Twelve
Dusa McDuff
Dusa McDuff is a highly accomplished mathematician who works in symplectic geometry, a relatively recent and somewhat esoteric branch of mathematics. She says, “Symplectic geometry is an even-dimensional geometry. It lives on even-dimensional spaces and measures the sizes of two-dimensional objects rather than the one-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry.”
McDuff claims that if it had not been for Miss Cobban, her high school math teacher who taught her geometry and calculus, she might not have become a mathematician. She did know that she had to do something to impress her father, a geneticist and developmental biologist, and fulfill the ambition of her mother. She also felt that she needed to live up to her grandmother, Amber, who as “Dusa” had a scandalous affair with the famous author H. G. Wells.
Her mathematical success is shown both by her election in 1994 as a Fellow of the Royal Society of London, the second female mathematician to be so honored after Dame Mary Cartwright’s election almost fifty years earlier, and also by her later elections to memberships in the National Academy of Sciences (U.S.) and the American Academy of Arts and Sciences. She now holds an endowed professorship at Barnard College, a college of Columbia University.
Her PhD thesis in functional analysis from Cambridge University was published in the Annals of Mathematics, one of the most prestigious mathematics journals in the world, a rare event for any doctoral thesis. A short time later she went to Moscow where she came under the influence of the famous mathematician I. M. Gelfand. Over the next six months she learned much from him and returned to Cambridge determined to work in a new field. In the interview that follows, McDuff describes the difficulties in shifting fields, including the special problems that a woman encounters in succeeding as a mathematician.
Figure 12.1 Dusa about 1948.
MP: Thanks very much, Dusa, for meeting in your MSRI [Mathematical Sciences Research Institute] office today to chat about yourself and how you became a mathematician. Let’s start at the beginning. I understand you were born in October of 1945 in London, and moved to Scotland when you were a child.
McDuff: Yes. My father, Conrad Hal (Wad) Waddington, got a professorship at the University of Edinburgh in 1947 when I was two. A geneticist and developmental biologist, he worked during World War II in Patrick Blackett’s lab devising strategies to defend against U-boat attacks. He told me that this group started the field of operations research but couldn’t publish until the 1970s because of secrecy laws. When the war ended he traveled extensively on war-related work. Demobilized in 1947, he took up his position at Edinburgh as Director of the Institute of Animal Genetics.
Mortonhall—An Amazing House
After the war, there was a serious housing shortage in Britain. My father’s solution was for everyone connected with his lab to live together. We moved into Mortonhall, a large stone mansion on the outskirts of Edinburgh, whose grounds had statues, clipped yew hedges, a large walled kitchen garden, and stables. The stables had no horses, but I remember collecting the chicken eggs. Although people lived in separate flats and rooms, we ate communally, with one dining room for the grownups and one for the children. Rations were shared. Much of the work of the house was done by a household staff that (at least in the later years when I got to know them) included many displaced people from countries like Estonia and Latvia. My father set up the household at Mortonhall almost as a social experiment; my mother, trained as a town planner, was always very interested in the influence of architecture on communities.
Although these arrangements suited our family very well since my mother worked full time as an architect in the Scottish Civil Service, they made some of the other wives rather unhappy. There were many other tensions in Mortonhall. It was different for the kids, but for the grownups, it was not necessarily so easy to live that way, partly because there was no escape from the lab but partly also because life in Britain at that time was very austere. There was little food, and houses were cold because of lack of fuel. Children got rationed orange juice and milk and other things like cod liver oil, but grownups didn’t.
Squabbles over Porridge
Edith Simon, one of the wives, wrote a novel about life in Mortonhall called The House of Strangers. One of the few things in it that rang absolutely true was stories of the grownups at the breakfast table squabbling over the porridge. There was a porridge rotation and scientists are not necessarily good cooks. They took the precious oatmeal and made porridge that was lumpy, or weak, or burnt, or too salty. Then they had to share their meager butter and sugar rations and would look to see exactly how much their neighbor took. It was difficult.
Figure 12.2 Dusa and Caroline playing in Mortonhall, about 1948.
Figure 12.3 Dusa (on left) and her sister Caroline, about 1949.
MP: It sounds that way.
McDuff: But it was wonderful for the kids.
MP: How many kids were in the house?
McDuff: Well, counting myself and my sister Carrie—I guess six or eight. My mother worked from nine to six every day and a half-day on Saturday, and couldn’t come home at lunchtime because her job was so far away. My sister and I went to nursery school and had Irish nannies. But for a lot of the time we were running around in gangs outside with very little supervision, able to do what we wanted—climbing trees, playing hide and seek. I also remember talking to the cook in her enormous old fashioned kitchen, the boiler man as he stoked the basement furnace, and the gardener working in the kitchen garden.
