Joan Birman grew up with her three sisters in Lawrence, Long Island. Three out of the four sisters majored in mathematics in college, though only Joan went on to become a research mathematician. After having three children of her own, Joan went back to do graduate work in mathematics at New York University, receiving her Ph.D. at the age of forty-one. It is remarkable in mathematics to get a degree so late and then to go on to become such a productive and successful mathematician. Birman exemplifies one way to integrate family with a research career; her life gives us insight into some of the advantages and disadvantages of such a path. Like many of the women in this book, she challenges the myth that “mathematics is a young man’s game.”
Joan’s research is in knot theory. She teaches at Bernard College–Columbia University and gives talks in mathematics all over the world.
From tinkertoys to topology
My father, George Lyttle, was a dress manufacturer. Neither of my parents finished high school. About the time I was married, which was 1950, my father retired because of economic problems in the dress industry in New York. He intended to take up some other area, because he was then only in his mid-fifties, but he never did. I think my father, under different circumstances, could have been a scholar of some kind.
Was either of your parents interested in mathematics?
No, they didn’t finish high school. My father had to work for a living from his very earliest days, and he just didn’t have the opportunity to think about other possibilities. Both my parents came from a generation where women did not think about such things. But I’m not sure my mother was inclined that way anyhow. But both my parents had a strong idea that we would all go on to college.
I am one of four girls. I am the third child. One sister, the one who is just older than me, and I had almost parallel careers. We both brought up our children and then began school afterwards. She studied plants and I studied mathematics. We both got our Ph.D.’s when our children were grown. I was forty-one; she was about the same age, a little bit older. She was four years older than me. We both had successful, recent academic careers in research. She was a plant physiologist. She died in 1989; her name was Ruth Lyttle Satter. Both she and my oldest sister were math majors in college. So three out of the four girls majored in math.
Did your father show interest when you were in school?
Always. Neither one of them ever had any interest in the specifics of mathematics. But if I came home with a 98 on schoolwork, my father would say, “What happened to the other two points?” — and at the same time indicate how pleased he was.
In some ways neither my mother nor my father understood that mathematics was not a woman’s world, because they didn’t understand enough of the subject. As long as you were studying, that was good, that’s what was important. In fact, as time went on and my various nieces and nephews went to college, my parents never discriminated against the one who was a photographer or doctor or the one who was working on a publication on women’s health — as long as you were at the books, it was good.
And doing well.
That was not an issue; we all wanted to do well. One way my parents encouraged me was with the toys they bought for me when I was very young. They bought me an enormous set of Tinkertoys with wooden sticks and connectors that you could use to build large structures, and also an erector set where you put pins into hinges. They got me other toys as well, and I would put things together. I’m a topologist, and all those toys involve shapes and structures just as topology does. I had a chemistry set too. Maybe my parents recognized my interest by choosing toys like that. They didn’t push what should be a girl’s choice on me. As long as I was really interested, it didn’t seem to bother them that this was not the usual choice for girls.
Did you and your sisters influence each other in mathematics?
Hard to say. I like to think that I made my own choices in later life, but I have to say that the evidence is that we probably did influence each other, sure. I was the third.
Do you remember them talking to you about mathematics? No. My early memories are that I was good at math and could understand more than other people. And I liked it. I’ve always wanted to understand things.
Mathematics and other subjects?
Mostly mathematics. I remember mathematics, specifically, when I was very young. I remember being able to figure out something about the sum of two numbers. It all came back when I went to visit my children’s school and I heard the teacher giving a lesson about whether the sum of two odd numbers was odd or even and what happens with the products. I remembered it from when I was a child, and I remembered how I understood it right away and nobody else did. And I remembered how beautiful it seemed.
So elementary school was where it started for you?
Yes. But I don’t remember having any particularly good teachers in elementary school. I do remember a good one in high school.
What do you remember about that high school teacher?
We had a group in high school — myself and three or four other girls who loved math — and this teacher was really positive. It was a girls’ school. We would just sit there, and our arms were almost coming out of the sockets trying to answer the question. We didn’t need encouragement. We were competitive with each other, and she was just a good teacher, and the material was interesting. Geometry was a course that I loved in high school. This group of friends and I went around the school to recruit candidates for a solid geometry course. We succeeded, so the school gave us a solid geometry class.
