“I am a mathematician. Mathematicians do exotic research so it’s hard to describe exactly what I do in lay terms. I work on partial differential equations which were originally derived from the need to describe things like electromagnetism, but have undergone a century of change in which they are used in a much more technical fashion to look at even the shapes of space.”...
“[But] My first love is the outdoors — I enjoy mountain climbing, back packing, hiking, canoeing, swimming, and bicycling. Many of these interests I inherited from my parents who, at age 83, are still hiking and back packing. I am at home in nature and, when I can't be out in the wilderness, I can often be found in my garden at my home in Austin. That's the real me. My day-to-day life is something very different.”
THAT is how Karen Keskulla Uhlenbeck, who on March 19 became the first woman to be conferred the prestigious Abel Prize in mathematics, described herself in a brief autobiography that she wrote in the journal Math Horizons way back in 1996 when she was at the University of Texas at Austin and had already become one of the top-ranking mathematicians of the day. Yet, it was only 17 years after the award was instituted (2002) by the Norwegian Academy of Science and Letters that the Abel Prize Committee recognised her work, which spans a diverse range in mathematics and includes topics that underpin modern theoretical physics such as gauge theory and string theory in particle physics. Her contributions have also opened up an entirely new area of mathematics called geometric analysis.
The award recognises contributions to mathematics that are of extraordinary depth and influence. It is presented annually by His Majesty King Harald V of Norway and includes a cash award of six million Norwegian krone (about $700,000). The selection of the laureate is based on the recommendations of the Abel Prize Committee, which comprises five internationally recognised mathematicians.
The committee chose Karen Uhlenbeck, 77, for the 2019 award for, to quote the citation, “her pioneering achievements in geometric partial differential equations [PDEs], gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics…. [She] is a founder of modern Geometric Analysis. Her perspective has permeated the field and led to some of the most dramatic advances in mathematics in the last 40 years… and her ideas and leadership have transformed the mathematical landscape as a whole.”
Geometric analysis, which has become a standard tool for mathematicians, is a field of mathematics that combines the powerful techniques of analysis—which is basically a generalisation of calculus—and differential equations with problems in geometry and topology (the study of shapes). Specifically, this involves the study of mathematical objects such as curves, surfaces, fields and connections (geometrical representation of transporting quantities from one point in space to another in a consistent manner) and their relation to quantities such as “energy” (in the mathematical sense) and volume. (Energy in mathematics is an abstract generalisation of the familiar concept of energy in physics.)
The other major awards she has won include the Steele Prize of the American Mathematical Society in 2007, the National Medal of Science in 2000 and the MacArthur Fellowship (popularly referred to as the “Genius Grant”) in 1983. She was elected to the United States’ National Academy of Sciences in 1986 and to the American Philosophical Society in 2012. Much of Karen Uhlenbeck’s career was spent at the University of Texas at Austin where she occupied the Sid W. Richardson Foundation Regents Chair in Mathematics from 1988. Currently, she is a Visitor at the Institute of Advanced Study (IAS), Princeton, New Jersey.
This is not the first time she has broken the glass ceiling. In 1990, she was only the second woman in history to give a plenary address at the quadrennial International Congress of Mathematicians in Kyoto, Japan, 58 years after Emmy Noether was bestowed that honour. “Such a shocking statistic reflects just how hard it is for many women to achieve the recognition that they deserve in a male-dominated field,” said Jameel Sadik “Jim” Al-Khalili, a professor of theoretical physics at the University of Surrey, United Kingdom, in a brief biography he wrote for the Norwegian Academy. “But by that point in her career,” he said, “Uhlenbeck had already established herself as one of the world’s preeminent mathematicians, having overcome many hurdles, both personally and professionally.... Yet for many, the recognition of her achievements should have been far greater, for her work has led to some of the most important advances in mathematics in the last 40 years.”
Sun-Yung Alice Chang, a mathematician at Princeton University who was a member of the 2018-19 Prize Committee and who regards Karen Uhlenbeck as her mentor, is reported to have told the journal Science that the committee stuck strictly to research in choosing the winner and gender did not have any bearing on the selection. As Science also notes, the citation does not refer to her mentoring efforts or her involvement in the Women and Mathematics Programme that she, along with Chuu-Lian Terng of the University of California, Irvine, founded 25 years ago at the IAS. According to Science , that programme today brings together about 60 women undergraduate, graduate and postdoctoral students for two weeks of lectures, panel discussions and informal interactions.
