# In Bubbles, She Sees a Mathematical Universe

PRINCETON, N.J. — On the evening of March 19, the mathematician Karen Uhlenbeck gathered with revelers at the Institute for Advanced Study for a champagne reception. Some hours earlier she’d been awarded the Abel Prize — the first time a woman had won it — for her discovery of a phenomenon called “bubbling,” among other effervescent results.

Dr. Uhlenbeck is a professor emerita at the University of Texas at Austin, where she spent the better part of her career (having declined a professorship at Harvard). She retired in 2014 and moved to Princeton. At the institute, she keeps a desk piled with boxes of books. She describes herself as a messy reader, and a messy thinker, and she is stylishly disheveled, with a preference for comfy, colorful clothing with pockets and Birkenstocks with socks.

As a procession of speeches and toasts lauded her life’s work, Dr. Uhlenbeck stood to the side of the lectern and listened, eyes mostly closed. When it finally came time to make her own remarks (unprepared), she began by simply agreeing: “From the perspective of my late seventies, I find myself as a young mathematician sort of impressive, too.”

She went on to note that, for lack of mathematical candidates, her role model had been the chef Julia Child. “She knew how to pick the turkey up off the floor and serve it,” Dr. Uhlenbeck said.

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Jo Nelson, a mathematician at Rice University and a friend of Dr. Uhlenbeck, was thrilled that her visit to the institute coincided with a celebration for one of her mentors. “It’s amazing to hear a woman’s mathematical achievements celebrated and discussed in such detail,” she said.

Even Robert MacPherson, a topologist and faculty member in mathematics at the institute, made a rare social appearance. “This is wonderful in so many ways,” he said, holding a mini cosmos of brut rosé bubbles.

A decade ago, Dr. MacPherson and a collaborator formulated an equation describing how, in three and higher dimensions, individual bubbles evolve in live foams — the fleeting foam at the meniscus in his champagne flute, for instance, or the more enduring head on a pint of beer.

Researchers of all stripes have written “many thousands of papers” on bubbles, Andrea Prosperetti, a mechanical engineer at the University of Houston, has estimated. Bubbles entice for their seeming simplicity, which approaches the existential.

“Bubbles are emptiness, non-liquid, a tiny cloud shielding a mathematical singularity,” he wrote. “Born from chance, a violent and brief life ending in the union with the nearly infinite.”

And bubbles are everywhere, on every scale, once you start looking: high-tech drug-delivery mechanisms, emulsified salad dressings, soapsuds, black holes and beyond. In architecture, the Beijing National Aquatics Center is a box of bubbles. It is an application of the Weaire-Phelan foam, the most efficiently packed foam of equal-volume polyhedral bubbles, discovered in 1994 by Irish physicist Denis Weaire and his student Robert Phelan (first using a computer simulation, then created in a lab in 2012).

Dr. Uhlenbeck’s contribution is less practical. The Abel Prize cited “her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” The whimsical name for her work — “bubbling” — belies its prickly technicalities.

“It’s much more abstract and theoretical, and metaphorical,” Dr. Uhlenbeck said.

## “Can the poetry of bubbles survive this?”

The definitive piece of literature on bubbles and soap films — a film is the wall of a bubble, and a bubble is one cell in a foam — is “Experimental and Theoretical Statistics of Liquids Subject to Molecular Forces Only,” published in 1873 by Belgian physicist Joseph Plateau.

He experimented with soap bubbles for decades, capturing their behavior in what are now called Plateau’s laws. In a review of the book in Nature, Scottish physicist James Clerk Maxwell lamented, “Can the poetry of bubbles survive this?”

A soap bubble is the physical world’s solution for a mathematical challenge: to minimize a surface area — in this case, one that surrounds a prescribed volume of air. Nature is always seeking to optimize, to maximize gain at minimal cost in energy cost. So “minimal surfaces” problems are all around, even in higher dimensions, and all kinds of researchers are working to describe the governing rules.

“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.”

Last December came news of another bubbling lab creation, this time involving ordinary dishwashing solution and inspired by the scutoid, a new geometric shape discovered last year in a study of tissue cells. “We took up the matter because we had always been interested in the close resemblance of some biological cell systems to soap froths,” said Dr. Weaire.

A scutoid has been compared to a twisted prism — an odd shape for a living cell, but sometimes an optimal one when tissues are growing, curving and developing. Dr. Weaire wondered: Is the scutoid’s optimization imposed by genetics, or merely geometry and the physics of the moment? Is it “simply attributable to the bare bones of the theory of bubbles, that is, just the elementary laws of surface tension, as laid down by Plateau? For this purpose, are cells just bubbles?”

