“It is a timeless subject,” said Dr. Uhlenbeck, at her dining room table. On the afternoon of prize day, she had been hiding out at her house, conserving her strength for the party. A founder of a field called geometric analysis, Dr. Uhlenbeck approaches minimal surfaces esoterically, under the banner of “variational methods in geometry.”
“I’ll give you a problem,” she said. “Take a fixed length of string, lie it down on the plane”— such as a table — “and enclose the largest area you can inside. I wobble the string a little bit, I vary it just a little, to see whether the area increases or decreases.” The optimal answer is not a square but a circle. You might try a square, she said: “But by unkinking and smoothing out the corners, you can enclose more area. That’s a variational process.”
In a classic paper from 1976, Cyril Isenberg declared the soap film “an analogue computer.” He used wire frames of polyhedra, such as a tetrahedron or cube, and dipped them in a soapsuds solution in order to compute — faster than a mathematician, certainly — the solution to those particular three-dimensional minimization problems. (Minimization problems are even important these days in deep learning. “The techniques involve minimizing lots and lots and lots of small problems all over the place,” Dr. Uhlenbeck said.)
This method was popularized in the 1940s by Richard Courant, the founding director of the Courant Institute of Mathematical Sciences at New York University. Recently, Courant researchers in the applied mathematics lab conducted a study that concluded that “there is more than one way to blow a bubble.”
Their experiment involved blowing large bubbles of olive oil (Bertolli) in a water tunnel, creating what investigators called “a mathematical river.” This led to a “master formula” describing the critical flow speed needed to blow and pinch off a bubble.
But they also discovered another way to blow a bubble: impose a gentle flow, below the critical speed, on a film that is already somewhat inflated.
“We found this second result surprising,” said Leif Ristroph, one of the study’s authors. “This might explain how we often blow bubbles as kids. A quick puff bends the film outward, and thereafter the film still inflates even as the flow of air slows.”