by Allyn Jackson
Born in 1942, Karen Keskulla Uhlenbeck received her Ph.D. in 1966 from Brandeis University, under the direction of Richard Palais. After postdoctoral positions at the Massachusetts Institute of Technology and the University of California at Berkeley, she held faculty positions at the University of Illinois at Urbana-Champaign (1971–1976), the University of Illinois at Chicago (1976–1983), and the University of Chicago (1983–1988). In 1988, she was appointed to the Sid W. Richardson Foundation Regents Chair in Mathematics at the University of Texas at Austin. Now retired from Texas, she is currently a visitor at the Institute for Advanced Study in Princeton.
Over the past half-century, Uhlenbeck has been one of the outstanding figures in analysis and has made especially profound contributions in geometric analysis. She is probably best known for her work in mathematical gauge theory, which laid the foundations for some of the most significant advances in mathematics in the twentieth century. For this achievement, the American Mathematical Society awarded Uhlenbeck the 2007 Steele Prize for a Seminal Contribution to Research. Among her other honors are a MacArthur Fellowship (1983) and the National Medal of Science (2000). She was elected to the U.S. National Academy of Sciences in 1986 and to the American Philosophical Society in 2012. In 1990, she became the second woman, after Emmy Noether in 1932, to give a plenary address to the International Congress of Mathematicians.
What follows is the edited text of an interview with Uhlenbeck, conducted in Princeton by Allyn Jackson in March 2018.
Interested in everything
As a child I read a lot, and I read everything. I’d go to the library and then stay up all night reading. I used to read under the desk in school […] I read all of the books on science in the library and was frustrated when there was nothing left to read.
Jackson: You have such a beautiful maiden name, Karen Keskulla.
Uhlenbeck: Thank you. I still sign myself Karen Keskulla Uhlenbeck. My grandfather was Estonian, although he was born in Riga, which is in Latvia. “Keskula” was the original spelling. It comes from the Finnish–Estonian language and means “center of the village”. If you go to Helsinki and drive around, you will see signs for “Keskulla” all over! That’s on my father’s side. My grandmother was German.
Jackson: Was your father born in the United States?
Uhlenbeck: Yes. I’m not sure about my grandmother, but I think she was born here too. I do not know much about my background. We are not a history-keeping family.
My mother was one of 12 children of a Methodist minister. He died when she was 5 and while her own mother was pregnant. My mother had an older brother who was able to help the family through. But she did not grow up with a lot of money.
Jackson: Where was this?
Uhlenbeck: In Farmingdale, New Jersey. There was a lot of sisterly help too — my mother was very attached to at least one of her older sisters. It wasn’t a question of their mother taking care of 12 children. I knew my maternal grandmother, and she wasn’t one to take care of anybody. She was a remarkable woman. She must have been at least 5 feet 11 inches [about 1.8 meters] tall, which at that time made her a very tall woman. She was always interested in everything and very talkative. I come from a line of strong women on the maternal side.
Jackson: Your mother became an artist.
Uhlenbeck: She was supported by at least one older sister in that. She started college around 1928, at the Art Student’s League in New York City. After about a year, she realized that she wouldn’t get a job as an artist. So she transferred to the City College of New York and got a degree in education. I think for the next few years of her life, until she met my father, she taught art in high school, and maybe grade school too. She always liked teaching little kids.
I don’t know how those first years of the Depression were for her. I think there was a time when she did not get enough to eat. She graduated from college in 1932, as my father did. That was not a good year to graduate.
One of my father’s aunts had a job as a buyer for Macy’s and was fairly well into the upper middle class. She financed my father to study mechanical engineering at MIT. In 1932 the only job he could find was in the gold mines of California. After a year or so, he went to Utah and worked in the mines there. He fell in love with the mountains out West.
The Depression mentality hung over our childhood. My parents were both upwardly mobile, educated, intellectual people who at a critical time of their lives, when they were starting out, had really severe economic problems. There was a background of real worry about money, but also an intellectual background.
Some of my mother’s siblings went to college, and her father was somehow associated with Drew University.
Jackson: Was your father the first in his family to go to college?
Uhlenbeck: Yes, I’m pretty sure he was. He grew up in Ossining, New York. After he had been out West, he got a job with the Aluminum Company of America, which after a while became Alcoa. He and my mother spent most of their married life here in New Jersey, until they retired to Santa Fe.
Jackson: What kind of paintings did your mother paint?
Uhlenbeck: My mother, through all her life, was really interested in scenery. But a lot of her early work is very much 1930s work. I trace it when I go to art museums and can see things like my mother painted. Through my childhood she mostly painted scenes around New Jersey. She tried painting her kids but that didn’t work very well.
Jackson: She didn’t paint people so much?
Uhlenbeck: Right. About the time I went away to college, she switched from oil painting to water color, I suspect because of the chemicals involved.
Jackson: You were one of four kids in your family. What did the other three end up doing?
Uhlenbeck: Well, they had difficulties. My father insisted my brother try to get a degree in engineering, and he did study engineering, but then he went to divinity school. He has not had an easy life. My youngest sister was badly handicapped, and she died more than 20 years ago. My middle sister is a poet. She lives with her husband in Carlisle, Massachusetts. They are very interested in wildlife and ecology.
Jackson: You were the oldest?
Uhlenbeck: Yes.
Jackson: Were there important influences in your youth in mathematics and science? Was your father an influence?
Uhlenbeck: Certainly there was indirectly an influence from my father. We were great readers, and my father took out from the library books by people like Fred Hoyle and George Gamow, which I ended up reading. He was more worried about my brother being the scientist than he was about me. My mother’s friends were painters, and they did not have your average, normal lifestyle. So when I was young, I was introduced to a lot of people who did not live normal, middle-class lives. I have very striking memories of that.
Jackson: Your father wanted your brother to become interested in science because he was a boy?
Uhlenbeck: Yes.
Jackson: Did your brother exhibit much interest in science?
Uhlenbeck: No. His interests were always somewhat mysterious at the time, but he certainly hated science. It was kind of a miserable upbringing for him. He is now a serious musician.
Jackson: So you were freer to do what you were interested in.
