#### by Claudia Henrion

In asking mathematicians around the country who they thought should be interviewed for this book, almost everyone named Karen Uhlenbeck. Uhlenbeck is considered one of the top mathematicians in the country; she has received many distinguished awards, including the MacArthur “genius” award, and a nomination to the National Academy of Sciences. Her pioneering work in mathematical analysis has earned her tremendous respect as a talented and creative mathematician.1 One of the ingredients of Uhlenbeck’s success is that she is extremely independent. She is proud of the way she was basically self-taught, pursued her own interests, and for a great deal of her professional life “didn’t need anyone.” 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.

At the same time, for many women, doing math is not “an automatic thing.” It is more difficult for women to envision themselves in such a role, and this lack of vision can have an effect on many levels. It makes it less likely for women to pursue mathematics, and even those who begin such a path are more easily diverted if they do not have a clear sense of where it will lead or how they would fit in. Consequently, women are more likely to choose alternative trajectories that present themselves. Moreover, even those women who do stay in math have trouble seeing themselves as mathematicians. As Karen Uhlenbeck says, “Even when I had had my Ph.D. for five years, I was still struggling with whether I should become a mathematician. I never saw myself very clearly.” This difficulty in imagining oneself as a mathematician arises in part from the strong social stereotypes about women, as well as from the lack of role models to present alternatives to these traditional expectations.

Most striking is the fact that even some of the most talented women in mathematics — those who are clearly gifted and who love mathematics — can still feel like outsiders in the mathematics community. As Uhlenbeck said, “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.”

What are the factors that contribute to this feeling of being “on the outside looking in”? And how does this sense of marginalization complicate women’s relationship to public awards and recognition?

#### Early childhood and early independence

From an early age, Uhlenbeck had a healthy disregard for the social expectations of women. The roles for women in the 1950s, the decade in which she came of age, were quite limited. In the previous decade, women had been exposed to a wide range of opportunities because so many men were away at war. Their skills were needed in factories and management, and they discovered that they could be welders, truck drivers, professional baseball players — almost everything that needed to be done was done by women. Later, in the sixties and seventies, the women’s movement and sweeping social change opened many doors for women, allowing them to envision and pursue new paths. It became increasingly common for women to become doctors, lawyers, scientists, or business executives. But the dominant, and unquestioned, roles for white middle-class women in the fifties were clear: their focus was on getting married, having and raising children, and caring for the home. Women were not supposed to be active in sports, except as cheerleaders, and being intellectually oriented made one an outcast. As Uhlenbeck said, “I did not feel like I was supposed to do anything interesting except date boys. That was what girls did.”

Karen, however, preferred to play football and climb trees. “I was very much a tomboy. The boy down the street and I played football and baseball for the better part of my life, right through high school. It was not a very respectable thing to do.” So entrenched were these roles and expectations that even her mother, who was far from the paradigm of convention, had difficulty accepting her daughter’s unwillingness to conform to gender roles.

But despite her mother’s discomfort with Karen’s unconventional ways, Karen’s whole family in many ways paved the way for her being different. In the rural community in northern New Jersey where she grew up, her parents were noticeably different — her father, an engineer, and her mother, an artist, were Democrats and went to a Unitarian church, while most people were conservative, Republican, and either Catholic or Protestant. Her parents were also ardent conservationists. For their honeymoon they went hiking in the mountains of the West, and later, even with four children (of which Karen was the oldest), the family would go camping every summer for two weeks in the Adirondacks. This background in the outdoors was an essential part of Karen’s development.

Her family provided many models of strong and independent women. Her grandmother, an imposing six feet tall, raised twelve children almost single-handedly and lived to the age of 103. Karen’s mother, also extremely active and energetic, “a kind of superwoman,” had a fiercely independent streak as an artist, an intellectual, a maverick in her own right. She believed in doing everything herself, and as Karen says, “I think I get a lot of my character from her.”

Growing up in a rural environment helped Uhlenbeck avoid intense peer pressure to conform. She found alternative forms of companionship in the unstructured world of the country; by playing in the fields and helping in the garden, she grew familiar with birds, flowers, and trees. And she found alternative visions of the world by immersing herself in books. She read everything she could find, particularly biographies (there were very few on women, “mostly presidents’ wives”), and books on math, science, and frontier life. These early influences began to shape her vision of what she wanted to do with her own life, a vision that would enable her to escape the uncomfortable expectations of who she was supposed to be.

I was either going to become a forest ranger or do some sort of research in science. That’s what interested me. I did not want to teach. I regarded anything to do with people as being sort of a horrible profession. … I felt that I didn’t get along with people very well. I always had a lot of girlfriends, but I never had very many boyfriends; I didn’t feel comfortable. I never felt like I was really a part of anything. I went my own way, without really wanting to, but I never did understand the trick of doing things like you were supposed to do.

“Not doing things like you were supposed to” may have been a liability in adjusting to social norms, but it would ultimately become an asset in her pioneering research as a mathematician.

#### Becoming a mathematician: “Not an automatic thing”

By the time Uhlenbeck finished high school, she had no idea that she would go on in mathematics. Her early schooling had not been particularly stimulating; indeed, she had spent most of her time reading novels under her desk. College, however, turned out to be a striking contrast.