MP: It certainly sounds like a memorable childhood experience.
McDuff: Yes.
McDuff: Five years.
MP: That’s a big chunk of one’s life at that stage. You moved there when you were two.
McDuff: Yes. At that time it was unusual for young children to go to nursery school.
Early Interest in Math and Grandfather’s Influence
My mother, who cared deeply about our education, chose a very progressive school run by the Parent’s National Educational Union (PNEU). I used to love school, especially doing math. I was allowed to do whatever math I wanted. By the time I was seven, I was way ahead of the others.
MP: In one of your writings, you mention the influence of your grandfather on your mathematical development. You said that he had done math before turning to the law.
McDuff: My grandfather G. R. (Rivers) Blanco White was the Second Wrangler at Cambridge (i.e., he placed second in his class) in 1904 and eventually became a divorce court judge. A private in the First World War, he served in the artillery, using his mathematical knowledge to calculate trajectories.
MP: So, he influenced you when you were quite young.
McDuff: I met him once when I was four and then again when I was about eleven. We were not a close family. So, this visit when I was about four was very special. I remember him showing me the multiplication tables.
MP: Were your mathematical interests apparent before he told you about multiplication tables?
McDuff: Probably. Since I liked math, I certainly knew how to add and multiply. But I had never seen a full ten by ten multiplication table before he showed it to me, explaining its various symmetries and patterns, how if you look down the nines column, the numbers change regularly, one digit going up each time and the other going down. He showed me its beauty.
MP: You said that a nice aspect of the school was that you were able to do as much math as you wanted. Do you recall what it was about math that was so attractive to you when you were little?
McDuff: I just loved mathematics, I don’t really know why. My mother was good at math (Rivers was her father), and she was always eager to encourage us intellectually. She told me once that she was thrilled when my “first word” was two words with two ideas. I was a bit precocious and quick at doing some things. I liked the way my sums came out correct. I remember doing an entrance exam when I was six to get into a proper school. They asked me to add two and three, or something like that. I said, “This is far too easy.” I did a sum with four digits, and all the teachers gathered round astonished.
Figure 12.4 Dusa with “Mousie” (age ten).
MP: So, your interest in mathematics was early and strong, and it’s never abated.
McDuff: I’ve had other interests. I always wanted to be a mathematician (apart from a time when I was eleven and wanted to be a farmer’s wife) and assumed that I would have a career. Luckily I had a very good math teacher in my high school. I went to the same girls’ school from seven to sixteen, the best my parents could find in Edinburgh. I despised the science teachers because they could not answer my questions, but I respected the math teacher Miss Cobban; she taught me Euclid and calculus. Otherwise, I don’t know whether I would have become a mathematician.
MP: What careers did your siblings pursue?
McDuff: My sister Caroline Humphrey is Professor of Social Anthropology at Cambridge University and a fellow of King’s College. My half-brother, Jake Waddington, whom I hardly knew while I was growing up, is an astrophysicist.
Figure 12.5 Dusa aged about sixteen.
Father Thought Mathematics Was Boring
MP: You said that your family very much valued creativity, and yet you always felt that in their view the really creative people were males.
McDuff: My parents would never have said that explicitly. Unusually for the time, I was brought up to think I would have a career and that women could do just what men do. But my mother had also subordinated her career interests to those of my father, justifying that by the fact of his brilliance and the needs of her family. When I became a teenager, about fifteen or so, I felt that the kind of intelligence I had did not count for much, and what was really creative was a more artistic kind of talent. I might be very good at reasoning, but that was ultimately not important.
I had some artistic interests; I played the cello, and I loved reading. My boyfriend at the time, David McDuff, who became my first husband, was a poet and linguist. (He knows an astonishing variety of languages, including Russian, Finnish, Icelandic, and the computer language C++.) We met through music. I thought that he had a brilliant, creative mind, while I didn’t see myself as creative. For example, although I was quite good at painting as a girl and now get great pleasure from going to art galleries, my sister was much better at painting than I was, with a wonderful sense of design and color. Although some people suggested that I study to be a cellist, I decided not to because I felt I had more talent as a mathematician.
MP: So, although you had these strong mathematical interests and were doing very well at it, there was no apparent feeling by your mother and father that mathematics was a particularly creative area?
McDuff: My father didn’t like mathematics; he thought it was very dry. When I was thirteen, we had many conversations about the book he was writing called The Ethical Animal, about the development of the moral sense in humans through evolutionary processes. He gave me philosophy to read, along with The Voyage of the Beagle and Freud, greatly broadening my outlook. He prided himself on being a scientist, a philosopher, and an artist. He wrote a book about modern painting, Behind Appearance, that even today some people find worthwhile. He knew Alfred North Whitehead and Bertrand Russell from Cambridge in the 1930s, but knew Russell as a philosopher, not as a mathematician. His attitude about mathematics was that it was boring—though in his later years he was very interested in developing a theoretical approach to biology and was open to the importance of mathematics in that connection.
