On Tuesday, the King of Norway presented Karen Uhlenbeck with the Abel Prize, the counterpart of the Nobel Prize for the field of Mathematics. The award was announced back on March 19, and a number of wonderful expositions of her research were published in the time since then.
Quanta Magazine’s Erica Klarreich prepared the most well-crafted exposition of the remarkable work that earned Uhlenbeck her award. Uhlenbeck created the rich subfield of Geometric Analysis and made enormous contributions in other subfields, including Gauge Theory. The frontier of research mathematics is always a challenge to explain, but this excerpt captures the flavor.
Uhlenbeck and Sacks’ work showed that the topology of this space informs what singularities a harmonic map can have, since bubbles can form only around holes. And conversely, the existence of harmonic maps can illuminate the geometry and topology of the space. Their work was instrumental to the birth of a new field of mathematics: modern “geometric analysis.”
The bubbling analysis “has been revolutionary, in a sense,” Labourie said. “There was before the Sacks-Uhlenbeck paper, and after.”
Since that time, bubbling phenomena have been discovered in a wide range of settings in mathematics and physics, said Sun-Yung Alice Chang, a mathematician at Princeton University. “[Uhlenbeck’s] influence crosses the different branches of mathematics,” she said.
There are several other good expositions, and of course many on Twitter expressed their delight.
The remarkable mathematician Terry Tao wrote of Uhlenbeck’s influence on his own work.
I was pleased to learn this week that the 2019 Abel Prize was awarded to Karen Uhlenbeck. Uhlenbeck laid much of the foundations of modern geometric PDE. One of the few papers I have in this area is in fact a joint paper with Gang Tian extending a famous singularity removal theorem of Uhlenbeck for four-dimensional Yang-Mills connections to higher dimensions. In both these papers, it is crucial to be able to construct “Coulomb gauges” for various connections, and there is a clever trick of Uhlenbeck for doing so, introduced in another important paper of hers, which is absolutely critical in my own paper with Tian. Nowadays it would be considered a standard technique, but it was definitely not so at the time that Uhlenbeck introduced it.
Uhlenbeck is the first woman to be awarded the prize, although it has been awarded every year since 2003. There has been a lot of reflection on how long it took for this barrier finally to be broken, and how limited the opportunities were for women even to participate in the highest levels of STEM fields until recent years.
Claire Maldarelli has a great discussion of this in a piece in Popular Science.
Unfortunately, Uhlenbeck’s win is noteworthy for reasons other than the importance of her findings: She’s the first woman to receive the award. The fact that this is the first time a woman has won such an honor is not really a hint but rather a bright, flashing light at major issue in STEM: Even today, few women pursue advanced degrees in mathematics….
The first step to making people confident in their math skills is making them believe they should be the ones doing that work in the first place. Showing the world at large how integral all sorts of individuals are to new research in math and the sciences—by giving those experts well-deserved awards, for example—is one small step toward fixing this disparity.
It is only in recent years that other remarkable achievements by women that were lost in our historical narrative finally received some recognition. The book and subsequent movie Hidden Figures did this for Katherine Goble, Mary Jackson, Dorothy Vaughan, et al, the Black women of NASA who calculated the trajectory of spacecraft on the way to the moon. Just recently I learned that two women, Ellen Fetter and Margaret Hamilton, were the computing experts whose work on computer weather simulation were instrumental in the foundations of Chaos Theory. Their supervisor pursued the discoveries and has received all the acclaim.
Uhlenbeck herself has been very forthright about her good fortune to come on the scene just as opportunities were opening for women. She credits Title IX and the waves of feminism during her life for making her career possible. She earned a reputation for being a good mentor working hard to keep her department inclusive despite the insidious forces of broader society that always loom.
With that, I will leave you with the transcription of her wonderful acceptance speech from Tuesday. She gave a further technical lecture of her work along with several of her collaborators on Wednesday, and I unfortunately still have not been able to listen. Many thanks to Eileen Clancy for contributing to the transcription, though the errors are all mine.
Abel Prize Acceptance Speech of Karen Uhlenbeck
May 21, 2019 in Oslo, Norway
Your majesty, your excellencies, ladies and gentlemen, thank you very much. I am astonished to find myself in the company of a very distinguished group of mathematicians, the 19 previous winners of the Abel Prize. Very unexpected. My sincere thanks to all involved. Pure mathematics is a wonderful subject, and I feel very privileged not only to have been a research mathematician, but to have enjoyed it and to now be rewarded for it.
