Siobhan Roberts

Einstein himself apparently had no special plans for what he knew could be a momentous day. He was home in Berlin. He wrote a letter admitting a “blunder” in an ongoing debate with Theodor Kaluza, a German mathematician with a new notion of space-time that required five dimensions.

He betrayed no jitters about the fact that, on that day in May, two scientific expeditions were finally putting his theory of general relativity to the test. In Sobral, Brazil, and on Príncipe Island, off the western coast of Africa, two teams were viewing a total solar eclipse; in measuring the deflection of starlight by the sun’s gravitational field, they proved Einstein right.

Einstein first received word of their preliminary results in September, and wrote his mother with the “happy news.” The confirmation was officially announced in November at the Royal Astronomical Society in London, triggering headlines about the ensuing excitement. (“Men of Science More or Less Agog Over Results of Eclipse Observations,” one headline in The Times noted.) Einstein bought himself a congratulatory violin.

One hundred years later, relativists the world over are rejoicing, each in their own way.

Katharine Leney, a physicist at CERN, in Switzerland, and the purveyor of @PhysicsCakes on Twitter, created a rotating solar eclipse diorama, featuring cake pops of the sun, Earth and moon.

On Wednesday, two actors, Ben Livingstone, playing Einstein, and Colin Uttley, as Arthur Eddington, the astrophysicist who led the Príncipe expedition, gave a special performance at the Royal Astronomical Society in London.

And there are weeklong conferences in Sobral and Príncipe featuring lectures, seminars, parades, dancing and fine dining. (On Wednesday, the scientific gatherings will merge for a special joint session via teleconference.)

In Sobral, in a lecture Wednesday afternoon, Luis Crispino, a physicist at the Federal University of Pará, in Brazil, aims to set the record straight about the importance of the Brazilian expedition.

The Sobral measurements commonly are assumed to have played a minimal role in verifying Einstein’s theory, in part because the photographic plates from the main Sobral instrument were blurry. But in fact, the backup Sobral instrument provided the crucial images, because many of Eddington’s photographs from Príncipe were obscured by clouds and revealed few stars.

The Sobral photographs allowed Frank Dyson, the astronomer and co-organizer of the expeditions, to conclude that the results favored general relativity, said Crispino: “This is common sense to everybody in Brazil. Outside Brazil it is not.”

Crispino and Daniel Kennefick, of the University of Arkansas, recently laid out this argument in a paper in Nature Physics. “The role of other astronomers involved has been, so to speak, eclipsed by Eddington’s (and Einstein’s) fame,” they wrote.

Kennefick also is the author of a new book on the subject, “No Shadow of a Doubt,” and Wednesday, at a conference in Paris, he will offer a similar retelling. Anna Curir, from the Astrophysical Laboratory of Torino, Italy, on Tuesday gave a talk titled, “Arthur Eddington and the Gestalt theory.”

On Príncipe Island, about 20 relativists gathered on Bom Bom beach to discuss a century’s worth of exploration about Einstein’s equations of general relativity.

“His equations allowed cosmology to become a science,” John Barrow, the cosmologist, wrote in an email. “Before him, cosmology was like a branch of art history. You could imagine any type, shape or form of universe you liked.”

But Einstein’s equations, he added, “are more sophisticated than any others in science. They describe whole universes. Every solution of Einstein’s equations describes an entire possible universe that is consistent with the laws of physics.”

Alessandra Buonanno, a director at the Max Planck Institute for Gravitational Physics in Potsdam, was also on the beach in Príncipe, where she discussed gravitational waves, another prediction of general relativity.

“The waves are like fingerprints of the gravitational-wave sources,” she said. Buonanno’s research focuses on improving the accuracy of the models of the fingerprints for upcoming observations with the Laser Interferometer Gravitational-Wave Observatory and Virgo detectors, and for future gravitational-wave detectors, such as with the Laser Interferometer Space Antenna, or LISA, which will be launched in 2034.

“LISA seems very far into the future,” she said. “But theoretical models and the required hardware and software need to be developed from now.”

The Príncipe proceedings opened with a talk by Clifford Will, a Canadian mathematical physicist at the University of Florida, who in 1986 published the popular book “Was Einstein Right?” His new talk was titled, “Is Einstein still right?”

Ever more so, it seems. The past few decades have seen “an amazing array of experimental tests of general relativity, all of them in agreement with the predictions,” Will said. But the quest continues: “There are still things we don’t fully understand. And that’s probably likely always to be the case.”