Figure 12.6 McDuff’s father, Wad (Conrad Hal Waddington) (about 1965).
My mother was an architect. For her, the artistic side of it came through design. She was passionate about research and pure thought (mathematics did qualify there!), but she didn’t know enough about mathematics to emphasize its creativity. I first got to know people who I thought were truly creative mathematicians when I went to Moscow in my third year as a graduate student. I discovered there that mathematics could grow and develop. Before I had seen its compelling beauty, but it was somehow static; I was unaware of how it was created.
“I Had to Do Something to Impress My Father”
MP: When you were at Cambridge, you wrote a thesis that ended up being published in the Annals of Mathematics. That isn’t chopped liver, as some would say.
McDuff: Right.
MP: It got a fair amount of attention in the mathematical community.
McDuff: I wasn’t so aware of that because I changed fields so quickly.
MP: You must have felt very good about it at that time.
McDuff: Yes, but I was totally divided. I was deeply in love with David, a poet, and he was math-phobic. I had a separate life as a mathematician, with no mathematical friends. When I was an undergraduate at Edinburgh, I didn’t talk to any of the other students. I didn’t know any of them, except I remember playing bridge with them one afternoon.
My only friends, and I had very few, were through David, and they talked about poetry, art, and politics. I was learning Russian and German to keep up. When I was little, everybody thought I was brilliant; I got a lot of attention for always coming out top on exams and that kind of thing. At some point, I decided that was irrelevant, and I turned my back on it, trying to live a different life. For one thing, I had to do something that would impress my father, and mathematics was not it.
“I Had to Live Up to My Grandmother Dusa”
McDuff: For another thing, I had to live up to my grandmother Dusa. I don’t know if you are aware of that aspect of the story.
MP: Well, you did mention in one of the articles that you have written that she was apparently very colorful and that she had a long, somewhat sensational affair with H. G. Wells. She bore a daughter, Anna-Jane, by him. Wells was a well-known writer then, and certainly his fame persists. Apparently Dusa was the name that Wells gave her.
McDuff: The story that my mother told me was that he gave her that nickname because of her long black snaky hair. Then, later, I read about her in H. G. Wells’ own book, H.G. Wells in Love: Postscript to an Experiment in Autobiography, a book about all the women with whom he had had serious affairs. One of the chapters is about my grandmother, Amber Pember Reeves. (She was the model for his very appealing “new woman” heroine Ann Veronica.) Her father, Pember Reeves, was the Governor General of New Zealand and then the first director of the London School of Economics. Wells said that Dusa was her private name for herself, chosen because she was fascinated by the image of the Medusa head held by Perseus in Bernini’s bronze statue. I much prefer that version of the story.
Wherever the name came from, it is somewhat puzzling. As far as I knew, the name Dusa meant a terrifying monster that rendered others powerless. Recently, I discovered that in Turkey she is a guardian figure, her head often portrayed on Athena’s shield; in prehistory she must have been an earth goddess because Medusa is the feminine form of the name Medon which means ruler. My mother never told me about those aspects of the name.
MP: It’s certainly a distinctive name.
McDuff: I felt I had to live up to it. Being called after Medusa made me feel unique. The other schoolgirls made fun of me, pretending that, like the mythic character, I would turn people to stone if they looked at me. And then, as I realized much later, I thought I had to do the equivalent of running off with H. G. Wells: I would not be able to hold my head up if I didn’t.
MP: I have read a bit about your grandmother in Shadow Lovers by Andrea Lynn. She portrays her as a progressive feminist and important author of books on social issues, whose admiration for Wells continued for more than thirty years after their affair commenced in 1907. She quotes from a letter that she wrote to him in 1939: “What you gave to me all those years ago—a love that seemed perfect to me, the influence of your mind, and Anna-Jane—have stood by me ever since. I have never for a moment felt that they were not worth the price.” I’m beginning to understand why you hold her in such high regard.
Figure 12.7 Amber Pember Reeves.
McDuff: At that time, the early 1900s, women were only beginning to have careers, and it was very hard to have both a family and a career. You had to choose. She had a job in the First World War in the Ministry of Labour promoting women’s employment but had to leave it at the end of the war. She then wrote some novels and a book on economics for the Left Book Club as well as helping her husband “nurse” a constituency for the Labour party (i.e., stand for Parliament in a hopeless seat). After more war work during the Second World War, she became interim president of Morley College (part of London University for adult education) for a short while until an unmarried woman could take it over. She served there as Tutorial Lecturer in Moral Science for many years, giving evening courses in psychology and writing a book Ethics for Unbelievers that used Freudian principles to argue for a kind of Confucian restraint. Despite all this accomplishment (much of which I learnt about later), my family’s attitude was that she had not lived up to her potential.