Uhlenbeck’s family assumed that Karen would go to college, but where she went was less important. She chose the University of Michigan because it was relatively inexpensive; spending money on a woman’s education was not considered a high priority. The year was 1960.

Karen enrolled in the honors program at the University of Michigan, which provided her with an excellent education; her strong training in mathematics there was to prove very useful in graduate school. Interestingly, this honors program produced a large number of women mathematicians. Because it was so rigorous, many people dropped out of the mathematics portion, but for Karen it was a period of blossoming, both personally and intellectually. She had lots of friends, and felt more comfortable with women from the Midwest than with those from the East Coast because they seemed to her more open and friendly. There was a large contingent of New Yorkers as well. She also began to have boyfriends, most of whom also studied mathematics or science. During her junior year, she went abroad to study in Germany. The world unfolded beyond her wildest dreams. The discipline of the classes and the polished lectures (given in German) were stimulating to her. She traveled around Europe, learned to ski, went to the opera, and got to see parts of the world that were dramatically different from her home town in New Jersey. She especially enjoyed being totally on her own.

It was during college that Uhlenbeck first discovered how much she enjoyed mathematics. Her first math class in the honors program at the University of Michigan was an extremely challenging analysis course. Karen flourished and decided to switch her major from physics to mathematics. “I got to college and discovered that I could do mathematics, and I never even saw myself as doing it, but I recognized that I was partaking in something that I enjoyed. … I just thought the idea of dividing things up into infinite amounts seemed really far out.” She particularly liked the excitement that came from connecting with mathematics directly instead of “just doing what the book says.” And she “found it really neat that you could think through these arguments and get them right just like the book.” Perhaps most important, she discovered in the discipline of mathematics an incredible sense of freedom, a creative freedom not unlike that expressed by poets who work with haiku or other highly structured forms of poetry. As Uhlenbeck says, “If you obey the rules, you could do almost anything you wanted.”

However, despite the fact that Uhlenbeck loved mathematics, she did
not assume that she would pursue it in graduate school or go on to be a
mathematician. As Uhlenbeck points out, for women, pursuing mathematics
is not “an automatic thing.” And she was no exception. She simply
could not *imagine* herself in such a role. Ultimately, it was the fact that her
boyfriend and other undergraduate friends were going to graduate school
that drew her into it — an early indication of how significant ties to a
community can be. But she still had no vision of where she was going or
what steps were involved.

I think if someone from industry had come and interviewed me and wanted to hire me, I might have been inclined, because I don’t think my parents were particularly enthusiastic about my going to graduate school. No other option came up and said we want you, and I was being pressed by people in math. All the people I knew were going to graduate school. And I met my future husband as a senior and he was going to graduate school (in bio-physics). … It wasn’t something that I had been geared up to doing all along. It was just that my life grew into that, especially when I started seeing him seriously. He was going to graduate school, so what was I going to do?

Karen received several fellowships, including a National Science Foundation and Woodrow Wilson Fellowship, which made it easier for her to pursue mathematics. Not only did it provide the finances to go on, it also encouraged her psychologically. While she couldn’t imagine herself as a mathematician, the fellowships at least implied that other people thought she had potential.

Despite the fact that she had received highly coveted fellowships and that her boyfriend was going to Harvard, Karen never thought seriously about applying to the most elite schools, such as Harvard or MIT (Princeton was still not admitting women). She chose instead to attend Courant Institute in New York City, which has produced a number of women Ph.D.’s. Although this was a strong program, its focus was on applied mathematics, which was considered less prestigious than the “pure” math focus of the Ivy League schools.

When Uhlenbeck married, she decided to transfer to a graduate program closer to her husband. Again she did not apply to Harvard or MIT, but instead chose Brandeis. It was not so much a lack of confidence in her mathematics ability as a sense of how problematic the atmosphere would be for her as a woman.

It was self-preservation, not lack of confidence. I was pretty sharp, without being conscious of it, of how difficult things were for professional women. (I knew all about being socially awkward!) At the time, I may have thought that if I were brilliant enough, I would succeed at Harvard. Now I do not believe that — I believe the social pressures of surviving in an environment that would question every move would have done any woman in, unless she were particularly interested in the combat. I knew I was not interested in the battle of proving social things, so I (wisely in retrospect) avoided it.

At this point Karen had no real role models: no other women with whom she could identify, and who could help her envision herself as a mathematician. Transferring to Brandeis did help, however, by at least providing her with young role models. “For someone like me, Brandeis was a super place to be a graduate student because the faculty were all extremely young. My thesis advisor was barely over thirty years old. … It was exciting to hob-nob with the junior faculty in my last year.” Brandeis was just beginning a new and rigorous graduate program modeled after Princeton’s. Karen was in the first year of the new program. Many graduate students with weaker backgrounds had trouble surviving, but Uhlenbeck’s training at the University of Michigan and her one year at the Courant Institute served her well. For such a small school, Brandeis produced many strong graduates, and women in particular.

During graduate school, Uhlenbeck still had no clear vision of where she was going with mathematics, or even what subject area she would pursue. “I think that my career was more marked by a wandering. … I would decide that I really liked something and I’d be kind of bored with it by the end of the course, and then I would decide that I really liked something else and I would get bored with it. It sort of wandered around.” In the end, however, this wandering was good training, for her research crosses the boundaries of many mathematical fields.

