertise is flying wing steady Ricci solitons—self-similar singularities that sustain a steady shape over time, which were conjectured to exist by Dr. Hamilton. “I confirmed this conjecture to be true, and I found a family of flying wings,” she said.

A 4-D flying wing soliton, left; and a photograph of a real-world flying wing, sent to Yi Lai by Richard Hamilton after seeing her research. 
 

Of numerous paths forward, some are promising while others are more circuitous.

 

“When you go into four dimensions with the Ricci flow, there are questions that if you even get close to them then it might be completely hopeless,” Dr. Bamler noted. “We don’t have any tools, so there is no way to attack them.”

 

“The good thing is that what I want to do doesn’t touch on any of these questions,” he said. “We may face obstacles, but it doesn’t seem like we won’t get around them”.