“The more we keep testing it, the more confidence we have in the theory,” he said. “And of course on the other hand, any sort of deviations from his predictions would surely tell us that there is something new to be investigated.”

On Wednesday at the Institute for Advanced Study in Princeton, where Einstein spent his twilight years, the centenary of the eclipse is being celebrated with an afternoon of talks featuring a new book by Graham Farmelo, “The Universe Speaks in Numbers.”

Edward Witten, Nima Arkani-Hamed, Freeman Dyson and Karen Uhlenbeck, the recent Abel Prize winner in mathematics, among others, will address dueling methods — experimental versus mathematical — of investigating our own cosmic diorama.

Those in favor of experiments have been known to view the mathematically inclined “as self-indulgent, without an obvious payoff for understanding the real world,” Farmelo said. And the mathematical cosmologists traditionally have viewed their opposites as “ambulance chasers,” rabidly pursuing every new experimental clue at the expense of overarching physical ideas.

“Einstein was convinced that the royal road to the laws of physics was not through looking at experimental data, but by developing the mathematical content of well-established theories,” said Farmelo.

This article originally appeared in <a id="8ca56018-3d8d-354a-a272-3de776db3857" href="https://www.nytimes.com/2019/05/28/science/solar-eclipse-einstein-physics.html">The New York Times</a>.

A century ago, on May 29, 1919, the universe was momentarily perturbed, and Albert Einstein became famous.

The eclipse that made Einstein famous

Karen Daniels, a physicist at North Carolina State University, had tweeted notice of the meetup earlier that day: “Are you a physicist into knitting, crocheting, or other fiber arts?” she asked. “I’ll be the one knitting a torus.” (A torus is a mathematized doughnut; hers was inspired by a figure in a friend’s scientific paper.)

At the bar, amid tables cluttered with balls of yarn, Daniels absorbed design advice from a group of specialized knitters, among them Elisabetta Matsumoto, an applied mathematician and physicist at the Georgia Institute of Technology and a co-host of the gathering.

For Matsumoto, knitting is more than a handicraft hobby with health benefits. She is embarking on a five-year project, “What a Tangled Web We Weave,” funded by the National Science Foundation, to investigate the mathematics and mechanics of “the ancient technology known as knitting.”

Some of the oldest examples date to the 11th century in Egypt. But despite generations of practical and experiential knowledge, the physical and mathematical properties of knitted fabric rarely are studied in a way that produces predictive models about how such fabrics behave.

Matsumoto argues that “knitting is coding” and that yarn is a programmable material. The potential dividends of her research range from wearable electronics to tissue scaffolding.

During the happy-hour meetup, she knitted a swatch illustrating a plastic surgery technique called Z-plasty. The swatch was for a talk she would deliver at 8 a.m. Wednesday called “Twisted Topological Tangles.” Scores of physicists turned up, despite a competing parallel session on “The Extreme Mechanics of Balloons.”

“I’ve been knitting since I was a kid,” Matsumoto told her (mostly male) audience. “That was the thing I did to get along with my mom when I was a teenager. It’s just been a dream to take all of this stuff that I learned and played with as a child and turn it into something scientifically rigorous.”

As a first step, her team is enumerating all possible knittable stitches: “We know there’s going to be uncountably many; there’s going to be a countably infinite number. How to classify them is what we are working on now.”

The investigation is informed by the mathematical tradition of knot theory. A knot is a tangled circle — a circle embedded with crossings that cannot be untangled. (A circle with no crossings is an “unknot.”)

“The knitted stitch is a whole series of slipknots, one after the other,” Matsumoto said. Rows and columns of slipknots form a lattice pattern so regular that it is analogous to crystal structure and crystalline materials.

By way of knot theory, Matsumoto essentially is developing a knit theory: an alphabet of unit-cell stitches, a glossary of stitch combinations, and a grammar governing the knitted geometry and topology — the fabric’s stretchiness, or its “emergent elasticity.”

How ‘Floofy’ Is It?

When discussing the emergent properties of knitting, Matsumoto sometimes makes reference to a butterfly, the vibrant blue morpho. Its color is optically emergent, the result not of chemical pigment but of structure. In effect, each wing is a metamaterial: covered in layers of nanosized scales, arranged in a pattern called a gyroid surface, the wing absorbs most wavelengths of light, but reflects blue.