Karen Uhlenbeck, the eldest of four children, was born in Cleveland, Ohio, in 1942. Her father was an engineer and her mother an artist and a schoolteacher. “I can’t say that I was really interested in mathematics as a child or adolescent, mostly because one doesn’t really understand what mathematics is until at least halfway through college,” she said in an autobiographical essay she wrote in Math Horizons in 1996. “As a child I read a lot. I read everything, including all the books in our house three times over. I’d go to the library and then stay up all night reading. I used to read under the desk in school…. I was particularly interested in reading about science.” She read Fred Hoyle’s books on astrophysics and found them very inspiring. “I also remember… One, Two, Three… Infinity by George Gamow, and I remember the excitement of understanding this very sophisticated argument that there were two different kinds of infinities. I read all of the books on science in the local library and was frustrated when there was nothing left to read.”
She joined the University of Michigan though she had wanted to go to the Massachusetts Institute of Technology (MIT) or Cornell University, “but my parents decided that those institutions were too expensive”. Initially wanting to graduate in physics, she switched majors “when they started taking attendance in the physics lecture. I also had trouble with labs—I could not learn to… fudge the experiments…. So I switched to math and have been interested in it ever since.”
After graduating in 1964, she spent a year at the Courant Institute of Mathematical Sciences of New York University and then, after her marriage to her biophysicist boyfriend, Olke Uhlenbeck, moved to Brandeis University for postgraduate study with a handsome National Science Foundation scholarship. Although already aware of the predominantly male culture in academia, she initially chose to ignore these issues and, for that reason, avoided prestigious Ivy League universities such as Harvard where her husband had joined for his PhD because it would have been more competitive for a woman to succeed there.
“There was a handful of women in my graduate programme…. [But] It was evident that you wouldn’t get ahead in mathematics if you hung around with women. We were told that we couldn’t do math because we were women. If anything, there was a tendency to not be friendly with other women. There was blatant, overt discouragement, but also subtle encouragement. A lot of people appreciated good students, male or female, and I was a very good student. I liked doing what I wasn’t supposed to do, it was a sort of legitimate rebellion,” Karen Uhlenbeck wrote. She completed her PhD from Brandeis in 1968 under Richard Palais.
Her PhD was in the subject calculus of variations, which is the study of how small changes in one quantity can be used to find the maximum or minimum value of another quantity, for example the shortest distance between two points. It is, of course, intuitively obvious that it is a straight line in the familiar space around us, but it is possible to prove that with full mathematical rigour, which technique is then amenable to generalisation to arbitrary spaces, and it need not always be so straightforward. One of the oldest and most famous problems in calculus of variations is the so-called “brachistochrone” due to Johann Bernoulli in 1696, which seeks to find the curve down which a ball will roll in the least possible time from one point to another.
Minimal surfaces
A similar problem in two dimensions would be surfaces formed by soap films, for example. The soap film that forms on the circular toy wire frame used to blow bubbles leads to similar minimising problems; the soap film minimises the surface area it encloses and, equivalently, surface tension and energy. For mathematicians, this leads to the generalised question, can you find a surface of minimum area produced by a given arbitrary frame? As Marianne Freiberger, a mathematician and editor of the online mathematics site Plus Maths (https://plus.maths.org/content/abel-prize-2019), points out: “Finding minimal surfaces is an incredibly difficult problem.” Until the 19th century, only three of them were known: the trivial case of a plane, the catenoid and the helicoid (see figure).
Another way of looking at minimal surfaces is in terms of “energy”. Just as in the case of the minimisation of distance problem, minimal surfaces in arbitrary geometrical spaces can be considered as problems in the calculus of variations that minimise the energy-like quantities in that space. Karen Uhlenbeck’s thesis was “The Calculus of Variations and Global Analysis”.
After her PhD, she began to look at the problem of minimal surfaces, working with Jonathan Sacks in the 1970s. This was to generalise what her PhD supervisor, working along with Stephen Smale, had formulated called the Palais-Smale condition for minimising certain mathematical constructs or functionals. But this was seen to fail in certain minimal surface problems because of singularities on the surface. Karen Uhlenbeck and Sacks introduced a new concept of what is now termed “bubbling” to solve the singularities problem and obtain a generalisation of the Palais-Smale condition. The ideas and the methods of this revolutionary paper are today used as standard mathematical tools to solve similar problems arising in other areas of mathematics as well.
“Even when I had had my PhD for five years, I was still struggling with whether I should become a mathematician. I never saw myself very clearly,” Karen Uhlenbeck told Claudia Henrion, who wrote about her in the online publication Celebratio Mathematica in 1997. “I’m not able to transform myself completely into the model of a successful mathematician because at some point it seemed like it was so hopeless that I just resigned myself to being on the outside looking in.”
After a brief postdoctoral stint at MIT, she moved to University of California, Berkeley, where she studied general relativity and space-time geometry, topics that would shape much of the future work she engaged in. “I was told, when looking for jobs after my year at MIT and two years at Berkeley, that people didn’t hire women, that women were supposed to go home and have babies,” Karen Uhlenbeck said in her Math Horizons autobiography.