In choosing an advisor, Karen based her decision on what mattered most to her at the time: who was doing what seemed like exciting mathematics. Dick Palais was working on material that seemed new and different. Indeed, it was his field and his clarity that drew her to him; personal compatibility was less important.

I thought he was an extraordinarily clear lecturer. I still remember I went in and said, “Tell me about the heat equation,” and I got an hour lecture. That was all I needed to know about the heat equation for twenty years. He really is very clear. For me, I’m much muddier, and I appreciated that kind of a teacher. So he was an extremely good choice for me.

For a long time we were very uncomfortable with each other. He frankly admits that he didn’t want to take me as a student when I appeared. I think his initial reaction was that I would just have children and give up. But in fact after me he had a lot of women students. I don’t think [his response] was a personal one, it was just an automatic response of the time. He wasn’t negative. But I didn’t choose him because I got along really well with him. I liked the kind of mathematics he did, and he was really a clear lecturer.

Thus, Uhlenbeck’s graduate years were characterized by an almost paradoxical combination of traits — her lack of vision about where she was going, indeed the almost accidental way she stumbled into mathematics, alongside a strong independent spirit. She pursued what she wanted, with whomever she wanted, and with little guidance from external authorities. Even in her thesis work, she chose a subject that her advisor knew little about, so that in the end she was able to teach him some of the material, rather than the other way around. But both this tentative commitment to mathematics and her highly individualistic nature contributed to the problems she began to face as a young professional.

#### Beginning life as a professional: On her own and alone

Up until this point, being a woman did not conflict with her role as a student, particularly at Brandeis, where there was a significant presence of women. If there was any discrimination, she was (“perhaps stubbornly”) oblivious to it. The clearly structured roles of student and teacher made it easy for Karen to plug into student life as both an undergraduate and a graduate student. Though she was unusual, she did not feel like an “outsider,” or what she would later describe as being “other.”

But in going from being a student to being a professional, Uhlenbeck found her life changing in important ways. Being a woman was to play a significant role in how she saw herself and how others saw her. She was beginning to hit territory where very few women had gone before. She became more of an anomaly, and social expectations of what it meant to be a woman would begin to conflict with her role as a professional.

It is this stage of her career that also begins to demonstrate the significance of community in shaping the life of a mathematician: what happens when there is a lack of professional community, the different forms of community that are important in one’s professional development, and the unexpected ways that community can sustain one through difficult personal and professional times.

When Uhlenbeck finished graduate school in 1968, she followed her husband (a bio-physicist), Olke Uhlenbeck, for two years, taking a one-year appointment at MIT and then a two-year appointment at the University of California at Berkeley. When she was offered a permanent appointment at the University of Illinois at Champaign–Urbana, her husband agreed to move there, rather than to Princeton or Palo Alto, where she would not have a good job. Though it was unusual for a man to take his wife’s career so seriously, it was beginning to be seen as prestigious to have a professional/intellectual wife, and he was proud of her in that way. But even as they followed this non-traditional path, traditional norms and expectations still had a powerful effect on Karen’s life. Though she was a full-time faculty member, she was perceived, and perhaps perceived herself, primarily as a faculty wife. She did not yet have a clear vision of herself as a professional.

I felt like I was in a cage. I did not like being a faculty wife. I remember that feeling very well. We mostly socialized with people in my husband’s department. I remember eating dinner in the faculty club one time, and I went in the ladies’ room and cried. I really didn’t feel at home.

I had an assistant professorship, but I was still trying to be a good wife. Maybe I thought of myself as that way. It’s so hard to know what is going on, but it didn’t work at all, and I think it was partly professional. I think it was much more professional than I realized at the time. I was very good friends with my officemate and one or two others in the faculty, but I didn’t feel like it was a place where I could live. At that point, I really didn’t know what I was doing, mathematically or personally. I didn’t like teaching that much; I never saw it as a career. And I was trying to work by myself, really in isolation.

It never occurred to me that this was not the place for me. I always did what people expected of me and kept some part of myself for what I really wanted to do. I’ve never been one to fight external battles. It’s a waste of time. [At Champaign-Urbana] I really didn’t have anybody. I did get discouraged. Before that I had always thought that I could overcome all the obstacles.

Personal and professional issues are intimately entwined for women.
Even Karen had difficulty determining how much of her unhappiness was
tied to her role as a wife, and how much of it came from dissatisfaction
with how her mathematical life was going. Moreover, all the signals of the
time were suggesting that women’s lives are about being wives, mothers,
and, if they are professionals, at least something like teachers. But these
roles did not mesh with Karen’s orientation. She was not inclined to play
the role of the “good wife,” she did not particularly like teaching, and
because they did not have children, she was not defining herself as a
mother. But she had not yet developed a strong enough identity as a
mathematician to fall back on that for a sense of strength and meaning in
her life, and to use it as a way to interact with the community. The cage she
felt trapped in by these personal issues permeated her math life, and so she
felt caged mathematically as well. At the same time her career
unhappiness may have contributed to the personal unhappiness.2
Karen and her husband ultimately split up. In 1976 she decided to leave
the University of Illinois at Champaign-Urbana and start a new job at the
University of Illinois in Chicago, after a semester at Northwestern
University. “That was a good move. I never had any doubt. __\( \dots \)__ It was hard to
decide whether it was professionally better or personally better. I think the
city seemed like more freedom.” During this period, Uhlenbeck’s
relationship with her professional community began to change. For one thing, it
was the first time she was on her own and needed to support herself
economically. Fortunately she received a Sloan Fellowship, which played
an important role in the transition she was going through. “It’s possible
that lots of people’s careers wouldn’t survive if they didn’t have some sort
of support during a bad period in life.”