Knitted fabric is also a metamaterial. A length of yarn is all but inelastic, but when configured in slipknots — in patterns of knits and purls — varying degrees of elasticity emerge.

“Just based on these two stitches, these two fundamental units, we can make a whole series of fabrics, and each of these fabrics has remarkably different elastic properties,” Matsumoto told the audience.

She first combined her math-y and woolly mindsets as a Ph.D. student, after admiring a friend’s crocheted interpretation of the hyperbolic plane (curly kale is a vegetable example) and wondering how to do it differently.

“It irritated me that it wasn’t isotropic,” she recalled on the day before her talk. She could see where the crochet had begun, whereas a true hyperbolic plane should betray no starting point and no direction.

She thought, “I can fix that.”

She crocheted a network of lace-like heptagons that produced a more uniform rendering. The hyperbolic plane has been her constant companion ever since. In April, she had a hyperbolic helicoid — a fantastically swirly helix, somewhat like a seashell — tattooed to her left shoulder.

During her talk, Matsumoto passed around her hand-knit swatches: stockinette (standard jersey, fairly stretchy, used for T-shirts); garter (stretchier); ribbing (stretchiest); and seed (not so stretchy, but one of her favorites).

A sizable fraction of her audience also flaunted their hand-knits — sweaters, hats, a water-bottle cozy, indeterminate works in progress. Matsumoto’s most prized hand-knit creation is her “dragon of happiness” shawl (from a design by knitter Sharon Winsauer, aka the Crazy Lace Lady).

Knitting away for two months, Matsumoto encountered one stitch in the dragon’s beard that she had never seen before.

“In the pattern for the dragon, there are all these crazy stitches,” she said — stitches that took up not just a single cell on the pattern grid, but stretched across numerous cells, seeming to follow a horizontal array rather than the usual, vertical orientation.

Her team’s knit theory will incorporate these and other stitch morphologies, as well as intentional stitch defects and constraints, such as how a yarn bends, twists and compresses; how many plies it has; how thick it is; and how “floofy.”

Floofiness refers to a yarn’s “halo area, where ephemeral fuzzy fibers stick out,” Matsumoto said, and it changes the way two pieces of yarn interact with each other, their friction and energy exchange. “I’d love to write a paper using the word ‘floofy’ as a technical term.”

Decoding the Knittable Knots

Matsumoto’s presentation opened a three-hour session titled “Fabrics, Knits and Knots” — the first time that the subject had been addressed at the American Physical Society’s annual meeting.

“Sabetta is spectacularly creative, and she is doing really mathematically sophisticated work,” said Pedro Reis, the session organizer, who leads the Flexible Structures Laboratory at the Swiss Federal Institute of Technology in Lausanne. “She is also attracting a lot of people to the field who might not otherwise even think about science.”

During his introductory remarks, Reis had a vexing encounter with an intertwined microphone cord. “This is a good example of why we really care about this topic,” he said.

Reis grapples with the likes of long-overhand shoelace knotting, climbing knots, basket weaving, surgical sutures, and how to pass on the art of surgical-knot craftsmanship to robots. During the session, his lab mates described how they had used a CT scan to probe the internal structure of knot filaments and the friction that arises where filaments touch. After the meeting, Matsumoto sent him home with some of her swatches.

Derek Moulton, of the University of Oxford, mentioned variants of sailor’s knots, DNA and protein knots, and worms that tie themselves into knots in order to minimize dehydration. He went on discuss “whether a knotted filament with zero points of self-contact may be realized physically.” That is, can a knot exist wherein none of its crossings touch? (It can; try it at home with a strip of paper, or a cord.)

And Thomas Plumb-Reyes, an applied physicist at Harvard, presented his research on “Detangling Hair” to a standing-room-only audience.

“What is going on in tangled hair?” he asked. “What is the optimal combing strategy?”

Shashank Markande, a Ph.D. student working with Matsumoto, reported on their stitch classification work so far. Together, they had derived a conjecture: All knittable stitches must be ribbon knots. (A ribbon knot is a very technical tangle.) And they pondered the corollary: Are all ribbon knots knittable?

Back in February, Markande (who started knitting only recently for the sake of science) thought he’d found an example of an unknittable ribbon knot, using a knots-and-links software program called SnapPy. He sent Matsumoto a text message with a sketch: “Tell me if this can be knitted?”