“So the places interested in my husband—MIT, Stanford, and Princeton—were not interested in hiring me…. I want to be valued for my work as a mathematician, not because I’m a member of a particular group…. Prejudice is very rude because it treats you as a member of a class or group instead of as a person…. I ended up at the University of Illinois [at Urbana-Champaign with a faculty position in 1971], because they hired me and my husband,” she wrote. She subsequently held faculty positions at the University of Illinois, Chicago (1976-83), and the University of Chicago (1983-88) after which she moved to Texas.
Her stint at Urbana-Champaign was not a very happy one; she felt isolated and undervalued, she was often, as she has remarked, viewed as “faculty wife”. At the University of Chicago five years later, however, things began to look up again. There were other female professors, who, according to Al-Khalili, offered her advice and support, as well as other mathematicians who took her work more seriously. It was in Chicago that her interests diversified into non-linear PDEs, differential geometry, gauge theory, topological quantum field theory and integrable systems. Given her inherent passion for physics, at Texas she began to interact with Steven Weinberg, one of the founders of the unified electroweak theory in physics, which combines electromagnetism and weak interactions into one unified mathematical framework based on the mathematical principles of what is called gauge theory.
Actually, Karen Uhlenbeck got interested in gauge theory after hearing a talk by Michael Atiyah (who passed away recently) in Chicago. The general class of equations that govern these gauge theories in physics are called Yang-Mills equations, which are basically non-linear extensions of equations of electromagnetism. Karen Uhlenbeck pioneered the study of these equations from a rigorous analytical viewpoint, and it has become one of her most noted works. Her papers inspired the mathematicians Clifford Taubes and Simon Donaldson and led to the work that won Donaldson the Fields Medal, the most coveted prize in mathematics for mathematicians under 40, in 1986.
Similarly, physicists had predicted the existence of mathematical objects called instantons, which describe the behaviour of surfaces in four-dimensional space-time. Karen Uhlenbeck became one of the world’s leading experts in the field. Instantons and Four-Manifolds , the classic textbook she co-authored with Daniel S. Freed, has inspired a whole generation of mathematicians, writes Al-Khalili.
Karen Uhlenbeck grew up as a very private person. But this changed as her career in mathematics brought her fame, particularly after the MacArthur “genius” award. “As a young academic,” she wrote in her 1996 autobiographical essay, “I worked by myself a lot. In fact, that was one of the attractions of mathematics…. I consider dealing with my siblings the hardest thing I’ve ever done in my life. That had a great impact on my choosing a career—I wanted a career where I didn’t have to work with other people. I’ve always been competitive, but I find it difficult to cope with the attitudes of people who lose.... As my career advanced, however, I found I had a lot to learn from other people of all sorts. I have found it very rewarding to deal with younger mathematicians, and I now truly enjoy collaborative projects.”
According to Claudia Henrion: “Over time, however, Uhlenbeck gradually became aware that such determined individualism can lead to isolation, and can have a negative impact on professional growth. As her career developed, community became essential to success.” Also, she began to get involved with gender issues and problems of women and minorities in academia, mathematics in particular.
“Starting in the 1990s I got involved in a lot of non-mathematical things,” she told Allyn Jackson in a 2018 interview to Celebratio Mathematica . “I was one of the handful of people who started the Park City Mathematics Institute (PCMI) [in Utah—a “vertically integrated” mentoring programme that brought together schoolteachers, undergraduate students and researchers]…. Through the PCMI I got involved with women’s issues. I had never been involved in them before because I wasn’t interested in politics. But at some point, I thought…Where are all the women?
Mentoring programme
“In the early 1990s… we didn’t see large numbers of women coming after us. I felt I did owe something for my success. So Chuu-Lian and I started working together on integrable systems, and we also started working in the Women in Math programme. It was originally associated with the PCMI, and later the IAS took it on and supported it.” While at Texas she also ran a mentoring programme for women in mathematics. Now, being at the IAS itself, Karen Uhlenbeck remains deeply involved in gender issues in the fields of mathematics and science.
“I am aware of the fact that I am a role model for young women in mathematics, and that’s partly what I’m here for,” she wrote in her Math Horizons essay. “It’s hard to be a role model, however, because what you really need to do is show students how imperfect people can be and still succeed. Everyone knows that if people are smart, funny, pretty or well-dressed they will succeed. But it’s also possible to succeed with all of your imperfections. It took me a long time to realise this in my own life. In this respect, being a role model is a very unglamorous position, showing people all your bad sides. I may be a wonderful mathematician and famous because of it, but I’m also very human,” she wrote.
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