[Having the Sloan Fellowship] made a big difference to me during this period because I think I always felt that I owed something to the profession. When your personal life gets all shot, you’re glad that you have something professional. That’s when you suddenly realize that this isn’t a game, that studying mathematics and going along and doing the next step is for real, and if you didn’t have a way to support yourself you would be in a very interesting position.

Indeed, this period was the first time Uhlenbeck articulated any sense of responsibility to the profession; she also suddenly began to see herself as a professional. This link between economics and one’s attitude about one’s work is significant; it is one of the factors that have contributed to women’s more tentative commitment to their work. To a large degree, many of the women who pursued academic work came from relatively privileged backgrounds. Those who married often did not see their income as essential to their family’s security, which gave them the option of seeing themselves as either “amateurs” or “professionals.” This pattern is only now changing in fundamental ways, as even married women’s salaries are increasingly seen as essential to the economic stability of their families.

Though a lot of Uhlenbeck’s unhappiness during the period in Champaign-Urbana came from trying to conform to roles that were not suited to her, much of it was also due to the stifling of her mathematical development. She had access to a pen, paper, and library, but mathematics is much more than an individual pursuit. Some kind of community is essential for cross-fertilization and the sustained stimulation of the mathematical imagination. And community is what Uhlenbeck began to find both at the University of Illinois and at the Institute for Advanced Study.

#### Breaking out of the cage: Giving birth to her professional and personal self

Leaving Champaign-Urbana was a major turning point in Uhlenbeck’s life. It was to signal a significant shift, a final letting go of trying to conform to an external image of a woman’s life: wife, mother, teacher. As she let go of these identities, she was free to explore and embrace different parts of herself, and she began to blossom professionally.

At the University of Illinois in Chicago, Uhlenbeck was very happy with her new environment. A primary factor was the professional and personal relationships she established on her own. She no longer spent most of her time with her husband’s colleagues, creating instead a community of her own. A particularly important part of that was the camaraderie, both professional and social, that she developed with other women mathematicians.

I had a female mentor [Vera Pless] for the first time. I don’t know whether she ever realized it, but she sort of saved me. She would give me paper clips and tell me what to do in trivial situations. It’s really the only time I had somebody help me out. I remember that very much — being very relieved. She helped me over all the little details of a new job in a new place where you don’t know anyone. I would go across the hall and bug her at least two or three times a week. She would tell me about the people on the faculty. I don’t know if I would have sought her out, but she was right there, and she was there a lot of the time.

Later she says of Vera, “I felt like we were living on the same planet anyway.” This kind of intimate support is extremely important, a kind of invisible support that men often take for granted, and that women often have less access to — it helps one identify the unwritten rules of a department, a university, a professional community. It is also a bond that can traverse the boundaries between personal and professional life. For women this often happens more easily with other women.

Karen also very much identified with Louise Hay, another mathematician who had gotten a divorce, and because of that she very much appreciated her Ph.D. in mathematics, and her ability to be economically independent. For Hay, as for Uhlenbeck, the seriousness of being able to support herself became real only after her divorce. With Hay, Uhlenbeck could discuss details of her life that would not arise for her male colleagues, small but significant issues such as whether to keep her married name.

Uhlenbeck stayed at the University of Illinois in Chicago for seven years. During that time she also had visiting appointments at several research institutes, including the University of California at Berkeley, the Institute for Advanced Study at Princeton, the new Mathematics Research Institute at Berkeley, and Harvard University. Both the University of Illinois and especially Princeton brought her in contact with other colleagues she greatly enjoyed working with.

The middle year when I was at the University of Illinois, I got invited and went for a year to the Institute of Advanced Study, which was a special year in differential geometry. Shing Tung Yau, Rick Schoen, Leon Simon, and J. P. Bourguignon visited. I learned a lot of mathematics that I hadn’t learned, of a different kind, and got more in the mainstream of mathematics. I remember that it took me a few months before I would talk to anybody. I felt very much out of it when I came, but after I was there for a whole year — I worked with Rick Schoen during that period of time and during that summer — that was really the beginning of my success.

This group is one she continued to work with for a long time. During this period a graduate student from Harvard, Cliff Taubes, also came down to work with her, and thus began another important mathematical relationship. As she says of this period, “I don’t think I became a better mathematician, but I became better able to give seminars and could say things that weren’t totally incomprehensible to everybody.”

Although Karen had always had some contact with other mathematicians, relatively speaking, she worked quite independently. This limited math isolation, in combination with what Karen would describe as her “messy thinking,” made it difficult for her to communicate clearly with other mathematicians. She describes her interaction with a young mathematician at the University of Illinois named Jonathan Sacks, who would beat her door down until he understood what she was saying. “I was hard to understand. I still am hard to understand. I was not socialized.” The year at the Institute, therefore, was very useful in teaching her how to communicate and work with other mathematicians. It was the beginning of her transition from a kind of monkish math life to working, communicating, and interacting with others in her field. In addition to teaching her how to communicate, this exposure helped steer her to “mainstream” problems in her field. In the end, this interaction was nourishing for her intellectually as well as emotionally.