The bubbles answered in the affirmative. The experiment successfully produced “scutoid configurations in a dry foam sandwich” (with a computer simulation to match, embodying only the surface tension forces in a foam).

Theorists also contemplate bubbles at the larger end of the scale spectrum. At the University of Cambridge, Adriana Pesci and other applied mathematicians study soap films that form on the frame of a Möbius strip. “The interesting thing about this investigation is that it all started with solar flares,” Dr. Pesci said.

Astrophysicists have long postulated, if only symbolically, that galaxy clusters have a soapsuds structure. Minimal surfaces also are important in studying black holes, their dynamics driven by a “soap-bubble law.”

## The bubbling of instantons, and other secrets

Over the 2018-2019 academic year, researchers from around the world gathered at the Institute for Advanced Study for one of its annual thematic deep dives — a “Special Year in Variational Methods in Geometry,” devoted to minimal surfaces and related tributaries. “It’s the world’s biggest bubble fest around here lately,” said Helmut Hofer, an institute mathematician.

Dr. Uhlenbeck’s contributions were cited often in the seminars and workshops. Hiding out at home on prize day, she missed a talk by Xin Zhou, of the University of California, Santa Barbara, who began with a prefatory reference to one of Dr. Uhlenbeck’s 40-year-old theorems at the top of the chalkboard.

Dr. Zhou, who had just proved one of the remaining open problems in minimal surfaces, invoked her work as an inspiration. “If there had been no Uhlenbeck theorem, people would not dare to try,” he said.

Fernando Codá Marques, a mathematician at Princeton, had been invited to organize the special year around some of his particularly spectacular work — beginning with a 2013 result achieved with his collaborator André Neves, at the University of Chicago.

Since then, Dr. Marques, Dr. Neves and others have been searching out minimal surfaces in closed spaces of various dimensions.

Previously, the best result, long forgotten, dated to 1981 and showed that in any three- or higher-dimensional space, there is always at least one minimal surface. “We were curious if there were more,” said Dr. Marques.

There are, according to the bubbles. A new proof by Antoine Song, a doctoral student of Dr. Marques, rounded out a series of results showing that there are actually infinitely many minimal surfaces, and that they are densely packed and equidistributed.

It’s hard to fathom, even for Dr. Uhlenbeck.

“I’m impressed with the statistical theorems that say that minimal surfaces are all over the place,” she said. “Those kinds of theorems are quite impressive, and a little bit hard to believe.”

One way to understand, or at least contemplate, the nuances of Dr. Uhlenbeck’s work is to consider the challenge of scale. She started drawing about ten years ago — outdoor scenes, mostly — and this led to an unexpected revelation: “I discovered the fascinating fact that the problem of scale occurs both in mathematics and in drawing.”

In drawing, you try to capture both the large scale (the expanse of the forest) and the small scale (the grasses and flowers). “In mathematics, there is very much the same thing,” she said. “The hardest part with both is fitting the two scales together. You need the right tools.”

In physics, she noted, quantum theory deals with the very small, while general relativity deals with the very large, and physicists don’t yet know how to reconcile the two.

Dr. Uhlenbeck’s bubbling dealt with a similar challenge: she observed intricate phenomenon at the small scale, and then she invented tools to investigate regions of interest at a larger, more accessible scale: “You simply blow them up and look at them as if under a magnifying glass, and then you can see what’s happening.”

With this approach she also enabled other theorists to tackle some messy, turkey-on-the-floor situations.

“I’ve had several run-ins with bubbling,” said Edward Witten, a physicist in the institute’s school of natural sciences, lounging on the sofa in his office before his speech at the Abel Prize party.

He tried to explain how the “bubbling of instantons” has various important applications and implications in both mathematics and quantum field theory. (An instanton is an event in space-time, nothing like a champagne reception.)

For mathematicians, Dr. Witten explained, the bubbling of instantons was a technical obstacle in understanding four-dimensional spaces. “For physicists it isn’t just a technical obstacle,” he said. “It also contains secrets. It was the key mystery to understand.”

After bubbling, Dr. Uhlenbeck moved on to other mathematical mysteries for a few decades, but over the last year she has returned to minimal surfaces. Every Friday, her collaborator Penny Smith visits from Lehigh University to talk math. After the Abel Prize whirlwind, Dr. Uhlenbeck, exhausted, took two Fridays off. But when their sessions resume, they’ll dive deep into some even higher-dimensional bubbling.

Here, Dr. Uhlenbeck said, “It gets much messier.”

Earlier reporting on mathematics