A decades-old conjecture about the best way to minimize the surface area of a three-bubble cluster seemed unprovable—until a breakthrough result.
Then last fall, Milman came up for sabbatical and decided to visit Neeman so the pair could make a concentrated push on the bubble problem. “During sabbatical it’s a good time to try high-risk, high-gain types of things,” Milman said.
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 up, quadruple bubbles in dimensions four and up, and so on. “It was really only when we gave up on getting it for the full range of parameters that we really made progress,” Neeman said.
Using perturbations to study bubbles is far from a new idea, but figuring out which perturbations will detect the important features of a bubble cluster is “a bit of a dark art,” Neeman said.Joel HassBut recognizing the perturbations as natural is much easier than coming up with them in the first place, Maggi said. “It’s by far not something that you can say, ‘Eventually people would have found it,’” he said. “It’s really genius at a very remarkable level.
Milman and Neeman hoped at first to find an overarching principle that would cover all these cases. But after spending a few months “breaking our heads,” Milman said, they decided to content themselves for now with a more ad hoc approach that allowed them to handle triple and quadruple bubbles. They’ve also announced an unpublished proof that Sullivan’s quintuple bubble is optimal, though they haven’t yet established that it’s the only optimal cluster.
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