Uhlenbeck: Yes. We lived in West Millington, near Basking Ridge. West Millington wasn’t even a small town. Millington was a town, and West Millington was the name for a few houses. It was out in the country. That was my mother’s influence. She always liked the scenery. My parents did not interact much with the local people. My father had some friends through his profession, and my mother had her artist friends and her extended family of eleven brothers and sisters. One was in Michigan, and one was in Alaska, but the rest lived in New Jersey.
Jackson: In high school, were there teachers who encouraged you to study mathematics?
Uhlenbeck: Not really. I was a good student, so my teachers encouraged me. My most vivid memories are of my Latin teacher. A good number of students at that time were taking Latin, and I took four years of it. My interpretation of that now is that Latin was the only hard course I had. You couldn’t just do it, you had to work at it!
Jackson: You didn’t have to work at math classes?
Uhlenbeck: No. Because of my schedule I couldn’t take the “honors” mathematics class. So the honors math teacher had me in a couple of times after school and taught me the rudiments of calculus. This helped tremendously when I took entrance exams for the honors college at the University of Michigan.
I once had a date with some really weird kid who was the son of friends of my parents. He lent me a calculus book, and I made my way through it. My father had some textbooks from MIT, including one on optics that I went through without really comprehending it. I was interested in everything. I don’t think I was very a comfortable kid to be around.
Jackson: Why?
Uhlenbeck: I was always tense, wanting to know what was going on and asking questions. I was not properly socialized.
I didn’t have to learn it, I could create it myself
I learned about mathematics in my freshman honors course at the University of Michigan. I still remember the thrill of taking limits to compute derivatives, and the little boxes used in proving the Heine–Borel theorem. The structure, elegance and beauty of mathematics struck me immediately, and I lost my heart to it.
Uhlenbeck: When it came time to go to college, there was no question of going to college in New Jersey. Rutgers and Princeton were all-male, and Douglas was a woman’s college. My mother did not want me to go to a woman’s college. I would have liked to ask her why; it just never came up.
So I picked out three places. We had cousins in Michigan, so University of Michigan was one of the schools. Actually large numbers of students from the metropolitan area of New Jersey and New York went elsewhere to college, including the University of Michigan. My father had gone to MIT, so I applied there. And for some reason I picked out Cornell; it sounded like a good place. I got in everywhere. I knew that my parents were worried about money. They didn’t put pressure on me, or if they did it was subtle, but I chose University of Michigan because it was a lot cheaper than the others. It was a very good choice for me. About ten years later, I taught at MIT, and the women there were miserable! Two women in every calculus class, kind of huddling together. University of Michigan was intellectually great, but also socially great because the girls from the Midwest were much more open and lively than the ones in New Jersey. In New Jersey it was very clique-ish, and I was a “brain”.
I don’t know anybody who likes high school, so I don’t know whether it’s worth dwelling on the fact that high school wasn’t much fun! I was in the band, and I played clarinet. I was very active in sports, though girl’s sports did not exist then as they do now. Since we lived out in the country, we had transportation problems if I wanted to stay after school.
I had two or three sessions with that honors math teacher in high school, and what he taught me was very formal and rote. I remember the moment at which I really grasped something about calculus, which was in college.
Jackson: Can you tell me about that moment?
Uhlenbeck: I had a very advanced course, similar to an undergraduate real analysis course. We had done limits. I went to a help session, and the teaching assistant showed how to take a derivative. I remember getting up and looking at the guy next to me and saying, “Are we allowed to do that?” It was a moment where suddenly I realized there were all sorts of things in mathematics that you “were allowed to do”.
Years later, I met the teacher in this course, Maxwell Reade, and he said that I used to sit in the last row and read the New York Times. I honestly don’t remember that! But I do remember finding the course boring because he would do something one day, and then come back the next time and do the same thing again. As a professor and as a teacher, I know that’s what you’re supposed to do. But as a student who was fascinated by the mathematics, I just saw this as repetition. It was an honors math class and started out with 80 students. By the end of the second year, there were about eight left.
Jackson: How many women were in the class?
Uhlenbeck: I don’t think there were many women in the original class. I think there were three women among the eight at the end. Both of the other two also have Ph.D.s in math. I’m not sure I can remember their names.
Jackson: That’s pretty amazing, three out of eight.
Uhlenbeck: Oh, yeah. It was a great environment. Of course it was a completely male atmosphere. The libraries in midwestern universities have a certain feel. An oak table, a lot of thick volumes — it’s somehow a male atmosphere. That was the way it was, and I accepted it, in the way young people accept most things. I was a sufficiently good student that I was very much encouraged. I took graduate courses as an undergraduate and had a lot of encouragement to apply to graduate school.
I remember another “aha” moment. I was taking a course from a rather famous guy named Raymond Wilder,1 who had been a student of R. L. Moore.2 There were fifty students in the class, but Wilder was still teaching like R. L. Moore.3 In this class I had an “aha” moment when I realized I could do what was in the books. I didn’t have to learn it, so to speak — I could create it myself. Those kinds of courses are tremendously valuable, for probably all kinds of students, because they make you realize that you can think your way through something.
Jackson: Although you started as a physics major, you ended up majoring in math.
Uhlenbeck: Yes. I did my junior year abroad. I had a boyfriend who was a math graduate student and was Polish. He thought I needed more of an education — I was too American or something! I applied for a junior year abroad program through Wayne State University. The program had students from all over the United States — from Yale, Princeton, Stanford, Wheaton College in Massachusetts. I went to Munich for a year, and it was a fascinating experience. I learned to speak German reasonably well. And I sat through German math lectures, which was an eye-opener. The lectures were completely polished. They were so easy to follow because everything was worked out so clearly. I’d been used to professors who prepare for fifteen minutes before lecturing. I learned to ski, I learned to like opera, and I was introduced to the world of the theater. It was a wonderful experience.
Jackson: Do you remember any of the professors in Munich?
Uhlenbeck: I took some algebraic topology from Rieger,4 and I took differential equations and complex analysis from Stein.
Jackson: This was Karl Stein?5
Uhlenbeck: I think it was him. I might know his name from mathematics, from Stein manifolds. Of course at that time I didn’t know that, I wasn’t far enough along. His lectures were beautiful.
There were also exercise sessions, Übungen, and I remember that the other math students in our program were not as advanced as I was. They were from places like Yale and Princeton, so it was an encouraging discovery that the education I was getting in the honors program at the University of Michigan was really good. I owe a lot to the University of Michigan.