Julia Richmond was a great big high school, but it had a separate little school within the school, called the country school, which was for academically strong students. When I first got there, I was not in the country school; I was admitted to it afterwards. It was a very nice school. Within that group it was possible to love math and not feel like an oddball. It wasn’t until later that I began to think there was something a little inconsistent between being a woman and being a mathematician.
How much later? In college, when I became interested in boys, which I wasn’t in high school. They weren’t around in high school, there was no opportunity, and I thought that the girls who were interested in boys were silly. I guess I matured rather late.
So by the time you were in high school, you were really excited about math.
Yes. And I wasn’t concerned about the fact that this set me off in some ways from most of the other students. There weren’t that many girls who were into mathematics, but I had a good circle of friends, and that was enough.
At that point did you think that you might pursue mathematics later?
Sure I did. Then two things changed that. The first was that college mathematics was initially quite disappointing. I started with calculus, and I didn’t like it or have enough sense to understand that it was the course and not me. It just got to a point where I felt like they could tell me anything and I’d have to believe it. All my confidence in my knowledge and understanding of math was gone. Most of the students that I teach now are happy with that kind of a calculus course; in fact, we have a hard time getting students who want anything different from that. But I found it very dissatisfying, and I didn’t have any idea what it was that I didn’t like. It seemed like mathematics had changed. That was the first thing that led me to question a career in mathematics. But later, when I understood things better, a second issue arose. I became aware of the fact that mathematics required enormous concentration, and that if I was going to do it badly, then I might as well not do it at all. That’s when I decided not to go on to graduate school. I knew that it required a kind of concentration that I felt was going to interfere with the rest of my life [as she describes later, her focus on relationships: marriage, children, etc.].
Were there other subjects that you were particularly interested in?
I was a good student. But I knew I had no talent for languages whatsoever, and I had a hard time remembering history. With mathematics, once I understood it, I didn’t have to memorize it. My interests were certainly in the direction of science and math. I liked biology. When I went to college, I somehow thought about physics, biology, astronomy — astronomy was very, very interesting to me. But in picking a career, I thought that with astronomy you have to live in a place where the sky is clear enough to look at it — that didn’t sound consistent with the city life I liked. I liked other things: sewing, cooking, things with my hands. I was clumsy and I knew that I wasn’t going to be any kind of an athlete, but I like doing things with my hands. I was never somebody who could take being with people all day long.
So I majored in math in college [at Barnard]. I didn’t go to graduate school right afterwards. I worked. In fact, I got an initial job in a place that was making electronic equipment. That job lasted about six months Initially it was very interesting; it involved solving a problem in geometry. But after the problem was solved, they had me making measurements on the oscilloscope, and that was terrible. I figured that there weren’t too many jobs like the first one I had, so I’d better learn something more practical. That’s when I started to go to graduate school [in physics].
I took a lab course in electronics at Columbia in the Physics Department, which I liked very much. I like working with my hands. Then one day — and this was really an accident — I met my old physics professor from Barnard, and he asked me what I was doing. He said they had an emergency and needed a teaching assistant for the physics lab at Barnard. He suggested I go back to graduate school and take the teaching assistant job. So that’s what I did. That’s how I got to graduate school.
What a coincidence.
Yes, it really was. I did finally get my master’s degree in physics. But by then I knew that I didn’t have the talent for physics. I scraped through, but I didn’t have the feeling of it. Again, I felt like they could tell me anything. I didn’t understand what the ground rules were.
Yes, you need some kind of an intuition, and I didn’t have it. The electronics laboratory involved a very precise measure of truth. I liked that. But when I got to problems in mechanics, I just didn’t understand what you could ignore and what you had to accept as given. There always seemed to be approximations. But I never knew which ones were acceptable, and the whole thing was hazy.