Matsumoto was just heading out for a run, and by the time she returned, having manipulated the yarn every which way in her head, she had worked out an answer. “I think that can be knitted,” she texted back. When Markande pressed her on how, she added: “It’s knittable by our rules, but it isn’t trivial to do with needles.”

Markande said later, “I was pretty surprised. With my limited knowledge, I thought it could not be knitted. But Sabetta managed to knit it.”

Taking Yarn to the Big Screen

For the Tangled Web project, most of the experimental knitting is produced by a replica of a vintage 1970s knitting machine, the Taitexma Industrial and Home-Based Knitting Machine Model TH-860, which is operated by Krishma Singal, a doctoral student. The machine can also be programmed by punched cards — as was the Jacquard loom, invented in 1804 by Joseph Marie Jacquard and sometimes called the first digital technology.

Matsumoto’s team likes to contemplate how stitch patterns provide code — more complex code than the 1s and 0s of binary — that creates the program for the elasticity and geometry of knitted fabric. The buzzword is “topological programmable materials,” said postdoc Michael Dimitriyev.

He is working on a computer simulation of knitted fabric, inputting yarn properties and stitch topology, and outputting the geometry and elasticity of the real-life finished object. “I’m the killjoy that brings in elasticity,” he likes to say.

The team’s first paper, currently underway, will verify Dimitriyev’s simulations against Singal’s hard-copy swatches. Once the computer simulation is refined, Matsumoto and her collaborators can pull out equations and algorithms for knitted fabric behavior, which in turn could be put into physics engines for computer game graphics or movies.

Pixar’s “Brave” and “Monsters, Inc.” showcased cutting-edge animation of hair and fur, but yarn has yet to have its time in the spotlight. Fabric animation is still very trial-and-error, and it requires time-intensive supercomputers to render.

“This could go in that direction,” Matsumoto said. It’s a good yarn, albeit just at the beginning, and still a bit floofy around the edges.

This article originally appeared in <a id="e0e97db9-cc0f-493f-b8aa-965420889dcd" href="https://www.nytimes.com/2019/05/17/science/math-physics-knitting-matsumoto.html">The New York Times</a>.

'Knitting Is Coding' and Yarn Is Programmable in This Physics Lab

(Science Times): BOSTON — On the eve of the American Physical Society’s annual March meeting, a Sunday “stitch ‘n bitch” session convened during happy hour at a lobby bar of the Westin Boston Waterfront hotel.

'Knitting is coding' and yarn is programmable in this physics lab

PRINCETON, N.J. — On the evening of March 19, mathematician Karen Uhlenbeck gathered with revelers at the Institute for Advanced Study for a Champagne reception. Some hours earlier, she had 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.

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, 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 chef Julia Child. “She knew how to pick the turkey up off the floor and serve it,” Uhlenbeck said.

Jo Nelson, a mathematician at Rice University and a friend of 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, 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, estimates Andrea Prosperetti, a mechanical engineer at the University of Houston. Bubbles entice for their seeming simplicity, which approaches the existential.

“Bubbles are emptiness, nonliquid, 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).

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,” Uhlenbeck said.

— Poetry of Bubbles

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 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 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, 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 3D 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,” 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.”

In December came news of another bubbling lab creation, this time involving ordinary dishwashing solution and inspired by the scutoid, a new geometric shape discovered in 2018 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,” Weaire said.

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. 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,” 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.”

— Instantons, and Other Secrets

During the 2018-19 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.

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 Uhlenbeck’s 40-year-old theorems at the top of the chalkboard.

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, Marques, 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 3D or higher-dimensional space, there is always at least one minimal surface. “We were curious if there were more,” Marques said.

There are, according to the bubbles. A new proof by Antoine Song, a doctoral student of 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 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 Uhlenbeck’s work is to consider the challenge of scale. She started drawing about 10 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.

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, 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, Uhlenbeck moved on to other mathematical mysteries for a few decades, but during the past 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, Uhlenbeck, exhausted, took two Fridays off. But when their sessions resume, they’ll dive deep into some even higher-dimensional bubbling.

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

This article originally appeared in <a href="https://www.nytimes.com/2019/04/08/science/uhlenbeck-bubbles-math-physics.html" id="e538623e-8ec7-3109-af2b-be7d9be9a56a">The New York Times</a>.

(Science Times)

The Bubble-Verse: A Window Has Opened

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Siobhan Roberts

Articles written by the author

The eclipse that made Einstein famous

'Knitting is coding' and yarn is programmable in this physics lab

The Bubble-Verse: A Window Has Opened