It was from this group, Shing Tung Yau in particular, that she also
received the kind of support she needed to gain confidence in herself as a
mathematician, and to begin to see herself as such. “I could tell that Yau
thought I was a good mathematician. I don’t think that had happened to
me before. He was obviously extremely bright. __\( \dots \)__ He’s a very remarkable
and energetic person__\( \dots \)__. I really credit him a large amount. I’m not saying
that other people haven’t tried to support me. My thesis advisor has
always been a large supporter. It was more real. I could tell that Yau
thought I was a good mathematician. That was hard for me to accept.”

But these connections did more than simply boost her confidence and bring her into the mainstream of mathematical ideas. They also became crucial advocates in her professional life. The significance of such advocates becomes apparent when Uhlenbeck talks about one of her two older mentors, a woman about five years older than Karen who is in roughly the same field:

We still have a very close relationship. She provides real support to me.

\( \dots \)She’s 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. I said Yau is really the person that I hold responsible for my success. You know, it’s true — to 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.

Her experiences have helped her recognize how important it is for young mathematicians to make contacts in the community — something that she missed out on at the beginning of her career. As she acknowledges, given her independent streak, this was probably inevitable. However, her more recent and fruitful interaction with younger colleagues and students is, in a sense, making up for what she missed at the early stage of her career.

I tell my students that the most important thing, if you want to keep doing mathematics, is that you establish mathematical contacts. Even if you don’t need to work with them, you’re going to get depressed sooner or later and you’re going to need some sort of input

\( \dots \). Whether people stay as research mathematicians or not, I think the big item is that they have some contact in the mathematics community of a personal nature. That sounds weird because mathematicians are crazy. They work by themselves and you sort of think of them as sitting in their room working by themselves, but every mathematician hits bad points and how do you get over it? Somebody has got to come along and say, “Cut it out, kid.” Or somebody has to come in with a new idea and hit you on the head with it.I see young people who always think they want to go to a place where there’s a lot of action and a lot of ideas going on. I think the only benefit they really get from that is that they make strong relationships. Some mathematicians are social. Some mathematicians work together, but a lot don’t. What happens to the people who go out and work in isolation? I think nothing, except that you’re bound to hit a bad point, and then how are you going to get over it? If you’re on good terms with your thesis advisor, you call your thesis advisor up and the thesis advisor says to cut it out or gives you some feedback. [On the other hand] I don’t want my students doing that. They need to find their own relationships.

\( \dots \)There are people who sail through and nothing ever goes wrong. Normal people aren’t like that. We have all sorts of awful things going on.

For women and minorities, this is particularly important, not because they necessarily have more problems, nor even necessarily different problems, but they often have less access to the support systems that help mathematicians through the tough periods. For those who feel more isolated, small obstacles can become enormous. Colleagues and friends can help put them in perspective.

We see, then, the subtle ways that the mathematical community can be very important in the development of one’s career: it exposes one to mainstream problems and new ideas, teaches one how to communicate with other mathematicians, instills confidence, provides support during difficult periods, and facilitates recognition and professional opportunities. We also see the many levels of mathematical community. There is one’s immediate work environment, and as Karen’s life illustrates, through her different experiences at the University of Illinois in Champaign-Urbana, the University of Illinois in Chicago, and later the University of Chicago, these communities can fundamentally affect one’s image of oneself, one’s vision of the math community, and one’s relationship to that community. In this way they can profoundly influence one’s image of oneself as a mathematician. Karen had the opportunity to go beyond these immediate communities and meet other mathematicians who worked in fields close to her own. The Institute for Advanced Study brought together top researchers in her field and greatly expanded her interaction with the larger math community. This exposure to Yau’s group was profoundly influential in the development of her career, leading to public recognition and prestigious awards.

#### Recognition and alienation

Though Karen was happy in Chicago at the University of Illinois, a number of factors made her decide to leave: money was limited, she wanted to be working with graduate students, and finally she “wanted her career to go somewhere.” She accepted a job at the University of Chicago, which was more prestigious; there was more money available for research, and the school had a first-rate graduate program. But in retrospect she says, “Leaving the University of Illinois and going to the University of Chicago was probably a mistake.”

At the University of Illinois she had had a number of close colleagues with whom she could work, mathematicians who were not necessarily directly in her field but were close enough that they had a lot to share and teach each other. They were also young, and she found them easy to relate to. The University of Chicago, on the other hand, was not a stimulating research environment for her; she did not find colleagues to work with — “It wasn’t congenial there.”

The university was going through a period of transition at that time. The older generation of well-known mathematicians had all retired, and there was a younger group who were not yet as well established. The graduate program had not had much success with women students. But the mismatch between Karen and the University of Chicago went deeper than these issues. The atmosphere and traditions of the university were alien to her on many levels.

I simply never became friendly with people. Most were educated in fancy institutions. There was this air of real elitism. I had gotten a degree from Brandeis. I’d been teaching at the University of Illinois. But they seemed to have lived their whole lives in prestigious institutions. A lot of people there hadn’t taught undergraduate courses. I’d been teaching remedial calculus to business majors. A different world. They also didn’t do the kind of mathematics that I did. This is not to say those things are in any way related to each other, but it was the double thing.