By then I had a bachelor’s in mathematics and a master’s in physics and this one year of job experience, and so I went out to look for another job. I got a second job in the aircraft industry working on early navigation computers for aircraft. That was very interesting. I did that for five or six years. In the meantime I had gotten married. I held that job until we had our first child. I had intended to go back to the same kind of work after we had children, but I found that it was just impractical. I didn’t want to. I did work part-time, which was important to me, but it got to be more and more difficult to make it meaningful. So after some number of years of working one, two days a week at the most, there was a crisis. My husband, who worked in industry but had leanings toward academia, had an offer to be a visiting professor at the University of Pennsylvania. It meant moving the whole family and leaving my part-time job. I was away from the job for six months, and when we returned, it just seemed impossible to go back to the same kind of part-time work.
While we were at Penn I took a course in digital computers, and that was interesting. While I was working, I had never felt like I knew enough mathematics; I wanted to learn more so that when I returned to work I would be better equipped. So after our third child was born, I started graduate school in mathematics. Throughout this period, then, I never really lost track of mathematics; I always kept some little thread of contact, even through the time the kids were growing up.
I took one graduate course in the evening at NYU, and that went so well that the next year I took two. There was an exam that determined who got a master’s and who was able to go on for a Ph.D. So I took the courses that were needed for the exam. By then my three children were a few years older. We had a babysitter take care of them during the summer so that I could study. I spent the whole summer in my bedroom office studying for the exam, thinking it would be for a master’s. But to my surprise, I passed it for a Ph.D. qualifier, and I was very pleased!
So it was the same exam? It just depended on how you did on it?
That’s right. I was surprised and pleased. At that point I thought if I really wanted to go on to get my Ph.D., I needed more help at home, and I couldn’t afford it, so I applied for a fellowship. By then our youngest child was in nursery school, and I was awarded the fellowship upon the condition that I would come back as a full-time student. So that’s what I did. I used the money that I got from the fellowship to get someone to come help. At first it was very good. We had a lovely woman. She really added a lot to the household.
Did she live with you?
No, she didn’t live with us. We never had anybody live with us. You know, studying mathematics, the amount of time I had to be in school was really relatively little, but what I needed was time to be free to work, so we used to have people who took care of the children and who cooked, varying hours, varying arrangements. I was forty-one when I got my Ph.D. As soon as I became a full-time student, I started to work with other students, and that made a big difference.
Did you work with women students?
No, the ones I worked closely with were all men.
How many years were you actually at NYU?
Our son, David, was born in January of ’61, and I got my degree in ’68. So it was seven years altogether. At first the pace was slow, and then it picked up.
That’s great that they gave you that support, because otherwise you wouldn’t have…
Couldn’t have made it — it really was important. I feel very grateful to NYU for that. First of all, they had this very open program for people working toward a Ph.D. You could come as a part-time student. That is not true at all graduate programs.
Who was your thesis advisor?
At what point, then, would you say you really got turned on to the research mathematics? Was it in graduate school?
No. I had worked on research problems in industry and I liked them. Those problems had the same quality as the mathematics that I’m doing now. But when I started to do research for my Ph.D. it grabbed me, because it seemed as if the purpose of the whole thing was much more beautiful to me than designing a new piece of aircraft equipment.
In industry, I had worked on very applied problems. For example, I worked on a Doppler navigation system in one of my jobs. I worked on another problem — this was on a bombing computer, and I really did not like the purpose of it, but it was a problem where I was very proud of my idea. The problem was to compute the effect that the wind and an aircraft’s up-and-down movement would have on a bomb when it was dropped from the aircraft. The trajectory of the bomb could be described with differential equations. They asked me to figure out how to correct the equation to account for the motion of the aircraft, but what I realized was that all we had to do was change the initial conditions — the differential equation would stay the same. So even though the change was a change in wind speed, you could simulate it by a change in the initial position. I had a hard time getting them to understand that. I had such an argument with my supervisor because he could not follow what I was saying. And I was right!
That’s an interesting difference between research work and working in industry. In industry you’ve got to convince somebody of your explanations, too.
Yes. So when I started to do research in mathematics, it seemed as if it was more meaningful to me to make contributions to mathematics than to navigation computers. People ask what the use of mathematics is. I had this whole long period when I was doing very useful things in the ordinary sense, but then I found it more meaningful to be contributing to [pure] mathematics. Most people feel the opposite.