Jackson: Do you remember professors from Michigan other than Wilder who were important to you at that time?
Uhlenbeck: I had a course in algebraic topology from Frank Raymond. I did not take a course from Paul Halmos although he was at Michigan at the time.
Jackson: There were no women professors there at the time?
Uhlenbeck: No. There were other women in the classes, and I know several of them now, for example Harriet Pollatsek and Martha K. Smith. There is a handful of women who were in the honors program at Michigan and went on to get Ph.D.s in math. One of the theories was that fathers would not send their talented daughters to elite schools; if they’d been male they would have been sent elsewhere. So a group of very talented women ended up in the honors math program at Michigan. I’m not sure I ever bought that theory.
The value of the imperfect role model
There were a handful of women in my graduate program [at Brandeis], although I was not close friends with any of them. It was self-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. So, if anything, there was a tendency to not be friendly with other women. There was a lot of blatant, overt discouragement, but there was also subtle encouragement. There were a lot of people who 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. There were no expectations because we were women, so anything we did well was considered successful.
Jackson: You got your bachelor’s degree from Michigan in 1964. When did you start thinking about graduate school in mathematics?
Uhlenbeck: I just fell into it. When I was an undergraduate, I applied for some internships at places like IBM and Bell Labs. I never got a position, and I always think my life might have been very different if I had. I was not interested in teaching — I thought people were way too difficult to deal with! So I fell into mathematics as something that I could do. My graduate student boyfriend had transferred to NYU. I don’t remember where I applied but I ended up going to NYU. This was partly because my mother had gone to college in New York and partly because NYU had a good reputation for women at that time.
Jackson: Cathleen Morawetz was there.
Uhlenbeck: She was the only woman I ever took, or even audited, a course from. I want to tell this carefully, as it is important to me. At the time, I was not very impressed by her. I was critical of her hair and clothes, her lecturing style, and even of her mathematics. I had no contact whatsoever with applied mathematics and was not interested. So I resolved to be better than my one role model, which serves some purpose. Many years later, I fell on this memory as a lifesaver. I was going through a rocky period, and the memory of less-than-perfect lectures by a highly respected mathematician, who I had grown to admire, kept me going. One does not have to be perfect! One can be human. Role models serve many purposes.
Joan Birman and I were graduate students together, and we got to know each other. She was a lot older than I was and already had three kids. She had a house north of New York City, and I went to lunch there. I had a roommate, Mary Anderson, who was married to somebody who went to the IAS for a year. Mary would come up in the middle of the week and go to her classes at NYU and go back to Princeton on the weekends. She did get her Ph.D. in math, and she lived a fascinating life. We are still in touch, and we have very different memories of NYU.
After a year at NYU, I married the boyfriend I had met in Michigan, who is a biochemist, and I took the name Uhlenbeck.
Jackson: He is the son of the physicist George Uhlenbeck. And then?
Uhlenbeck: We moved to Boston because my husband went to Harvard. I had an NSF Graduate Fellowship, and that helped me move. This was around 1967, not long after Sputnik. Throughout my college career, there were many programs encouraging students to go into math and science. They were encouraging everybody, and women counted.
I transferred to the math graduate program at Brandeis University. I did not even apply to MIT and Harvard. I have no idea where this kind of gut wisdom came from, except that I knew that I wasn’t interested in the woman question. I did not want to be an exceptional woman. It was a gut feeling; I can’t cite any terrible experiences I’d had. I just knew I would be isolated at those places. Brandeis was a smaller school, and I felt that I would be safer there. And it was true. I don’t think the professors were all that eager to see me. A lot of them didn’t want or approve of women students. But I was a sufficiently good student that whatever their feelings had been about whether women should do math, I was able to overcome them without any trauma, so to speak.
But I never did get along with the secretaries! They were much harder. I remember the early parts of my career being made difficult by having to get along with secretaries. By the way, in the later part of my career, especially at the University of Texas and the IAS, I got terrific support from the staff. Things have really changed.
Jackson: Your adviser at Brandeis was Dick Palais. How did it come about that he became your adviser?
Uhlenbeck: I chose him very consciously because he was involved in a new subject called global analysis — using topology to solve problems in analysis, or analysis to solve problems in topology. Doing differential topology in infinite dimensions was a big thing the second year I was in graduate school. There was a seminar every Friday night at MIT that I went to on this new stuff, and I was very taken with it. Dick Palais had been at the IAS and ran a seminar there, and he had published a book about the Atiyah–Singer index theorem.6 I consciously decided that I wanted to do this new mathematics rather than, say, prove a small change in some boundary value problem.
It was a good fit for me because I am by nature an analyst, while my thesis adviser was a differential topologist. He framed the problems for me and taught me the abstract thinking, which I am not so good at. He is a beautiful expositor and really filled in some part of my knowledge that I never could have gotten myself. And I was able to go to the literature and work out the technical details of the analysis, which was not something he knew how to do.
Jackson: Can you talk about the connection of this to the work of Eells and Sampson?
Uhlenbeck: Their work is foundational. There was a paper by Eells and Sampson that actually constructed harmonic maps into manifolds of negative curvature using a heat flow.7 That was one of the first times that anyone had actually used topological conditions in such a setting. The negative curvature is related to a topological condition. Their paper showed that you could get solutions in any homotopy class, and in many cases, if you had a little bit more negative curvature, they were unique.
That was one of the first papers that connected analysis on manifolds and topology. In survey talks where this analysis and topology connection comes in, I would always also credit the Bott periodicity theorem, which was proved in the late 1950s. That’s a case where the topology was investigated by looking at geodesics, and it definitely uses analysis. Another is the Atiyah–Singer index theorem, which connects the dimension of the solution space with topology.
Also fundamental was the Palais–Smale condition. Steve Smale and my adviser Dick Palais discovered it pretty much independently. It shows that the Morse theory that was developed by Marston Morse could be carried out in infinite dimensions. This is the point at which I began graduate school, in 1964. This area was opening up, connecting analysis and topology and geometry. I was attracted to that — the area in between things. It was like jumping off a deck where you didn’t know what was going to happen.
Jackson: In your response to getting the Steele Prize, you said that you were at Brandeis in the thick of global analysis and that the area fell into disfavor later. What happened?
