For her “pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems and for her “fundamental impact” on analysis, geometry, and physics, Karen Keskulla Uhlenbeck has received the 2019 Abel Prize — probably the most prestigious award a mathematician can ever claim, widely regarded as the “Nobel Prize” of math.

Uhlenbeck, a staunch supporter of women in science and math, was the first woman to have ever been awarded the distinction.

A mathematician’s trajectory

Image credits: IAS.

In 1990, in Kyoto, Japan, Karen Uhlenbeck gave a plenary lecture at the International Congress of Mathematician (ICM). She was only the second woman to ever do so, after Emmy Noether in 1932. By the time she addressed the ICM, Uhlbeck was already a well-established name, and one of the most prominent mathematicians in the world.

Although she did receive awards and accolades throughout her career, many of her colleagues felt like her recognition should have been far greater — and she was barely able to have a career in mathematics at all.

As a child, she was always curious about everything. She enjoyed reading everything she could, regularly shutting herself away so she could read, often late at night, sometimes even during her school classes.

Her love affair with mathematics happened rather spontaneously. She had wanted to do physics, but math provided her with more satisfaction, and it also meant that she didn’t have to do any lab work — which she resented greatly. Even as her career progressed, she enjoyed the peace of solitary work.

A brilliant and promising mind coupled with a great desire to succeed in research seemed to put her on a trajectory for success. But things weren’t nearly as simple.

The two body problem — and nepotism

In science, it’s not uncommon for both members of a couple to be active researchers, either in similar or different fields. This makes a lot of sense — that’s how the life of a scientist often goes about. But this also leads to some very practical problems when it comes to finding a job. Finding a research job is hard enough, but finding two in the same city is a very difficult challenge. Among scientists, this is often informally referred to as the “two-body problem.”

The two-body problem

Academics tend to move a lot chasing job offers — and many of them have life partners also working in academia. It’s estimated that than 70% of academic faculty have a working partner, while more than 30% have an academic partner.

This leads to substantial difficulties in finding two jobs at the same university or within a reasonable commuting distance from each other. This forces many couples to either split up or one of the partners to give up on his or her career. This is called the two-body problem, in allusion to the insolvable three-body problem in classical mechanics.

Uhlenbeck experienced this firsthand as she tried to find work along with her then-husband, Olke. She was turned down several times under so-called nepotism regulation: an alleged ban on both husband and wife working at the same institution. Many, including herself, felt this was just a pretext to avoid hiring women as part of the faculty. Decades later, Uhlenbeck tried raising this issue, the universities simply denied it.

“I was told, when looking for jobs after my year at MIT and two years at Berkeley, that people did not hire women, that women were supposed to go home and have babies. So the places interested in my husband – MIT, Stanford, and Princeton – were not interested in hiring me. I remember that I was told that there were nepotism rules and that they could not hire me for this reason, although when I called them on this issue years later they did not remember saying these things.”

Thankfully, she was able to find a generous scholarship at Brandeis University, which would prove to be a turning point for her. It was there that she completed her PhD with Richard Palais as her adviser. Palais was trying to find a link between analysis (a generalization of calculus) and topology and geometry (the structure of shapes). He was also supportive of her, providing a very fruitful collaboration for Uhlenbeck.

Uhlenbeck’s work has applications in surfaces like soap bubbles.

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She then moved on to MIT, Berkeley, and the University of Illinois at Urbana-Champaign, where both she and her then-husband were professors. She felt underappreciated there, a mere “faculty wife”. So she kept moving on, reaching the University of Chicago. It wasn’t until Chicago that she received the support of other faculty — there were other female professors there, who gave her valuable guidance. She finally felt like she belonged.

She had already made a name for herself and received the praise of colleagues, but in Chicago, Uhlenbeck was able to make a true mark, establishing herself as one of the most outstanding mathematicians of her generation. Her interests were broad and varied, spanning over fields such as nonlinear partial differential equations, differential geometry, gauge theory, and quantum field theory.

Ultimately, she became a household name in top mathematics but she was a rather unusual presence — not just because a woman, which was and still is relatively rare in top scientific circles, but because of her way of thinking. Mark Haskins, a mathematician at the University of Bath, UK, who was one of Uhlenbeck’s doctoral students, described her as having a non-linear and non-intuitive way of thinking. She would receive a question from someone and give them a baffling answer.

“Your immediate reaction was that Karen had misheard you, because she had answered a different question,” Haskins says. But weeks or months later, you’d realize “you haven’t asked the right question — and the answer to the right question was what Karen said.”

She definitely represents a very specific way of thinking about math, and about things in general. She is one of those mathematicians who appear to have “an innate sense of what should be true”, even if they cannot always explain why, Haskins recalls. The inquisitive mind of the young little girl is still very much present in mature Uhlenbeck.