First teaching job
Then I thought, why couldn’t I just do my research and attend seminars and keep in touch that way? I went to a meeting, and just by accident people had heard that I was looking for a job. Stevens Institute, which is an engineering school in New Jersey, just happened to need people. It was late in the year, after the academic hiring had been done; several people had left suddenly, so they had vacancies. So by almost an accident, I got that job. And it was a good job. It was an engineering school with good students; there was always somebody who responded and kept you on your toes if you did something stupid in class….
I was there for five years, with two interruptions — one was when my husband had a sabbatical leave and we all spent a year in Paris; the other was the year I was teaching at Princeton. That came about through my research. I had done some good work, and I was invited to give a seminar at Princeton. The following week I went to attend the seminar again, and, who was a very well known mathematician in knot theory which was what I was interested in most of that time, said, “How would you like to come and visit for a year?” I talked this over with my husband and he said, “If you want to do that, we can move to New Jersey and I’ll commute.” But the three children had just had a year of their schooling interrupted when we were in Paris, and I knew we’d have to make arrangements to rent a house, find a place where my daughter could take swimming lessons and music lessons, deal with transportation, etc. I couldn’t handle that, and the children were sensitive to these moves. So I thought I would commute.
I knew this was my chance to be a research mathematician, and it was. It made a tremendous difference — because of the prestige of being at Princeton for a year, the context, and everything else.
You have had many fortunate twists in your life.
Well, yes, I think that’s true. A lot of fortunate twists, but I think you have to be alert to them.
Yes, if it weren’t for those, it might have been something else.
I think so. I do a lot of joint work, and people ask me how I find somebody to do joint work with me, and it always seems like it’s almost accident, but of course it isn’t. I am alert and ready for it. I’m looking for it.
Choosing a field of research
When I went back to graduate school, I was fully intending to get back into the aircraft industry or some math-related job. I picked NYU because it was a school of applied mathematics. It was completely surprising to me that I became so interested in pure mathematics. That happened by a process of discussion and learning about things. After I passed my Ph.D. qualifier exams, I went to speak to different people on the faculty about what they were doing, andsaid something that was excellent advice to me. I had liked him. He was an excellent teacher. In fact, he taught a course that was really important to me. I said I didn’t know what area I wanted to work in. He said, “Do you like inequalities?” And I said no. And he said, “Well, you don’t want to work in differential equations.”
That was very good advice. At NYU there weren’t too many who were not in applied mathematics and in something related to ordinary or partial differential equations. One of them was Magnus, who was ultimately my thesis advisor. He worked in combinatorial group theory.
Then how did you get into topology? When did that transition happen?
The two subjects are really very closely related. By the time I’d gotten my Ph.D., I knew that low-dimensional topology was the thing that was more interesting to me than combinatorial group theory, and I just gradually worked my way into it. I made some contributions and used a little bit of topology. In my thesis I solved one problem, but it suggested another problem. This is what I was really ambitious to solve when I was at Stevens. In one of the nearby offices there was a young fellow who had just gotten his Ph.D. at Stevens, and he, like me, was a little bit beyond the usual age, though much younger than me. He had been an engineer and he didn’t like what he was doing, so he went back to graduate school, and he had just gotten his degree. Because he got his degree rather late in the year, they kept him on as an assistant professor. So we began to have lunch together. I talked to him about this problem that I had tried to solve. We talked about it through the whole year. I had a conjecture, but first the conjecture didn’t make any sense, and then gradually we began to understand some of the structure that I was guessing was there, and we began to understand it better and better. Then came a key moment, and I remember it very well — when he put something down on the blackboard and we started going on it. We solved the problem together, and that’s what I was invited to give a seminar on at Princeton. It was a very good, satisfying piece of work.
Was that your first joint work, also?
That was my first joint work.
That’s a nice way to do it — just sitting having lunch and working on it a little bit at a time.
Yes, that’s right. But all my joint work has been like that. I just talk to people and it just happens.
So that’s when you really started getting more and more into topology?
Yes. And then I gave a talk on our work at Princeton, and I discovered that there was this weekly seminar on knot theory. My work involved surface mappings, but surface mappings are very closely related to knot theory, so I began to attend this seminar on knot theory at Princeton and began to go to it regularly, and that’s how I learned topology.
