Then last fall, Milman took a year off and decided to visit Neeman so the couple could focus on the bubble problem. “During the sabbatical is a good time to try high-risk, high-gain things,” Milman said.
For the first few months, they got nowhere. Finally, they decided to give themselves a slightly easier task than the complete Sullivan conjecture. If you give your bubbles an extra dimension of breathing room, you gain an advantage: the best group of bubbles will have mirror symmetry in a central plane.
Sullivan’s conjecture deals with triple bubbles in dimensions two and higher, quadruple bubbles in dimensions three and higher, and so on. To get the added symmetry, Milman and Neeman restricted their attention to triple bubbles in dimensions three and higher, quadruple bubbles in dimensions four and higher, and so on. “Really, it’s only when we gave up on getting it for the full range of parameters that we really made progress,” Neeman said.
With this mirror symmetry at their disposal, Milman and Neeman proposed a perturbation argument that involves slightly inflating the half of the bubble cluster above the mirror and deflating the half below. This disturbance will not change the volume of the bubbles, but it might change their surface area. Milman and Neeman showed that if the optimal bubble cluster has walls that are neither spherical nor planar, there will be a way to choose this disturbance so that it reduces the surface area of the cluster, a contradiction, since the optimal cluster already has the smallest surface area. . possible area.
Using perturbations to study bubbles is far from a new idea, but figuring out which perturbations will detect important features in a bubble clump is “a bit of a dark art,” Neeman said.
In retrospect, “once you see [Milman and Neeman’s perturbations]They look quite natural,” he said. joel hass from UC Davis.
But recognizing disturbances as natural is much easier than thinking about them in the first place, Maggi said. “By far, it’s not something you can say, ‘Eventually, people would have found it,’” he said. “It’s really cool on a very remarkable level.”
Milman and Neeman were able to use their perturbations to show that the optimal bubble cluster must satisfy all of the central features of Sullivan clusters, except perhaps one: the stipulation that each bubble must touch each other. This last requirement forced Milman and Neeman to grapple with all the ways the bubbles could connect into a group. When it comes to just three or four bubbles, there aren’t as many possibilities to consider. But as the number of bubbles increases, the number of different possible connectivity patterns grows, even faster than exponentially.
Milman and Neeman initially hoped to find a general principle that would cover all of these cases. But after spending a few months “puzzling over our heads,” Milman said, they decided to be content for now with a more ad hoc approach that would allow them to handle triple and quad bubbles. They have also announced an unpublished proof that the fivefold Sullivan bubble is optimal, although they have not yet established that it is the only optimal group.
Milman and Neeman’s work is “a completely new approach rather than an extension of previous methods,” Morgan wrote in an email. It is likely, Maggi predicted, that this approach could be taken even further, perhaps to groups of more than five bubbles, or to cases of the Sullivan conjecture that do not have mirror symmetry.
No one expects further progress to come easily; but that has never deterred Milman and Neeman. “From my experience,” Milman said, “all the important things I was lucky enough to be able to do required just not giving up.”
original story reprinted with permission from how much magazine, an editorially independent publication of the simons foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.