Uhlenbeck: It was a subject that was suddenly very big for a year or so, but then there were no big payoffs. About ten years later, it came back as geometric analysis, with people like S.-T. Yau proving theorems. It was not a smooth trajectory. What was in the minds of the people who were working in it in the 1960s actually came to fruition in the 1970s and 1980s. It wasn’t very popular in between.
I should add something else about that period of my life. My in-laws lived in New York, and we visited them often. My father-in-law was a physicist who had moved from the University of Michigan to Rockefeller University. He was a combination of amused by me and encouraging to me. It was interesting and useful to get the perspective of an academic family. I valued them because I had come from outside academia and from outside this larger intellectual world. My in-laws knew a wide range of people in academia from various places. My mother-in-law taught me to cook. I did know how to cook basic stuff from my mother, but my mother didn’t think of cooking as an art. So my in-laws were very important to me.
Jackson: You said your father-in-law was amused but encouraging. Amused, meaning…?
Uhlenbeck: I think he just enjoyed the fact that I was different and wanted to do things in a way that he was very unfamiliar with.
Jackson: Some men would have been put off by that.
Uhlenbeck: That’s right. He was not put off.
Jackson: His wife was not an academic?
Uhlenbeck: No. They were Dutch and had both grown up in Indonesia. They’d seen a lot of the world. It was a real eye-opener to meet people from such a different background.
Jackson: After your Ph.D. you taught at MIT and then you spent two years at Berkeley, 1969 to 1971. What are your impressions of Berkeley from that time?
Uhlenbeck: Politics. The Vietnam War. I also learned a lot of mathematics.
Jackson: Who do you remember in the math department from that time?
Uhlenbeck: Interestingly enough, an old timer named Abe Taub.8 He was at that time a mathematical physicist, and I took some quantum mechanics from him and some general relativity from his colleague Ray Sachs. I also learned a little bit of quantum field theory, but that was interrupted when the university closed down because of demonstrations. Abe Taub impressed me quite a bit. He had worked on shock waves and fluid mechanics and had also been instrumental in the construction of one of the first computers, at the University of Illinois. He later moved to Berkeley. I was interested in how you could use very analytical means to study things like shock waves, and I was impressed by his breadth.
At Berkeley I didn’t have many people to work with. I got involved in proving some things about eigenvalues and eigenfunctions, and I worked on Lorentz geometry, which is geometry in spacetime. These things were not actually inspired by anybody; I just picked them up. I was a little bit frustrated that I couldn’t get anybody else interested in them! But I’m not being critical; it surely was part of the atmosphere. I listened to a semester of celestial mechanics from Steve Smale in my last term there. By the time I finished the course I had learned mechanics very well.
Jackson: After Berkeley you moved to the University of Illinois in Urbana-Champaign in 1971. Both you and your husband got positions there.
Uhlenbeck: We thought it would be a little bit like the University of Michigan, where we were both undergraduates and which we had enjoyed. Somehow or other there was a fuss about my job. I don’t know what it was — people seemed to enjoy telling you that you didn’t really get the job.
Jackson: That you weren’t really qualified?
Uhlenbeck: Yes. They said they wanted to make a lot of offers, and they could make only one, the implication being that somehow the administration had something to do with it — which could have been true.
Four or five wives of professors taught in the math department, but they were not part of the seminars. And there was a woman professor, M.-E. Hamstrom.9
Jackson: You weren’t happy at Urbana.
Uhlenbeck: It’s hard to know whether I was unhappy, or my marriage wasn’t working out, or whatever. I didn’t feel at home mathematically, for sure. It was a big change from Berkeley. I don’t remember that I talked mathematics so much to people in Berkeley, but somehow I had a lot more people to talk to. It might even have been just age. At Illinois there weren’t so many people in my age group, and none with my background.
The wives did all the entertaining in the math department. Years later, when I went back to visit, there were no parties because the wives were all working! But during my time at Illinois, there was really a division between women and men in the department. The atmosphere was stifling to me. Up to that time, my experience in academia had been an escape from feeling confined.
From regularity of maps to minimal surfaces
[Lesley Sibner is] a typical example of a woman who is a very good research mathematician but who is not recognized. She is at a technical institute. The only good thing about the job is that she periodically gets good graduate students. She is underpaid and teaches a lot […]. [T]o be really successful you have to be protected, and there is no way to do it any other way. I think about this all the time.
Uhlenbeck: I must have traveled quite a bit when I was at Urbana — I don’t know how I did it, but I must have. I became very friendly with Lesley Sibner,10 who was at Polytechnic University in Brooklyn. I first met her in Trieste at a conference in 1971 or 1972. I went to a talk on a subject I thought I would know something about. I was expecting some guy to get up and speak, because I didn’t know who Lesley Sibner was. And this woman in a purple suede pantsuit and high heels came charging in and gave a beautiful talk on stuff that I knew about and was interested in. I was so excited to meet her. We became friends, and I made quite a few trips to New York and would always see her and her husband Robert Sibner, who was a mathematician at the City University of New York. Lesley was an exceptional woman. She had actually started out studying acting and came to mathematics a little bit late. She was a very good mathematician and a wonderful person.
Jackson: Did you write any papers with her?
Uhlenbeck: We wrote a paper11 about ten years after we had met. I was by then at the University of Texas at Austin.
Jackson: You left your husband and, in 1976, took a position at the University of Illinois at Chicago. You liked the atmosphere there much better than Urbana.
Uhlenbeck: Right. I was very comfortable there. I also learned some Teichmüller theory from Howie Masur, who had the office next to mine. This proved useful when I tried to understand Thurston’s work.
Jackson: It was when you were at the University of Illinois at Chicago that you wrote a paper with Jonathan Sacks.12
Uhlenbeck: He came to Urbana as a postdoc and started talking to me there. I didn’t know very much about minimal surfaces, but we talked about them and worked together. He brought the knowledge of the subject of minimal surfaces, and I brought the main idea, which was really a fairly simple idea from complex analysis. We wrote a couple of papers together. I was very close to him for a number of years, but then I lost track of him.
Jackson: You said you brought the main idea from complex analysis. What was that idea?
Uhlenbeck: It was the idea of rescaling the problem. If you have something that doesn’t depend on scale, you can blow it up and look at it at a different scale.
Jackson: This rescaling idea is what you called “bubbling” in the paper?
Uhlenbeck: That’s right.
Jackson: Why did you call it bubbling?