“If I really understanding something, I’m bored” Haskins remembers her saying.

She also remained stunningly humble in her success. Haskins remembers that even as a professor, Uhlenbeck regularly attended other seminars, even junior seminars — something which for most mathematicians, is simply not a consideration.

Bubbles and the Abel Prize

Instantons are extremely important to physicians, and some have been used to describe the fundamental properties and evolution of the universe, such as the Hawking-Turok Instanton, depicted here. Image credits: Cambridge University.

In a phone interview for the Abel Prize, Uhlenbeck said she was “a bit overwhelmed,” adding that “I hope I can hold myself together for this.” The reasons why her work warranted the Abel Prize are worthy of a separate article — or rather, a book. Indeed, a book she co-authored called Instantons and Four-Manifolds has become a staple for mathematicians in the field.

Instantons were of great interest to Uhlenbeck — these are solutions to equations of movement which define motion as a function of time. They are also useful in quantum mechanics, to describe the probability for a quantum mechanical particle tunneling through a potential barrier.

Perhaps her most important work was on so-called ‘minimal surfaces’. Soap bubbles, beautifully spherical objects are capable of forming minimal surfaces in an instant. They do so in order to minimize the wall tension and energy, by pulling the bubble into the smallest possible shape — known for centuries to be a sphere. But computing these minimal surfaces can be a daunting task, and in more complex situations, the theory simply broke down.

For instance, if instead of blowing the bubble we would dip a deformed wire loop into the soap bubble solution, it would form a disc, with its boundary given by the wire loop and of minimal area, explains Arne Sletsjøe, assistant professor of Mathematics in the University of Oslo. In this situation, soap bubbles might still form the area of minimal surface instantly, but computing and predicting it is a very different matter. Uhlenbeck attacked this problem from a mathematical perspective and expanded our understanding of such surfaces.

A helicoid minimal surface formed by a soap film on a helical frame. Image credits: Blinking Spirit / Wikipedia.

Uhlenbeck also made her mark in gauge theory, which is used in powerful theories in physics. A key idea of Einstein’s work, something still regarded as the Holy Grail of physics, is that laws of physics should be the same in all frames of references — and this is also the general idea of a gauge theory: finding connections that compare measurements taken at different points in space and looking for quantities that don’t change. Working at the interface of mathematics and physics has been a point of focal interest for Uhlenbeck.

Of course, this doesn’t even begin to scratch the surface of Uhlenbeck’s fundamental contribution. She is a founder of geometrical analysis, and her work stretched to cover other fields such as integrable systems and harmonic mappings. Just a few months ago, Uhlenbeck worked with Penny Smith to submit another paper on gauge field equations.

The woman who could

Despite some progress, women in math and science still face great adversity.

“We all thought that once the legal barriers were down, women and minorities would walk through the doors of academia and take their rightful place,” Uhenbeck once said.

Uhlenbeck spoke how she felt she was the first generation of female mathematicians who could truly succeed in the corrosive culture of mathematical publishing — and scientific publishing in general. But Caroline Series, the president of the London Mathematical Society, said she would have said the same thing — despite being 10 years younger than Uhlenbeck; and if you would ask younger female mathematicians, you almost invariably get the same result: they feel like they barely get through.

“I think if I were five years older, I wouldn’t have made it. It’s better now, but I worry a lot about the other minorities. There are many women and minorities capable of being great mathematicians,” Uhenbeck said after receiving the prize.

Needless to say, Uhlenbeck did manage to break through, but many others might not.

In 2007, when Uhlenbeck was awarded the American Mathematical Society’s Steele Prize for Seminal Contribution to Research in 2007, she pointed a finger at the mathematical community, blaming it for the small number of leading women mathematicians. She summed it up by saying that “changing the culture is a momentous task in comparison to the other minor accomplishments I have mentioned”.

Uhlenbeck has inspired a generation of mathematicians, and as she wrote in 1996, she is painfully aware of this. It’s a hard job because “what you really need to do is show students how imperfect people can be and still succeed. … I may be a wonderful mathematician and famous because of it, but I’m also very human.” For all her immense talent and hard work, Uhlenbeck is, certainly, very much human.

Karen Keskulla Uhlenbeck is a founder of modern geometric analysis. Her work has opened the door for some of the most dramatic advances in mathematics (and physics) over the past 40 years. She has come a long way from the girl who would secretly read in courses, and in addition to having a stellar career in mathematics, she is also a relentless advocate of women’s’ and minorities’ rights in science.

Her research has already inspired a generation of mathematicians, and she now hopes that her winning the Abel Prize will inspire even more girls to pick up mathematics and science. We can only hope for the same thing.