Here are some thoughts on how I ended up here. First it is already significant that there is such a prize. Mathematics has been studied for several millennia and has been central to both scientific and philosophical thinking. I can identify two opposite facets: the usefulness of mathematics as the language of science and the intellectual pleasure of invention.
Mathematicians tend to be driven by the second and to lose sight of the first. And often there is a long delay from the formulation of the mathematics to its applications. Many thanks to the Norwegian government for establishing the Abel prize and to the Norwegian Academy of Science and Letters for awarding us recognition that we may celebrate both sides and remember mathematicians like Abel, whose mathematics we still use today.
While my parents had an intellectual bent, as a girl, I was not designated for a profession, except perhaps as a high school teacher. In 1957, during my second year of high school, the Soviets launched the satellite Sputnik. The United States woke to the realization that as a country we were not producing mathematicians and scientists. I benefited directly a few years later from a few of the many programs put in place to catch up. It was such an important goal that women and minorities were explicitly included in the target audience.
During my years at the University of Michigan, these programs fit my background and abilities. I was much encouraged by my professors and I received a National Science graduate fellowship which funded four years of graduate study.
My parents were pleased as my education had cost very little and I was qualified to teach high school. But, to progress in a research career, more had to happen–a post-doctoral position and then a professorship were necessary. At this time, it seemed very unrealistic. If I had been five years older, it would not have happened.
While the first wave of feminism resulted in women obtaining the right to vote, that was 1920 in the U.S.; but earlier starting out in 1913 in Norway.
The second wave of feminism started in the 1960s with demands for equal job opportunities and equal pay. We all read Virginia Woolf’s “A Room of One’s Own,” Simone de Beauvoir’s “The Second Sex” and Betty Friedan’s “The Feminine Mystique.”
I was having none of it. I wanted to do mathematics, not change the world.
How I would do this was unclear. But, for the most part, my friends, teachers, and husbands encouraged me. I only applied to programs which accepted women and was helped around discriminatory attitudes and laws.
In 1972, the U.S. federal rights law, known as Title IX, unlocked the already battered door. This law prohibits discrimination on the basis of sex in any federally funded US education program. By then, I was an assistant professor and the legal barriers were down. My understanding is that women mathematicians in Norway also began obtaining professorships in this time frame. This would not have happened without the second wave of feminism.
I started my thesis work with Richard Palais in 1965 in the rapidly developing field of Global Analysis. Roughly speaking, a function is a transformation between mathematical spaces. In the new view, a function is a point in an infinite dimensional space. Minimization, flow equations and cumbersome complicated partial differential equations could be put in a different framework. Ideas from geometry, topology and mathematical physics were reformulated.
I was admittedly entranced by this admittedly grandiose viewpoint but also set myself to learn the technical underpinnings needed to shore it up. The initial promise of this approach did not really begin to pay off for a decade or so.
I credit Shing-Tung Yao’s solution to the Calabi conjecture in 1976 with the start of the big payoffs. Global Analysis was renamed Geometric Analysis with Yau at the helm, and I was perfectly situated to tackle some of those problems. This is the story of the soap bubbles.
Meanwhile high energy theoretical physicists discovered that some of the ideas in Global Analysis were tremendously useful to them. I mention the Atiyah-Singer index theorem. Both Michael Atiyah and Isadore Singer have Abel Prizes. The resulting dialogue between mathematics and physics continues to this day.
I heard four lectures by Michael Atiyah on the Yang-Mills equations, dug out the very obtuse texts on fiber bundles, and I have to tell you they were hard for me but I did pity the physicists. And I had the right combination of technical background plus viewpoint to help lay the foundations of the field.
The connection between mathematics and physics remained enticing. From physics papers I learned about loop groups and began a long profitable collaboration with Chuu-Lian Terng on integrable systems.
Ed Witten pointed out to me the appearance of the KdV equation in what is now quantum cohomology. I sadly have never understood a convincing explanation of this phenomenon. As merely an aside, I wonder if this is not just part of the unreasonable effectiveness of mathematics. If there is a mathematical idea around, it seems as if somebody will use it.
Attitude change was an important ingredient in both the success of my mathematics and my progress professionally.
The idea of who will do mathematics and science is still changing. It gives me great pleasure -- in addition to thanking the Norwegian government and the many people who have encouraged and helped me -- to watch the younger mathematicians, many of whom are now women, create new ideas and directions. And while we are looking towards the future, each of us might consider contributing to the health of the planet by taking one less airplane flight in the coming year.