Uhlenbeck: You have a problem in \( \mathbb{R}^2 \), and something is happening at a particular point. You expand tiny discs to large ones, and what you find hidden at that point is a map to a sphere. So it’s like a bubble. Other kinds of problems can bubble too, but they usually have to have some scale-invariance in them.
Jackson: Later you used this idea of bubbling in the context of Yang–Mills equations, is that right?
Uhlenbeck: Yes. Scale-invariant problems all behave very similarly. Another problem that I didn’t even know about at the time was the Yamabe problem, which is also scale-invariant. A lot of problems that are totally geometric are scale-invariant. That is, if you have a problem where you are looking for a metric, and somehow or other, there is no intrinsic external scale on it, you can actually change the scale that you are looking at.
Jackson: Can you tell me about the problem you were working on with Sacks?
Uhlenbeck: Conceptually, it was a problem of finding minimal surfaces. I never learned and still don’t really know the Federer–Fleming approach using geometric measure theory. The alternative approach was to find them with conformal parameterizations. The problem is, these conformal parameterizations don’t see scale. That is, if you write down an integral, over a piece of \( \mathbb{R}^2 \), that represents a piece of the minimal surface, then you can actually get the same answer by changing the scale of the piece in \( \mathbb{R}^2 \). It’s a calculus of variations problem, and I had learned a lot about the calculus of variations as a graduate student. The idea was to write down the energy functional that would produce the minimal surface and make the small perturbation that would actually give it scale. And then you could prove it satisfied the Palais–Smale condition, so that there was a Morse theory. There were lots of critical points in that case. You let the perturbation scale go to 0, and then, although there are places where you would not get convergence, you just rescale around those places until you find something.
What you get when you rescale is a map from \( \mathbb{R}^2 \) into the manifold where you are looking for the minimal surface. The point is that, because this integral doesn’t see anything but angles, it can’t tell the difference between \( \mathbb{R}^2 \) and \( \mathbb{S}^2 \). You get a map from \( \mathbb{R}^2 \), but you can also think of it as a map from \( \mathbb{S}^2 \) with a singular point. Then there is a regularity theorem — which is also something that I worked on in my thesis — where if you have a map from a sphere to a manifold that satisfies the harmonic map condition, you can actually fill in the point where there is a singularity and prove that it’s actually regular over the singular point. This was forty years ago, and nowadays the technique would be thought of as very standard.
At the time I had not had contact with a large number of mathematicians. I had gotten my Ph.D. at Brandeis, not from Harvard or MIT or Berkeley. And when I was at Berkeley and MIT, I didn’t really work with anyone else. That was valuable in some sense, because I didn’t know what people were thinking about minimal surfaces. So when Jonathan Sacks taught me the problem, I didn’t have any built-in machinery to think about it. So I was able to think about it on my own.
Jackson: That’s an advantage.
Uhlenbeck: It’s definitely an advantage. With several problems, I worked in a context where I didn’t know other people who were developing machinery about the problem.
Jackson: Was the work with Jonathan Sacks more geometric than you had done before?
Uhlenbeck: Yes, more geometric. I owe that to Jonathan.
There is something else I should tell you about. When I was a graduate student, I ran across a very technical problem about proving regularity of maps that have a bad nonlinearity in the top-order term. It came up in my thesis. Rather than looking at the integral that would, say, give the minimal surface equation, I was looking at integrals that had a slightly different integrand. You could prove that there was a Morse theory for them. They are called \( p \)-harmonic maps now. The problem was, they were weak solutions only in a Banach space; they were not smooth. I worked on this problem for several years by myself, pretty much. I read some papers on elliptic PDE, and during one of the first conferences I went to, I met Jürgen Moser.13 He gave me several reprints of his papers on what’s now known as the Moser iteration scheme. The work I did on that problem came out of my meeting Moser and reading the papers he gave me. That’s the hardest mathematics I ever did.14
S.-T. Yau noticed that paper, and he invited me to visit him. He was at Stanford at that time. I was absolutely startled because no one had ever asked me about that work. I had never talked to anyone about it! I went to visit and talked with Yau and Leon Simon, and met Rick Schoen, who was their graduate student at Stanford. I’m not even sure I could read that paper nowadays. It was a very hard, very technical paper on a problem that I became consumed with. I worked on it for several years — one year as a lecturer at MIT, and two years at Berkeley. This was the start of a great relationship with S.-T. Yau.
Jackson: Was there a global analysis perspective in that paper?
Uhlenbeck: No, this was just straight PDE, but it was motivated by global analysis. I had some functionals that satisfied a Morse theory, and I wanted to show that their critical points were actually regular maps, that they weren’t very discontinuous. And in a way that’s what grew into the minimal surfaces papers. I was looking for 2-dimensional minimal surfaces, and the Dirichlet integral, which is what you usually used to minimize, almost satisfied the Palais–Smale condition. So if you add a little epsilon term, it would satisfy the Palais–Smale condition. The idea was to make this small perturbation, and then see what happened as the perturbation went to zero.
So the minimal surfaces work grew out of this very technical PDE problem that I had gotten sucked into when I was working on global analysis. Several other students who were also doing global analysis went on to do much more abstract things. And I went on to do very technical things. But since I was trained in a topological-geometric viewpoint, I’ve been able to handle topology. Although now I’m always lost whenever I go to a talk in topology!
Changing what geometry is
The next major development in the story of the so-far-only-physicist’s Yang–Mills equations was one by mathematicians! The Atiyah–Singer index theorem said something about the dimension of the space of solutions to the Yang–Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics communities began to change. It is here that the cooperation and cross-fertilization really began.
I don’t know how to describe the tremendous difference this made. The entire position that mathematics held within the scientific community changed. The mathematicians actually had something to say!
Jackson: What kinds of problems appeal to you? What does a problem have to have to grab your attention?
Uhlenbeck: It’s a combination of being fairly concrete — so one can understand concretely examples — and also connecting with a lot of other ideas. For example, you see the analysis in a minimal surface equation, but then you also realize it has connections with other geometric questions that are not just analysis. I am definitely very attracted to the idea that there are a lot of different facets in mathematics and seeing the connections.
Jackson: A lot of analysis comes from physics.
Uhlenbeck: I’m impressed by that. It seems to me that mathematicians in some ways are not very inventive thinkers. They get an idea, and then they sort of beat it to death, pushing it forward. The idea behind it is often lost to most of the people working on the problem.
