Then, last fall, Milman came on sabbatical and decided to visit Neeman so the couple could focus on tackling the bladder issue. “During the sabbatical is a good time to try high-risk, high-reward things,” Milman said.
They got nowhere for the first few months. Finally, they decided to set themselves a slightly easier task than Sullivan’s full guess. Adding an extra dimension of breathing space to your bubbles gives you a bonus: the best bubble cluster has mirror symmetry over a central plane.
Sullivan’s conjecture is about triple bubbles in dimensions two and up, quadruple bubbles in dimensions three and up, and so on. To get the bonus 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. “It wasn’t until 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 came up with a perturbation argument, which is to slightly inflate the half of the bubble cluster that is above the mirror and deflate the half below. This perturbation does not change the volume of the bubbles, but it could change their surface area. Milman and Neeman showed that there is a way to choose this perturbation to reduce the surface area of the cluster when the optimal bubble cluster has walls that are not spherical or flat—a contradiction since the optimal cluster already has the smallest surface area has area possible.
Using perturbations to study bubbles is far from a new idea, but figuring out which perturbations detect the important features of a bubble cluster is “a bit of a dark art,” Neeman said.
In hindsight, “as soon as you see [Milman and Neeman’s perturbations]they look pretty natural,” said Joel Hass of UC Davis.
But recognizing the disorders as natural is a lot easier than finding them in the first place, Maggi said. “It’s far from something you can say, ‘Eventually people would have found it,'” he said. “It’s really awesome 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 core features of Sullivan’s clusters, with the possible exception of one: the condition that each bubble must touch one another. This last requirement forced Milman and Neeman to consider all the ways bubbles could combine to form a cluster. If it’s only about three or four bubbles, there aren’t that many possibilities to consider. But as you increase the number of bubbles, the number of different possible connectivity patterns grows even faster than exponentially.
Milman and Neeman initially hoped to find an overarching principle that would cover all of these cases. But after spending a few months “racking our brains,” Milman said, they decided to settle for the time being with an ad hoc approach that allowed them to deal with triple and quadruple bubbles. They have also announced unpublished evidence that Sullivan’s fivefold bubble is optimal, although they have not yet determined that this is the only optimal cluster.
Milman and Neeman’s work is “a whole new approach rather than an extension of previous methods,” Morgan wrote in an email. It’s likely, Maggi predicted, that this approach can be pushed even further — perhaps to clusters of more than five bubbles, or to the cases of Sullivan’s conjecture that don’t have mirror symmetry.
Nobody expects further progress to be easy to achieve; but that never deterred Milman and Neeman. “From my experience,” Milman said, “all the important things that I was fortunate enough to be able to do just didn’t have to give up.”
Original story Reprinted with permission from quanta magazine, an editorially independent publication Simons Foundation whose mission is to improve public understanding of science by covering research developments and trends in mathematics and the natural and life sciences.