These various factors — the lack of colleagues to work with, the air of elitism, a long commute — all contributed to the distance and alienation she felt from the institution.

I just didn’t feel at home, I guess that’s the biggest thing. I kept telling myself that I should give it a little longer, that I should do something different, but it’s difficult to know how to change it. Chicago had a bad reputation among the women mathematicians in the Chicago area — it did not have a reputation as a friendly place. I just really didn’t think those things were important. I now think those things are really important.

For women especially, the atmosphere of the workplace can have a very significant impact on their work, their image of themselves, their image of the mathematical community, and whether they fit into it. In Karen’s case, the University of Chicago functioned to further polarize her from a kind of traditional, mainstream mathematical community. It reinforced a vision of “them” as “other,” an image she could not imagine conforming to.

Karen’s evolving vision of the significance of one’s professional environment, particularly for women, had a strong influence on later career decisions. She has been offered prestigious positions and declined them because she felt they would not be positive environments for her professionally. And she ultimately chose instead an offer from the University of Texas at Austin, where she now holds a Sid Richardson Foundation Regents’ Chair in Mathematics.

#### Teaching and research

##### Teaching

Karen has a fairly unusual record with respect to teaching. In her early years she taught mostly low-level calculus and finite math courses. Later she taught primarily graduate courses. She has had little experience with anything in between, i.e., standard undergraduate courses or upper-level (math major) courses; therefore many of her thoughts about teaching emerge primarily from working with graduate students.

Teaching has always been difficult for Karen. Even when she was young and planned to be “a forest ranger or a scientist,” she did not see herself as a teacher. The intensive social skills involved in teaching and the ability and patience needed to explain complex ideas in a simple manner were not Karen’s strengths. In fact, her quick mind in many ways hindered her ability to teach. She discovered early on that the way she learned, and the way she thought about math, did not help her at all in teaching other people. “As far as teaching goes, it really is true that it takes me years to understand the difficulties students have. I just never comprehend that you have to say something twice. It took me a long time to understand that saying something once is essentially not saying it.” Furthermore, Karen was in many ways self-taught, both as a student and as a mathematician. This made it harder to know how to teach other students; and there were no teachers that she was striving to emulate. Finally, the very traits that are assets in her research — including her non-linear way of thinking — can be problematic in the context of teaching.

The way I learned is totally useless for teaching. So you have to start again. I have no love of organization. The appeal that mathematics has for me is not that I can organize it; I think many successful teachers enjoy this organization. They like getting the material in a straight line, and I think that many students enjoy that kind of presentation. In fact, when I’ve had teachers who do that, I often find that it is a very efficient, neat way to learn material. But the kind of mathematics I do is very sloppy mathematics. I was discussing this with a colleague who said that non-linear analysis seemed so wild or untamed compared to linear analysis, and the kind of mathematics I do is really not a very organized kind of thing. So I think I have difficulties in teaching, but that kind of mathematics is going to be like that. Students who like things to be orderly and neat would be crazy to go into this kind of mathematics, where you have to learn an immense amount, not quite understand all sorts of stuff, and put a lot of things together that are completely different. I think it’s probably hard to lecture on.

What emerges from Uhlenbeck’s style and experience is a very personal approach to working with her students. Since each student has a unique voice, she interacts with each of them differently. Rather than trying to find a common denominator, or to teach students in the same way, she focuses on what is different about them. “First I decided that the students were so different that the idea that one would impose an outside theory of how they should learn was crazy. Then I decided that imposing an outside theory of how I should teach was also crazy.”

When I started having graduate students it was an eye-opener, because I consider myself a sixties liberal, and we were all interested in teaching differently and giving people the right ideas and not being so formal. But when I started having more than one graduate student, maybe around my fourth graduate student, I realized that theories of learning are probably just nonsense because people think so differently. My first four students were all Americans. They all went through school, they all came from a similar culture, they were all interested in mathematics, they were all interested in roughly the same kind of mathematics, and they were all completely different.

Some of them think abstractly, so much so that I don’t ever understand anything they say because I write differently. Some of them were so concrete; they had to start with the simplest case and work up. There’s just this terrific variety between thinking abstractly, thinking by examples, thinking concretely. The brilliant student I had who was half physicist was a very loose thinker. Some of them are very tight thinkers. They all have something very different to contribute to mathematics if you prod them enough.

They all got interested. I never had a student who didn’t get interested. I can’t imagine trying to get a student through a Ph.D. when they weren’t interested. But they’re all different. When you’re going to teach kids, freshmen, anybody else

\( \dots \)if there’s that much variation of graduate students in mathematics, how much variation is there going to be in the general population?

Karen’s primary task as an advisor, therefore, is to help students find their unique styles and contribution. As she says, “In order to be a good mathematician, you’ve got to figure out what you can do and find out the way you think. I don’t think it’s so easy. You have to find your own way of doing things.”

At the same time, she also appreciates that help from others at critical times can make a tremendous difference. When her students get discouraged, for example, she reminds them that everyone gets discouraged and recommends tricks that can help: having an easy problem and a hard problem to work on, trying something different for a thesis topic, taking a vacation, reading an article if you’re tired of banging your head against the wall, or giving a seminar.