Jackson: Are physicists more inventive?
Uhlenbeck: They definitely think differently, but I’ve never been able to understand the different way they think about problems. Mathematics, especially in the last fifty years, has benefited tremendously from ideas that come in from outside. Somehow it doesn’t grow internally on its own.
I take that back. Separate areas within mathematics can influence each other. Examples of this would be the Langlands Program and the use of probabilistic methods in PDE. The point is, mathematics is not a machine where you write down definitions, theorems, proofs, and tack on an example. It is much more interesting than that.
If I were younger, I would be very interested in learning a lot more about things that are not core mathematics, but are mathematics connected with other areas.
Jackson: What do you have in mind there?
Uhlenbeck: All sorts of stuff in computer science, in biology.
Jackson: Physics as well?
Uhlenbeck: I tried that. I’d go in a different direction. It is interesting that somehow or other the ideas in physics lead to ideas in mathematics, which tend to get stuck. Physicists have a looser idea of constructions.
Jackson: You went to a lecture in the late 1970s by Michael Atiyah at the University of Chicago.
Uhlenbeck: He gave a series of four lectures at the University of Chicago on Yang–Mills and gauge theory.
Jackson: Is that when you got interested in gauge theory?
Uhlenbeck: Yes. Most mathematicians didn’t know anything about it. There were two textbooks, one by Steenrod and one by Husemoller, on fiber bundles.15 They were very abstract. The physicists would always come to us and ask what they could learn from. And we could only give them these very abstract textbooks. The language was surely difficult for them — it was difficult for the mathematicians who tried to learn it, and even harder for the physicists.
Jackson: It took the physicists to come up with the Yang–Mills equations. That wasn’t something mathematicians would have studied on their own.
Uhlenbeck: They could have, but they didn’t. They had all the tools, but it wasn’t one of the things they were thinking about.
Jackson: Why not?
Uhlenbeck: I wish I knew exactly why and what makes the difference between mathematicians and physicists seeing things in a slightly different context.
I had been working on variational problems. With the Yang–Mills equations, there’s a second-order equation that is the solution of a variational problem, which is very natural, and you could easily come up with it. But then there are the first-order equations, which need a little bit more topology to understand. Most mathematicians, when this was explained to them, could understand it immediately. It wasn’t totally mysterious.
I remember my thesis adviser Dick Palais encouraging me to get interested in Yang–Mills — this was after my Ph.D. I had no basic knowledge of fiber bundles, and I had to learn what connections and curvature are. But I got interested and worked it out. I could see what the problems were in the analysis. I just worked on what the obvious problem was, which was to make gauge changes so you could control the analysis.
In 1979–80, I was at the IAS for a geometry program organized by Yau. Cliff Taubes was then a graduate student at Harvard. He had a girlfriend who was in Princeton at the time and later became his wife. He came down to Princeton for a while and talked to me. The first thing he said was, “Tell me about the calculus of variations.” And the first thing I said to him was, “The calculus of variations isn’t going to do this for you!” That’s the kind of good advice an expert can give to a novice: No, this is not the right thing for what you are trying to do! That was the beginning of a long friendship with Cliff Taubes.
Jackson: You said that when you learned about gauge theory, you asked the “obvious question” about the analysis. Can you tell me about that question?
Uhlenbeck: I remember asking people what gauge theory was. The answer was that, for the abelian bundle, it’s just a case of Hodge theory. The connections are defined only up to some ambiguity, which you are allowed to change. It’s very much like the definition of a manifold. You have patches on the manifold, and you define how to go from one to the other on the manifold. In general relativity, you have a tremendous amount of ambiguity in how to write down the metric. You write it down in different coordinate patches, and you transform one to the other. This transformation is more complicated than for gauge theory. So there is no question about gauge theory being a simpler problem than general relativity and equations about Riemannian manifolds.
At first geometers were thinking about these problems in gauge theory, but people working in PDE just did not think about them as an aspect of partial differential equations. Then came the solution of the Calabi conjecture, where you ask the question of whether a manifold with the right topological condition can have a metric condition to pair with it. That was solved by Yau in 1977.16 There was a slow awakening to the fact that there were partial differential equations on manifolds that one could actually solve.
A manifold is a bunch of points, and you have coordinates, but you can change them. So if you write down a partial differential equation on a manifold, you have to understand how it can be moved from one coordinate to another. The work of Eells and Sampson is always considered one of the basic, fundamental breakthroughs because they actually solved a geometric equation on a manifold, and they did not use coordinates. They used the heat equation, which is very similar to the elliptic equations that we all learned to use, but theirs was just a new idea of mathematics that you could do. There was a tremendous change in the perspective of what geometry was. And it was certainly encouraged by the fact that physicists came up with questions that needed mathematical solutions.
Jackson: You mentioned the special geometry year at the IAS, organized by S.-T. Yau. He was an important figure for you at this time.
Uhlenbeck: I could tell that he admired my mathematics. He understood what I was doing, and he honestly thought I was a good mathematician. I’d been encouraged in a kind of fatherly way by other mathematicians, but Yau was different. I feel I owe a lot to him, because he made me feel like a real mathematician. Of course by then he’d produced a series of fantastic papers, with new ideas in all of them. I felt he was a fantastic mathematician.
Jackson: So when he thought you were good, it meant a lot to you.
Uhlenbeck: Yes, it meant a lot to me. It wasn’t just being encouraged because I was smart.
Gauge theory: An adopted child
How did gauge theory appear and become successful in mathematics in the space of a few years? The fundamental mathematical ingredients were in place [fiber and vector bundles, connections, Chern–Weil theory, De Rham cohomology, Hodge theory]. In hindsight, the Yang–Mills equations were waiting to be discovered. Yet mathematicians were in themselves unable to create them. Gauge field theory is an adopted child.
Jackson: Your work in gauge theory laid the analytic foundations for a lot of subsequent developments, including Donaldson’s work.
Uhlenbeck: That’s right. In a small neighborhood you can often choose a gauge, which is a choice of coordinates in the bundle, in which things work very well. It was clear from the PDE point of view that you had to impose an extra equation, and you got the equation to be true by choosing the right coordinates. A problem I struggled with for a long time was that, if you bound the curvature of a connection, you would like to get a bound on the connection in some coordinates. The paper in which I showed how to do this is still one of my more influential papers.17 Actually, it isn’t a really hard paper, but it was very hard to think of the idea. It took me several years to think of how to prove that this was true.