Did you stay in contact with people at Princeton and continue to work with them?
The year that I was there, sure. After the first year, Ralph Fox started to have health problems and ultimately had a second heart surgery. He died a week after the second surgery. After that I felt so bad. The whole seminar fell apart after his death. But by then I had a job at Columbia, so my contacts at Princeton ended because the graduate students who were working with him were no longer there. There wasn’t anybody else who was doing just what he was doing.
That must have been rough for the graduate students, too.
When he died, he had one graduate student who was still working with him. When he knew he was going to go in for open-heart surgery, he asked me, “Will you look after him in case I have any problems?” In fact, this graduate student applied for a job at Columbia, and I really wanted him to have him there. I was very touched that the department respected my wishes and offered him a job.
So you got a job at Columbia right after Princeton?
Yes. I was doing good work, and the year at Princeton was very helpful to me. People knew about what I was doing. The very process of my possibly getting a job at Princeton made my name known, so it was quite important. I was offered a position at Barnard–Columbia [two schools with one mathematics department].
A late start
When you were going to graduate school, you had an unusual background since you had worked for a while first. Can you talk about the ways in which that made your experience in graduate school different? Did you feel more mature than other students in your attitude about graduate work?
More mature in a sense that I knew what I wanted to do. I was glad to be in graduate school, and I wasn’t fussing like all of the other students were. I was past the adolescent angst. I n college I was very concerned about other aspects of my life, of my relations with people and where I was ultimately going to live, would I get married, what was I going to do with my life. When I was a graduate student, my personal life was somewhat settled. I had a good, secure marriage, and I had had a good shot at being a housewife. I couldn’t see that as a lifetime occupation. It didn’t interest me.
Mathematics did, and that was enough. The other students, on the whole, the younger ones, were much more preoccupied with their personal lives than I was. They had the luxury for that; I did not.
They had the luxury for it, but on the other hand you had the luxury of not having that distraction.
I didn’t have that distraction, but I had plenty of other distractions with the children. I had a lot of responsibility. The other women I knew could not understand how I was able to do it. But the moment everybody left in the morning, I sat down at that desk, and nothing interfered with my concentration. I really worked. The way I did it is that I was interested. It wasn’t hard. I was enjoying it.
But it was hard initially to get back into it?
I had always had a little thread of contact. There wasn’t a discontinuity like that. And we didn’t really need the money; our needs were simple, and my husband’s salary sufficed. When I began to work, I was damn glad to have it, and it added something quite extra that I didn’t think about, a certain independence that was very nice. But I liked what I was doing so I had no conflict.
Were there ever times in the beginning where you were discouraged and felt like you weren’t going to go on?
No. I didn’t have any real crises like that. I knew that the other students did, but I didn’t.
I’m wondering if that was because you entered research mathematics relatively late.
I think that was a good way. It filled an enormous need in my life at that time that I got started in research mathematics, and it’s continued to. Especially because my husband has had a very active career, so he’s always busy with his own thing, and I’ve got to have my own thing. Otherwise I would feel like an appendage to him. My work is necessary for me in order to be myself.
Did you have mentors in either college or graduate school that were particularly important to you?
People were certainly helpful. Magnus was, and Fox was. But I had enough confidence on my own. But I don’t think that made an awfully big difference. I tend to discount this role model idea. I think that if a person doesn’t have enough underlying feeling of “I’m going to do what I like” and not be so affected by what other people think, then they’re not going to be mathematicians anyhow. I certainly knew that being a mathematician was not what the average man on the street could comprehend. But I felt right at home with mathematicians.
Relationships, marriage, being a woman
You haven’t talked very much about college — high school and graduate school more significant in your mathematical development.
That’s true. In college my preoccupation was with other things, i.e., my relationships with people. That really took precedence over the mathematics.
Is that when you met your husband?
I met my husband after I finished college, but there were other people that I was dating in college, and that whole issue was very much on my mind. Not only did I know that I had to or wanted to get married, but there was even a feeling [in my family] that my sisters and I should married in the right order.
Did that happen?
Yes, it did.
I could see why in college relationships would be such a big thing. For one thing you were at an all-girls’ high school, so that was the first time you met men.
