In the mid-19th century, the Belgian physicist Joseph Plateau — who had been designing and conducting scientific experiments since he was a child — submerged loops of wire in a soapy solution and studied the films that formed. When he bent his wire into a circular ring, a soap film stretched across it, creating a flat disk. But when he dipped two parallel wire rings into the solution, the soap stretched between them to form an hourglass shape instead — what mathematicians call a catenoid. Different wire frames produced all sorts of different films, some shaped like saddles or spiraling ramps, others so complicated they defied description.
These soap films, Plateau posited, should always take up the smallest area possible. They’re what mathematicians call area-minimizing surfaces.
It would take nearly a century for mathematicians to prove him right. In the early 1930s, Jesse Douglas and Tibor Radó independently showed that the answer to the “Plateau problem” is yes: For any closed curve (your wire frame) in three-dimensional space, you can always find a minimizing two-dimensional surface (your soap film) that has the same boundary. The proof later earned Douglas the first-ever Fields Medal.
Since then, mathematicians have expanded on the Plateau problem in hopes of learning more about minimizing surfaces. These surfaces appear throughout math and science — in proofs of important conjectures in geometry and topology, in the study of cells and black holes, and even in the design of biomolecules. “They’re very beautiful objects to study,” said Otis Chodosh (opens a new tab) of Stanford University. “Very natural, appealing and intriguing.”
Mathematicians now know that Plateau’s prediction is categorically true up through dimension seven. But in higher dimensions, there’s a caveat: The minimizing surfaces that form might not always be nice and smooth, like the disk or hourglass. Instead, they might fold, pinch or intersect themselves in places, forming what are known as singularities. When minimizing surfaces have singularities, it becomes much harder to understand and work with them.
Mathematicians consequently want to know how common such non-smooth minimizing surfaces are, and what properties they might have. If singularities are rare in a given dimension, appearing only under contrived circumstances, then they’ll disappear if you wiggle your wire frame just right. You’ll be left with a smooth minimizing surface that you can study more easily, which will give you the chance to develop a thorough understanding of such surfaces in that dimension.
In 1985, mathematicians proved that in eight-dimensional space, singularities can indeed be wiggled away. But in higher dimensions, “all hell breaks loose,” Chodosh said. The singularities become much more difficult to analyze. For nearly 40 years, no one could make much progress on the problem.
That barrier has finally been broken. In 2023, Chodosh — along with Christos Mantoulidis (opens a new tab) of Rice University and Felix Schulze (opens a new tab) of the University of Warwick — showed that in dimensions nine and 10, smooth minimizing surfaces are the norm (opens a new tab). And earlier this year, the team, joined by Zhihan Wang (opens a new tab) of Cornell University, showed that the same is true in dimension 11 (opens a new tab).
The work marks a major advance toward understanding the strange kinds of minimizing surfaces that can arise in higher and higher dimensions. And mathematicians can now use the result to resolve a host of other math problems that have long been limited in scope to dimension eight or below — making those theorems even more powerful.
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