Jackson: How did you come up with the idea?
Uhlenbeck: I just banged my head against the wall. I think it might vaguely have been influenced by Yau’s solution of the Calabi conjecture. The way I solved it was that I put a condition on the curvature and showed that the space of connections having bounds on curvature was of a certain size; it was both open and closed. It’s the continuity method in PDE. That is, you are trying to solve an equation and you put some topology on it, and you prove that for all the elements in the topology, the space of solutions is open and closed, modulo the other conditions.
The continuity method is what you might think of as the most elementary technique in nonlinear analysis, and it’s probably one of the most valuable. It’s kind of amusing, because nonlinear analysis has all this fancy stuff like the Atiyah–Singer index theorem, all this fancy topology, all these fancy fixed-point theorems. But in fact, the first thing you should learn is the continuity method, which is that the space of solutions is both open and closed.
Jackson: It’s something that’s conceptually simple.
Uhlenbeck: It’s so simple that it was bypassed, in trying to solve the problem.
Jackson: That was one of the papers for which you received the 2007 Steele Prize. What was the other one?
Uhlenbeck: It was about removable singularities.18 This was an interesting question. It started with the physicists. They wrote down solutions to the Yang–Mills equations in \( \mathbb{R}^4 \). They were able to show that those solutions could be continued over infinity. These equations are what is called conformally invariant. This means that if you take a solution in \( \mathbb{R}^4 \) and you make a conformal change of coordinates — which means basically taking \( x \) to \( 1/x \) in four dimensions — then the equations are still satisfied. The question of whether every time you wrote down an equation in \( \mathbb{R}^4 \) it would continue over infinity was actually around for quite a number of years.
Jackson: And that was the question that you resolved in this paper.
Uhlenbeck: Right. I now know completely trivial proofs. I have some extra machinery that I’ve learned that makes it a lot easier.
Jackson: Is “Uhlenbeck compactness” part of these two papers?
Uhlenbeck: Yes. Uhlenbeck compactness is easiest to describe in dimensions two and three. There the statement is that if you have bounds on the integral of the curvature squared, then you can construct a subsequence that converges weakly. When you have a bound on norms of functions, it’s kind of a tautology in distribution theory that you can construct a sequence that converges weakly. But if you are working in gauge theory, you’ve got to fix the gauges, so convergence is not so easy to show. It’s most important for solutions of equations in four dimensions, though it’s often used in other settings. You have solutions of equations in four dimensions, and if the curvature is small enough in a neighborhood, you can construct a subsequence that converges. Then there is a counting argument that shows that if you have a functional like the Yang–Mills functional in four dimensions, and if you have a bound on solutions, then in fact a subsequence will converge off a finite number of points.
This was part of the solution to the minimal surface problem. That is, you have a perturbation and you look at limits of solutions, and they converge off a set of a finite number of points. Around those points you can blow them up, which is the phenomenon that happens both in Yang–Mills fields and in minimal surfaces. This kind of thing is actually used quite a bit in PDE now.
The compactness theorem has to do with a situation where you look at a solution of a partial differential equation, and you need to know something about when a solution can exist. And you need to know that the space of solutions is compact with an identifiable boundary. My part of Donaldson’s theorem was to prove that the space was compact, with a certain type of boundary occurring. Then Cliff Taubes proved that that kind of boundary really did exist. The most surprising thing to me is this kind of technique of constructing geometric invariants is so ubiquitous now. It was completely new. This perspective of “newness” limits older mathematicians. Younger mathematicians accept it as “well known” and use the techniques with more confidence.
Jackson: What did you think when you heard about what Simon Donaldson had done in his Ph.D. thesis?
Uhlenbeck: It was completely out of the blue. He was a graduate student, and if he’d been more sophisticated, he probably never would have thought you could do anything like this! I was very impressed. Given the machinery, it wasn’t that hard a theorem for me, or for people like Taubes, who had worked in the field. But putting it together and actually getting a result, was amazing. I can’t even wish I could have done it, because it was so outside of my thinking.
Jackson: You and Mike Freedman organized a semester-long program at MSRI to study what Donaldson had done.
Uhlenbeck: Yes. I met Dan Freed that year. He did a lot of the organization and was very instrumental in writing up the notes from the lectures.19 We have been friends ever since.
We had people coming to the seminar throughout the semester. In some sense, the topic got easier as the semester went on because the problem was the analysis and topological prerequisites. The actual proof itself, once you had all the pieces together, is very elegant. So it wasn’t a question of the material getting harder and harder. The number of people who knew all the technical business was very small, because you had to know analysis and you had to know topology.
I understood Donaldson’s proof right away. The only problem I had is, I didn’t know how to prove the solution space he constructed was oriented. During the time at MSRI, I found out how to orient the moduli space in a way that I really liked! It turns out there is a canonical way to orient a space that is given by a solution to equations and that carries from finite dimensions to infinite dimensions.
Jackson: This was 1982, and you still were on the faculty at the University of Illinois at Chicago. After MSRI you spent the second semester that year at Harvard.
Uhlenbeck: Yes. During that time I got to know Cliff Taubes quite well. I enjoyed it. I had never gotten a chance to teach graduate courses before. I found it difficult and spent many hours preparing.
Jackson: At University of Illinois at Chicago, you did not teach graduate courses?
Uhlenbeck: Right. I didn’t want to. There wasn’t an audience for the kind of topology and analysis that I was doing. And I’ve never been one to want to lecture and have nobody understand what I’m talking about.
Jackson: Was that part of the reason you then moved to the University of Chicago in 1983?
Uhlenbeck: Yes, I moved to the University of Chicago because I wanted Ph.D. students. And it was great. I got quite a few good Ph.D. students right away. My salary went up a lot too! But that was not the primary reason.
Jackson: But you didn’t stay so long at Chicago.
Uhlenbeck: No. The commuting was a lot harder. I lived in Evanston. My husband [Robert Williams] was a professor at Northwestern. I’d had more people to talk to at University of Illinois at Chicago. I’d never been in a place like University of Chicago. It’s a very elitist place. Everyone thought very well of themselves. I felt a lot of pressure, possibly for the first time. The graduate students, especially the women students, struggled.