That’s the first time I began to date.
Where did you go to college?
I started college at Swarthmore but didn’t like living in a college dorm. So I transferred to Barnard and lived at home. I was very happy to be able to do that. My sisters had done the same, so that model was there, I felt better with the privacy of living at home. I really have a limited ability to be with people.
Did you get married soon after you met your husband?
A year and a half later. I graduated from college in 1948. I was married in 1950. It was a stormy period. It’s a very difficult decision to make — is this the right person for me? It just took all my attention. I guess that one of my thoughts about women in mathematics is that it is just a much more absorbing issue for women at a time when men say this is when you do your work or you’ll never do it. And we women buy that hook, line and sinker. I see it in the graduate students. There’s just another pull that the women have; it has to do with home, family, human relationships, etc. And while the men are also involved in all those things, it doesn’t seem to take their attention quite the way that it does for women. I don’t know. You’re from another generation, and I always suspect that this is just my experience.
The bringing up of a child takes an enormous amount of attention, and you don’t want to put it aside. It goes so quickly, and if you’re not there you miss it. It was different for my husband. He was working so hard and had pressing responsibilities to “take care of us,” all four of us. Even when I started to work, the personal issues were still there. However, I did bypass many of the problems by starting graduate school later, when the children were older and the demands were much less.
Yes, this is a primary conflict for women, particularly in academia, because often women are having children around the same time they’re coming up for tenure.
Yes, and the men are working like lunatics on their mathematics and the women, they see this choice, and if you just give it a little less effort, your research is dead. I think that’s a big issue. I guess that if I could see any solution to the non-participation of women in mathematics, it would be, first of all, if women were able to think about going back to mathematics at a later point, and if there was a practical way for them to do this, and if you could reach the right women, and if the whole community was ready to accept this as another option — all of these things would help. But the fact is that the community has bought hook, line, and sinker this whole idea that if you are going to do research, you do it when you are young.
Do you think that there’s any merit to that? Why is it such a powerful myth?
I don’t think there is merit in it. I think doing math when you’re enthusiastic, yes, that’s what is important. Not your age.
Women, mathematics, and children
When you look back, in what ways has being a woman affected your career?
It affected it enormously. I took a fifteen-year break. I got my Ph.D. when I was forty-one, not twenty-five or twenty-six. That’s a great big difference. It’s affected my life in absolutely fundamental ways. It’s made it hard to be a mathematician in some ways. It was lonely out there in 1968. I was not crazy about that. It would be a different world if there were lots of women mathematicians. We would feel very differently. But I don’t know; maybe I like aspects of that.
Do you think it is easiest to do the children first and then go back, or are there ways women who are having children can be accommodated in the system?
I don’t know the answer to that. I think it’s something that each person is going to have to suffer through. It may be much easier now than it was in my day, because I think that a lot of the younger men are much more understanding of this problem and that the upbringing of a child is not a woman’s responsibility as it was when I was brought up. It’s so common now for women to be working.
So did you use nursery school?
Yes, we used nursery school, but in those days nursery schools were not designed to help the mother. It was impossible for me to have any kind of a career without having anybody really in charge at home.
How old was the youngest when you started?
When he was an infant, I started going to school at night. My husband was the babysitter. I found that easy. I found that I had enough time of my own so that without asking anybody to help out, I could handle three children and study for that one course. But then the second year, when I took two courses, that was more ambitious, and that was really hard without help.
By the time you were a full-time student, your kids were in school, so there were regular hours. Did that make things easier?
When I started being full-time at NYU, my youngest child, David, was starting kindergarten. But even kindergarten was challenging because it was only a half-day. And a full day of school was until 3:00 P.M., and no afterschool programs existed. Quite the contrary; it was a suburban neighborhood. There was a lot of carpooling and taking them around to visit one another, and I was very hard-boiled about that; I told my children that they had bicycles and feet!
So were you really unusual in this neighborhood, being a woman working outside the home?