But the encouragement and stimulation does not go just one way. Uhlenbeck considers her students her mathematical children; working with them is a very rewarding part of her life. Many of them have also been great teachers for her, and a major source of inspiration. Her early graduate students were particularly important; they would take courses, for example in algebraic topology, and would eventually learn more than she knew, so they could teach her, making the relationship a reciprocal one. “My graduate teaching is so successful partly because I have arranged it so that I learn from them.”

Indeed, when she is asked about who has been most influential on her mathematical development, she says, “I think I’ve been influenced by some of my students more than anything. I had a really good, bright student who was in between physics and mathematics, and when he left Chicago, I knew I was going to miss him. So I would go over to the physics seminars.” She recalls that he said of their discussions, “It’s really strange, you go in there and she talks and talks and talks and you never quite understand, but after a while something happens and it’s worth it.”

##### Research

Uhlenbeck’s research blends geometry, topology, physics, and analysis. At the time of the interview, it involved finding connections between mathematics and the new physics. For example, she combines geometric concepts with detailed analysis of partial differential equations to describe objects as varied as soap bubbles, black holes, and certain kinds of quantum tunneling.

She is often torn now about how to use her time: doing problems, developing ideas that she’s already been working on, or learning new material which could lead to new problems and a larger picture of how pieces fit together. It is her willingness to constantly explore new territory that has kept her mathematically vibrant and alive.

There is the pleasure of doing mathematics and this real desire to learn what the physicist Ed Witten is doing. They are pulling me in opposite directions. I would like to do mathematics, but I also want to know what’s going on over there, and the two are really different. I can’t leave these new ideas alone. This is where the action is, and I feel that I really have to learn all this. I’m in a field that has had a lot happen in the last fifteen years. For me this is very exciting. As one of my students said, “It’s like being a pioneer and walking on some area that nobody had ever walked before.”

But for Uhlenbeck, a new kind of pleasure has emerged from doing mathematics, one for which age and experience are assets.

I still get a kick out of doing mathematics. It’s harder to come by now because many of my standards are higher. But now there’s a new kind of pleasure of trying to fit things together, making something match something else. When I was younger I had no in-depth knowledge of mathematics. By now I know an awful lot of mathematics, and I’m really fascinated by connections.

One characteristic of Uhlenbeck’s research that is distinctive is not only the subject matter she pursues, but also how she thinks.

[How I think] is not linear. When I write a paper, it’s much better to just have the basic ideas, and then I can pick them out and fill them in. If I just write this thing that goes linearly, I get confused. I’ve discovered that there are basically two types of mathematicians, those that really do go from point to point and get real upset if papers aren’t written that way and write their papers that way and want the lectures that way, and then there are people like me who prefer the ideas to be given and the filling in [is secondary] — it’s just structured differently in my mind.

I think some people are very surprised that a mathematician would be like this. They think of mathematics as being ordered and careful and so forth. And indeed many other mathematicians are like this. Maybe more of them are very orderly as a whole. But it’s not the way I think, and it’s not the way I learn. In fact, there’s really no way to get into communication with modern physics without just sitting through a lot of it so that it stops sounding like garbage. You can’t logically work your way through this nonsense. You just sit through enough and suddenly what they’re saying seems logical and starts fitting together. It’s a different language.

#### Awards and honors

In the end, Uhlenbeck’s unique interests, curiosity, and style have all contributed to her success. And while Uhlenbeck is delighted by her mathematical successes, the external recognition that has accompanied this success has not been unmitigated pleasure — something that is hard for many people to understand. Awards such as the MacArthur Fellowship and election to the National Academy of Sciences are coveted by most everyone in the profession, but for Uhlenbeck they raise difficult issues about her identity.

I didn’t mind being a woman doing math, not supposed to be doing it, working on the fringes, succeeding in a small way, and sort of being incomprehensible and not having many students. In many ways that was much more comfortable. I was really sort of doing it for myself. Then [when I started getting awards and public recognition] I had to make a major reevaluation of who I am. Getting the MacArthur is really sort of traumatic in some ways. I just never thought of myself in any way like that.

There are several factors that contribute to the discomfort Uhlenbeck has with this recognition. There is of course the practical issue of how time-consuming such awards can be: partaking in ceremonies, giving talks, socializing with other award winners and academy fellows. But the sense of burden is much deeper and more complex for Uhlenbeck than just increasing demands on her time.

Like many mathematicians, Karen is a very private person who chose
to pursue mathematics in part because it was a world apart from the public
arena. “You choose to do your own thing [in mathematics], and what you
do is very private and personal, and three other people in the world may
understand that.” That privacy is what Uhlenbeck is most comfortable
with, and it is what she lost as she gained recognition. Because of the
awards she suddenly became a public figure, a far cry from what drew her
to mathematics.3
Moreover, the awards forced her to publicly see herself, and recognize
that other people saw her, as a *woman* mathematician, not just a
mathematician. Since so many people began to address questions of women in
science to her, she suddenly had to speak for *all* women. This was an
uncomfortable role for Uhlenbeck. She does not see herself as a typical
woman, nor does she think she makes a very good role model for women.
In fact, because she found her passage a difficult one, she does not want
her story to be exemplary.