S.-T. Yau was on the faculty at the IAS in Princeton at that time. His wife had a job in San Diego, and she didn’t want to live in New Jersey. So Yau planned a grand project to build a geometry group in San Diego. He involved lots of people, including myself, Rick Schoen, Richard Hamilton, Leon Simon. I spent a winter in San Diego. The mountains were close so you could ski. I took surfing lessons, although I must have been almost 40 at the time! I was all set to move to San Diego. And then everything fell apart. Maybe it was bound to fall apart. It wasn’t clear that everybody at San Diego was happy with this influx of people. This was around 1986.
Jackson: Didn’t Mike Freedman also go to San Diego?
Uhlenbeck: He was already there. Richard Hamilton was already there too and stayed for a few years. But the point is, I had kind of agreed to leave Chicago. I had not signed anything, but I had gotten it in my mind to leave. My move to the University of Texas was in the wake of this. At that time Texas was trying out all the big shots. Dennis Sullivan and S.-T. Yau spent semesters at the University of Texas around this time. They invited an awful lot of people to visit and made a big fuss about it. I guess I was attracted to the change. My husband grew up in Austin and had family there. That was not the reason, but it certainly was a positive reason that a lot of other people didn’t have to go there. It seemed like an adventure. The presence of Steve Weinberg and Harry Swinney in physics and Cameron Gordon in mathematics was attractive to both me and my husband.
And I really did not like commuting in Chicago. When I was at University of Illinois at Chicago, I got an hour of walking a day, to and from the train station, and I could work on the train. But going to University of Chicago, it was a question of driving. I would have stayed there if I’d been able to do all my commuting in the morning, but that late night commute, especially in the middle of winter, was bad. I’m an outdoor person. I did not want to live in Hyde Park. We had a great big yard in Evanston. Also, the two Chicago faculty members I had befriended, Peter Jones and Jerry Bona, both left. That certainly influenced me. They were not in my field but close enough that we could talk.
Why does KdV arise everywhere?
[D]ue to what was now an addiction to intellectual excitement, I tried to follow the influence of physics on geometry which is associated with the name of Ed Witten. My work in integrable systems grew out of this connection with physics.
Jackson: How did you get interested in integrable systems?
Uhlenbeck: It’s very immediate, because integrable systems play an important part in conformal field theory in physics. I did learn quite a bit about integrable systems and worked in the area, but my motivation was to understand things in physics, and that didn’t pan out. That is not to say that I didn’t enjoy working on it. My friend Chuu-Lian Terng and I have written some nice papers on integrable systems. She’s married to my thesis adviser, so I have known her for a long time. Although she is from Taiwan and grew up there, we have a lot of interests and experiences in common. We’ve worked together for quite a few years now. Our collaboration is also partly connected with the Women in Math program.
Starting in the 1990s I got involved in a lot of non-mathematical things. I was one of the handful of people who started the Park City Mathematics Institute (PCMI). Dan Freed and I had become very friendly, and he moved to Texas. He was young and energetic and wanted to get involved in all these things. So he instigated our involvement with the PCMI.
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, Here I am, in my 40s, successful — Where are all the women? We women mathematicians thought, yes, it has been a little difficult, people weren’t so friendly to us, but things are going to change. But in the early 1990s, many of the women roughly in my generation were at that point the last women hired in their department. We didn’t see large numbers of women coming after us. I felt I did owe something for my success. So Chuu-Lian Terng and I started working together on integrable systems, and we also started working in the Women in Math program. It was originally associated with the PCMI, and later the IAS took it on and supported it. Chuu-Lian and I did mathematics and also organized the Women in Math program.
Jackson: That’s nice to do both.
Uhlenbeck: Yes! I was motivated by one of the mysteries to me in mathematics, which is why KdV comes up all over the place, in all sorts of geometric and physical problems.
Jackson: Do you have any inklings of why that’s the case?
Uhlenbeck: Not really. The only guess I have is that it’s a natural construction on an infinite-dimensional torus somewhere. So it appears in a lot of places where you have an infinite number of commuting variables that are somehow involved in the structure of a torus somewhere. Another possibility is that, once it exists as a mathematical model, it serves to explain patterns that appear in calculations for a time. When this becomes inadequate, new models are found or developed that are more useful. One should never forget that mathematics is different from what it is used to model.
Jackson: Can you tell me about your work with Chuu-Lian on integrable systems?
Uhlenbeck: We came to a certain understanding of what scattering theory and inverse scattering theory were for different kinds of geometric PDEs, like KdV, sine-Gordon, and so forth. We discovered that, rather than just a lot of complicated formulas, there was actually a Riemann–Hilbert problem underlying most of the constructions. We were able to get definitions, which pleased us geometrically, of things like the Virasoro algebras and functions. So you could motivate a lot of the constructions that you could find in the literature by factoring complex matrix functions.
Chuu-Lian is really a classical geometer, not a PDE specialist, so she knows a lot about the constructions of geometric surfaces and geometric submanifolds that these equations describe. I was motivated by the fact that they appear in quantum field theories. The two perspectives helped our research.
Jackson: Is there an area of mathematics where you think, “I would have loved to have worked in that”?
Uhlenbeck: Algebraic geometry. When I was a student at Brandeis, the algebraic geometry that was taught there was abstract. My generation said there are two kinds of algebraic geometers: those who learned it from Hartshorne,20 and those who learned it from Griffiths and Harris.21 When I was a student, my opportunity was to learn algebraic geometry from Hartshorne, which didn’t make any sense to me, whereas the Griffiths and Harris approach has all these wonderful examples of spaces. If I were to do it again, I would make a big effort to learn algebraic geometry.
Jackson: What is it about algebraic geometry that appeals to you?
Uhlenbeck: The interesting geometric structures that come up.
Jackson: Do you think geometrically?
Uhlenbeck: Yes and no. My thinking is very concrete, meaning whenever I work on a problem, I always do baby calculus problems around it, to see what is going to happen. I don’t think terribly abstractly. I think a lot by examples, finding simple constructs that model complicated phenomena. I am willing to learn the complicated stuff, but I like to be motivated by something that I can understand that is basically very simple. I also work a lot with inequalities. I don’t name each term in an inequality with a word, but I think of them somehow as individuals. After you work at it a while, they are very personal. They are your good friends.