Yes, it was isolating, absolutely! My involvement with the community went way down when I started to be a serious professional. Still, I think that what I did was easy — the advantage I had was that I did not have a Ph.D. before we had children. So I got my training and I used it right away. I think the people who have the biggest problem of all are the ones who get their training and then feel that they’ve forgotten it all and lose their confidence when they are ready to go back to work. The way that I got myself back in was quite natural and gradual. I did it a little bit when I had a little bit of time, and I was slowly able to build it up. But somebody who was working at a peak and then found at a later point that bringing up a child is really time-consuming and takes a lot attention may feel like they are losing touch with the field, or feel like the job is being juggled all the time with child care. That kind of a person is going to have a very different feeling when the children grow up, whereas I was starting something new. I wasn’t going back to something old and feeling like I’d missed many years.
Right. And the enthusiasm you had because you were starting something new really carried you, too.
That’s right. And I also was able to build it up very slowly in a way that was quite easy. If it’s mathematics that you’re doing, a lot of your time is spent with books and paper. You don’t need a lot of equipment. You don’t have to be long hours in a laboratory. So I could be at home when the kids came home from school, and they would just want to come into the house and say hi and run out again. I could spend a lot of time just doing that, and it didn’t really interfere with my ability to study. So in some ways that made it very easy. I don’t know if that’s a model that would work for anybody else. I also wonder how to reach the people for whom this would be of interest. How do you find the women whose children are growing up and who feel they could get back to mathematics?
I think that NYU was really excellent in this respect. The graduate school was open to the community, and it’s an excellent school. The atmosphere is really stimulating and very good, and there was a constant flow of people from all sorts of different circumstances who were doing serious graduate work, and you were also with students who were full-time students. In 1996, Columbia still has nothing like this for graduate students. A graduate school that has a good master’s program at least has some way for people to do a little bit. You see, I never planned what I was going to do. I didn’t really set myself a goal, like wanting to be a research professor in mathematics. I had very small goals. I wanted to be able to learn the material of this linear algebra this semester, and then it built up after a while. That was very handy.
That seems very common among women, that they have small goals first and then they keep going.
You have to feel your way.
And it’s not a given to many women that they’re going to have a profession. It’s a given for most men. Women often do it because they love it, but they don’t necessarily think (at least at first) of math as their profession.
That’s starting to change a little bit, but it is still predominantly the case. What you say about NYU and their willingness to be a little bit more flexible is very important.
Encouraging women in mathematics
Over the years, you must have thought about how to get more women involved in mathematics. Do you have ideas about how to encourage them to do math, and then to stay with it?
They have to feel passionate about doing mathematics. Doing creative work takes a passion. You have to be driven. So what can people help? I really think that in some ways I’ve been helpful to others, but it’s never been through committees and policies. You can do something in other ways too. Last year, one day there was a knock on my door, and totally unexpected a young woman came in. She was a graduate student at another university and was depressed and unhappy about graduate school. All of a sudden all of the starch had seemed to go out of her. She used to love mathematics, and she was in the middle of writing her thesis. Why did she come to see me? I don’t know. Somebody said to her, “Go talk to Joan Birman; maybe she can help you.” I felt that I could remember myself at that time; it wouldn’t have been a great time to be writing a Ph.D. thesis. I don’t know what else was on her mind, but she wasn’t getting the pleasure out of math that she had in the past. She was thinking of dropping out of graduate school and ultimately did not, and I think maybe I helped her in a way that a man would not have been able to at that moment. I think the fact that we were two women did make that a little easier to her.
Do you remember the kinds of things you said to her to help her out at that stage?
Maybe just that she was able to cry in my office and wouldn’t have been able to cry in front of a man.
So it’s really through individual cases that you feel you can encourage women in mathematics?
Individual cases, being alert, being there. I think it’s a small thing.
The nature of mathematics
What do you think mathematics is all about? Are we discovering things, creating them? How do you think about what you are doing when you are doing mathematics?
My husband and I have had a big discussion about this, whether mathematics is real, and I guess I really do believe it is, very very strongly — that there is structure out there and we’re finding it. It seems to be endless. It’s amazing. It’s just amazing that there is always a deeper level that you can ask questions, and there is always a deeper structure. Maybe mathematics is almost the easiest of the sciences. When I was in Monterey, California, I visited a marvelous aquarium, filled with fish and creatures of the sea who have the most extraordinary patterns and colors on their bodies. When you look at this, you think that nature is so complicated, and that mathematics is picking out the very simplest of the patterns and analyzing it. And that we have the easiest science.
