And last fall, Millman decided to take a sabbatical and visit Niemann. .
For the first few months they weren’t going anywhere. Finally, they decided to give themselves a challenge that was a little easier than Sullivan’s full expectations. If you give the bubble one more dimension he will get a bonus. The best bubble clusters will have mirror symmetry about the central plane.
Sullivan’s conjecture is a triple bubble with two or more dimensions, a four-dimensional bubble with three or more dimensions, and so on. To get bonus symmetry, Milman and Neeman restricted their attention to his triple bubble over 3D, his 4D bubble over 4, etc. “The only time we really made progress was when we gave up on getting the full range of parameters,” Niemann says.
With this mirror symmetry at their disposal, Milman and Neeman came up with a perturbation theory that causes the half of the bubble cluster above the mirror to expand slightly and the half below the mirror to contract. This perturbation does not change the bubble volume, but may change the surface area. Milman and Neeman showed that if the optimal bubble cluster has walls that are neither spherical nor flat, there is a way to choose this perturbation to reduce the surface area of the cluster. This is contradictory because the optimal cluster surface is already minimal. possible area.
Using perturbations to study bubbles is by no means a new idea, but understanding which perturbations detect key features in bubble clusters is “a bit of the dark art,” Niemann said. .
Looking back, “Once you see [Milman and Neeman’s perturbations]they look very natural.” Joel Hass of UC Davis.
But recognizing a perturbation as natural is much easier than thinking about it in the first place, McGee said. ‘ he said. “It’s really genius on such an amazing level.”
Using perturbations, Milman and Neeman were able to show that an optimal bubble cluster must satisfy all the core properties of Sullivan’s cluster. This last requirement forced Milman and Neeman to grapple with all the ways bubbles could lead to clusters. With only 3 or 4 bubbles, there aren’t many possibilities to consider. However, increasing the number of bubbles increases the number of different possible connection patterns faster than exponentially.
Millman and Niemann originally wanted to find an overarching principle that would cover all these cases. But after spending months “getting my head around it,” according to Millman, for now I’ve decided to settle for a more ad-hoc approach that can handle triple and quad bubbles. We have also published unpublished evidence that a 5-fold bubble of is optimal, but have not yet established that it is the only optimal cluster.
Millman and Niemann’s study “is not an extension of previous methods, but an entirely new approach,” Morgan wrote in an email. This approach, McGee predicted, is likely to be pushed further, perhaps in the case of clusters of five or more bubbles, or the Sullivan conjecture, which does not have mirror symmetry.
No one expects further progress to come easily. But that never deterred Millman and Niemann. “In my experience, all the major things that I’ve been fortunate enough to be able to do have involved not giving up,” Millman said.
original story Reprinted with permission from Quanta Magazine, an editorially independent publication of Simmons Foundation Its mission is to advance public understanding of science by covering research developments and trends in mathematics, physical sciences, and life sciences.
