‘Epic’ mathematical proof solves the three-bubble problem

Then last fall, Milman took a leave of absence and decided to visit Neeman so the two could focus on advancing the bubble. “During the sabbatical, this is a good time to try high-risk, high-return things,” says Milman.

For the first few months, they got nowhere. In the end, they decided to give themselves a task that was a little easier than Sullivan’s full conjecture. If you give your bubbles an extra dimension of space to breathe, you get a bonus: The best cluster of bubbles will have mirror symmetry across the center plane.

Sullivan’s conjecture was about three bubbles in dimensions two and up, four bubbles in dimensions three and up, etc. To obtain additional symmetry, Milman and Neeman limited their attention to three bubbles in dimensions three and up, four bubbles in dimensions four and up, etc. “It’s really only when we give up on reaching the full parameters that we really make progress. ,” said Neeman.

With this mirror symmetry at their disposal, Milman and Neeman put forward a perturbation argument involving the slight inflating of half of the cluster of bubbles above the mirror and the deflation of the half below it. This perturbation will not change the volume of the bubbles, but it can change their surface area. Milman and Neeman have shown that if the optimal bubble cluster has any walls that are not spherical or flat, there is a way to select this perturbation such that it reduces the surface area of ​​the cluster—a contradiction. contradictory, because the optimal cluster already has as few surface areas as possible.

Using perturbations to study bubbles isn’t a new idea, but figuring out which perturbations will uncover important features of a bubble cluster is “a bit of dark art,” says Neeman.

With hindsight, “once you see [Milman and Neeman’s perturbations]they look pretty natural,” says Joel Hass by UC Davis.

But recognizing disturbances as natural is much easier than figuring them out in the first place, says Maggi. “So far, it’s not something where you can say, ‘Everybody will find it eventually,'” he said. “It’s really genius to a very remarkable degree.”

Milman and Neeman were able to use their perturbations to show that the optimal bubble cluster must satisfy all the core characteristics of the Sullivan cluster, except perhaps one thing: the rule that every bubble must touch. This last requirement forces Milman and Neeman to grapple with all the ways that bubbles can connect into a cluster. When there are only three or four bubbles, there are not many possibilities to consider. But as you increase the number of bubbles, the number of different possible connection types increases, even faster exponentially.

Initially, Milman and Neeman hoped to find an overarching principle that would cover all of these cases. But after a few months of “headbreaking,” Milman said, they decided to settle for a while with a more specific approach that would allow them to handle triple and quadruple bubbles. They also published unpublished proof that Sullivan’s iridescent group is optimal, although they have yet to determine that it is the only optimal group.

Milman and Neeman’s work is “an entirely new approach rather than an extension of previous methods,” Morgan wrote in an email. It is possible, Maggi predicts, that this approach could be pushed further—perhaps to clusters of more than five bubbles, or to cases of Sullivan’s conjecture without mirror symmetry.

No one expected further advances to come easily; but that never discouraged Milman and Neeman. “From my experience,” Milman said, “all the important things that I am fortunate enough to be able to do are just not giving up.”

original story Reprinted with permission from Quantum journal, an editorially independent publication of Simons . Foundation Its mission is to improve public understanding of science by covering research trends and developments in mathematics, the physical and life sciences.


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