Why are other sciences harder?
They’re harder because the phenomena are so much more complicated that they don’t admit the kind of sharp analysis that mathematical problems do. We reject anything that is too hard to understand. I just had a conversation with m gynecologist the other day, and it concerned hormone therapy. She described the different programs that have been prescribed for balancing how much estrogen and progesterone should be taken, and at what point in the cycle. She said you wouldn’t believe the number of different studies of which way to do it. Nobody understands it. They don’t understand fundamentally what is going on. Sometimes it’s dangerous. People get cancer from it. They do not understand what they’re doing. What do we do with a mathematics problem? We take something that’s very simple and we analyze it. We pick problems that lend themselves to analysis. Doctors don’t have that kind of a choice. Their subject matter is handed to them, and it’s by nature very complicated.
Right. And in mathematics you can abstract out any little piece of it that you want. You don’t have to look at the whole thing.
That’s right. I think that in other sciences they try to do that, too. My sister Ruth, the plant physiologist, studied motions of plants. She had a plant that responded to a twenty-four-hour cycle. It folds up at night, and then the leaves fold in on the stem and the stem folds on itself. What she studied was not this phenomenon, because everybody knows it, but how the motion comes about. She looked at a particular membrane — potassium is transported across the membrane. Well, what is the mechanism that gets the potassium moving from here to there, and why does it happen? What’s the chemical that gets it going? So it’s almost like a mathematics problem. She tried to isolate, but no matter how much anyone tries to isolate, they may not be isolating the right thing. They don’t know what the question is. But we, as mathematicians, can define our problems. We make our problems precise.
Jumping on the bandwagon
That leads to the whole question of how we decide what are important questions in mathematics.
There are a lot of fashions in mathematics. Fashions come and go, and it’s always a question of what do you do. Do you follow the fashion or not? Sometimes I tend not to because I dislike working under intensely competitive conditions.
Yes. It seems like you’ve been in it and out of it both.
Well, new things have happened, and when something big happens, then immediately there’s a big rush toward it. I’m a little bit intimidated by that. I was in the middle of two big developments like that, and they both posed big problems in my mathematical life because the question was whether to try to get in there with the crowd; they were crises for me. Twice this happened where there was this enormous new explosion from somebody’s work. Once was whenmade the discovery that earned him the Field’s Medal, and the second time was when did the same. The question for me was, Do I drop everything and rush in this direction, and if I do, will I contribute anything? Are other people better than me? Will they get there first? How do I handle it, and what should I do about my own research? Somehow after a period of anxiety, I kind of worked it out and came through the crisis.
When Thurston came along, there was a part of geometric topology which most people dropped. He brought a really new point of view, and the new point of view was geometry versus topology. Along with the new point of view came many new ideas that you had to learn, and the question was, Do I drop all the things that I know about very well and go follow the fashion or do I keep doing my stuff and pretend it didn’t happen? Or am I just going to be left out? Eventually I kind of got over that and learned a little bit of the new and found a way to contribute.
When Jones’s work came along, perhaps I made a bad decision because I was one of the first people who knew about it. The main discovery was made in my office, and I even played a role in it. At that time I was working with. We were finishing up a paper, and the question was, Do I just drop Carolyn and drop this paper and run in a direction where I know I’m a little ahead of the crowd and have some advance warning?
What did you do?
I stuck with Carolyn and finished our paper, and I missed out on a whole lot of mathematics where I really had an inside track. I don’t know if that was the right decision.
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Birman’s descriptions give us a sense of what life on the cutting edge can be like — the excitement and the intense competition, the tough decisions, and the unforgiving pace of discovery. In the end, she did ultimately return to work on the Jones polynomial and was able to “find her own niche” in the wave of activity that ensued. Indeed, Birman was able to find her own niche by paving a new path for integrating one’s roles as a mathematician and as a mother — both of which were central to her identity. Though clearly this integration posed challenges at times, it illustrates that the two roles are certainly compatible once we are able to let go of traditional assumptions about when and how a mathematician’s life unfolds.