Furthermore, being cast into the role raises conflicts in her strategy for
dealing with gender problems. To a large degree Uhlenbeck felt that she
was oblivious to gender bias, and her way of dealing with problems was
often to ignore them. But when she hears the discouraging experiences of
many other women, she feels a heightened sense of responsibility. “I’m
not sensitive to myself at all, but I am sensitive on behalf of younger
women.” In her public role, particularly when she is seen as a
representative of women in math, she could no longer ignore the problems. “I think
you’ll find lots of older women, even though they may not say they’re
feminists, who will get furious [about the kinds of problems women
confront]. You won’t find that in young women who have any hope in
succeeding. It’s self-preservation.” At the same time, taking these
problems on can be extremely frustrating, particularly when there is little one
can do about them. “Once I became a member of the mathematical elite I
found it a pain __\( \dots \)__ to be a woman. There are two choices for me. I can
either ignore the fact that I’m a woman or I can become a rabid animal.
__\( \dots \)__ I don’t see any in-between reaction to the situation.”

In her own life, though she did to some extent “ignore the fact” that she was a woman, Uhlenbeck developed mechanisms to create an environment in which she felt comfortable. She created a niche by defining herself in opposition to the very group she felt excluded by. In this way she was able to embrace what was different about herself, and in the process to liberate herself from the need for external affirmation. Woody Allen quipped that he never reads reviews of his work because if he took the good reviews seriously and was pleased by them, then he would have to accept the bad reviews and get depressed by them. Uhlenbeck, too, did not seek out or have much investment in affirmation from the larger math community. But this strategy made it confusing to maintain her identity when she suddenly found herself being defined as “one of them.”

I probably had always been looking outward in

\( \dots \). You say “those guys” and here you are, your whole career you’re looking from the outside in and saying “those guys” and then you’re suddenly one of them. But you’re not, because you’ll never be. I found it extremely hard to readjust everything.\( \dots \)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. It will take a long time, if ever, before I can see myself as being really successful because I’m so conditioned to do it because I want to do it and to get along with life.

Uhlenbeck’s life is a reminder that traditional measures of success can
be viewed quite differently by women. This is *not* to say that women
should not get these rewards, nor that they don’t want them. It does,
however, make us more sensitive to the fact that even “success” for
women in traditionally male fields is a complex issue. There are two sides
to these blessings, and while it is nice to have external recognition, the
attendant burdens and responsibilities are weighty.

Uhlenbeck’s life exemplifies the fact that what hinders women in mathematics is not necessarily lack of talent. Many factors contribute to the difficulty women face in feeling like equal members of the mathematics community. While Uhlenbeck has the talent and the independence to pursue mathematics on her own, her success was significantly tied to her connections within the mathematics community. But establishing these ties can be more difficult for women. In some cases, women are more likely to be seen as faculty wives rather than professionals in their own right. In others they may just be seen as different, not fitting the image of what a mathematician or a colleague is supposed to be.

For Uhlenbeck, establishing ties to the community was also made more difficult by her independent nature, and the fact that she thought quite differently than many of her colleagues. She was not socialized to “speak the same language.” While this very trait of independence is what enabled Uhlenbeck and other women to pursue mathematics in the first place, in some ways it can also later become a hindrance.

Many people have suggested that the difficulties women face in mathematics are partly due to women’s lack of confidence, but Uhlenbeck suggests that the problem is more one of a lack of fit. “I somehow feel that it’s not so much a lack of confidence as not feeling in the right place. Women have plenty of confidence. The women that I’m talking about can do anything. That’s the problem. They’re survivors of life. They know they can find something else interesting to do.” For this reason, Uhlenbeck argues that many very talented women choose to leave mathematics. But even those who stay can also experience that sense of not belonging, or not fitting in. This was true for Uhlenbeck, both at Champaign-Urbana and at the University of Chicago. In different ways gender played a role in making her feel like an outsider, a sense of distance that can be further exacerbated by other differences such as race, class, or ethnic background. In the end, even someone as highly self-sufficient as Karen Uhlenbeck discovered that one’s professional environment can play an important role in one’s sense of belonging. As she says, “There are all these trivial things which have nothing to do with your mathematical ability [which influence your mathematical life]. Mathematical ability is such a small part.”

#### Postscript on Karen Uhlenbeck’s research

#### by Daniel Freed

Karen Uhlenbeck works on a variety of problems which defy pigeonholing into a unique mathematical field. The terms “global analysis” and “partial differential equations” probably come closest, but they do not convey the importance that ideas from theoretical physics play in her work and in this field generally. “Differential equations” are expressed in terms of calculus, which was invented by Newton and Leibniz in the seventeenth century. Newton was motivated by the study of motion, particularly motion of the planets. Much of Uhlenbeck’s work deals with equations whose origins lie in modern versions of Newton’s theory of mechanics and gravity. Though I stress the physical origins of the equations, one must realize that mathematicians study these equations in a manner much different than physicists. Often we apply these equations in new contexts, and we use them to teach us about other mathematical structures. Such is the case with the “Yang–Mills equations,” whose study occupied Uhlenbeck during much of the previous decade. Her fundamental first work on these equations enabled other mathematicians, particularly Simon Donaldson, to use them to revolutionize the field of four-dimensional topology. Her later work had ramifications in the field of algebraic geometry. Without discussing the technical details, one can still appreciate the centrality of her work, which connects on the one hand to theoretical physics and on the other to more abstract fields of mathematics. Uhlenbeck has done seminal work on many other equations as well, and her current work is in yet another new direction — the theory of “integrable systems” — and promises to yield exciting results.