MP: Did you ever meet her?
McDuff: Oh yes. I met her several times, not very often when I was small, but when I was a teenager, I spent a few days in her house. We talked quite a bit then, and we also corresponded for a long while.
MP: What do you remember of your personal interactions with her?
McDuff: My grandmother liked telling slightly risqué stories about the people she knew. She was very good at that! She spoke in incredibly complicated well-formed sentences of the kind nobody uses now. She knew Lloyd George, Beatrice and Sydney Webb, and many other politicians and socialists. She once told me a story about Beatrice Webb getting a bee in her blouse at some garden party, which made her lose her poise. Beatrice was known to be so very correct.
We also talked about what I was doing. She liked me. I remember one day as I left the room I heard her mutter “she’s my favorite granddaughter.” She wouldn’t tell me that to my face. I really enjoyed my relationship with her. I wrote to her, and she regularly wrote back. As she got older, my mother wrote a letter to her every week.
MP: Wow. A letter a week!
McDuff: People did write letters in those days. The telephone existed, obviously, but we used it for practical things. We never telephoned each other to chat; we wrote letters.
MP: I’m convinced that your grandmother was a rare person. You said that you very much wanted to live up to her. Do you feel you have?
McDuff: Yes, I think so. I certainly have done the best that I could.
Mother’s Career Frustrations
McDuff: I knew that my mother always felt frustrated in her career. She had great ambition. In her mind, I had to ignore the fact that I was a woman and just succeed in my career. She was taken aback when I had my first boyfriend, David, because she thought that would be a hindrance.
MP: When your mother said that you had to ignore the fact that you were a woman, was she suggesting that it was better to be a man?
McDuff: On my mother’s side, I come from a line of strong women, starting with my great-grandmother Maud Pember Reeves, who wrote a pioneering book Round About a Pound a Week on the London poor. Nevertheless, there was definitely a feeling in my family that men were more important. When I had my daughter, Anna, the very first thing my mother said when I told her was, “Oh, what a pity it’s not a boy!” She thought that boys were better than girls. My grandmother didn’t have a proper career, and my mother didn’t have a full career, though they were both incredibly talented people. They didn’t do as much as they could have if they had been men. So, there was definitely a feeling that to be a woman was to be inferior. My desire not to be second, together with the fact that it is easier to earn a living as a mathematician than as a poet, was why I made the living during my marriage with David; but it did make things difficult, because then I had to do everything. In some ways very self-confident, I also had many feelings of inferiority that took a long time to overcome. There are many contradictions.
Figure 12.8 Justin (Blanco White) Waddington, McDuff’s mother, on veranda of the house she built in Italy about 1980.
MP: Earlier you said your parents greatly embraced and approved of creativity, and certainly poets are thought to be quite creative. And you referred to David as being a truly brilliant and creative person.
McDuff: Indeed, indeed, but even so, people are full of contradictions.
MP: Okay, fair enough. So, she wasn’t that pleased with him. Was your father happy with him?
McDuff: My father didn’t pay much attention. He was off in his own world. Anyway, part of the point of trying to live up to my grandmother was to do things that my parents wouldn’t approve of. You have to understand that my grandmother didn’t behave as the people around her thought she should.
I grew up in the 1960s. In the summer of 1961 I was home alone during the week of the Berlin Wall crisis, and I really believed that a nuclear war would annihilate everyone in a few days. So I joined the Campaign for Nuclear Disarmament. From then on, I too was not willing to be or do everything that people expected of me.
Figure 12.9 Dusa aged about seventeen.
MP: You have spoken about the importance of getting your father’s approval. Did he live long enough to see you achieve professionally at very high levels?
McDuff: Yes, he did. He died in 1975 when I was a lecturer at York. I heard indirectly that he was very proud of me, at his funeral from a close friend of his. As far as I know, his basic attitude about mathematics was still that it is too dry, but he was much more open to its value as a language for science. Towards the end of his life, very interested in sustainability and environmental issues, he wrote a book and developed an undergraduate course on Tools for Thought, how to understand scientific techniques of problem solving. He also organized several meetings on the theme “Towards a Theoretical Biology” together with mathematicians such as Christopher Zeeman and René Thom. But by that time I was under the influence of Gelfand, who thought that Thom’s approach was shallow, so my father and I did not even connect on that.
MP: Well, it would have taken a bit of time before he would have reached a point where he could understand what you had actually done.
McDuff: I remember one time trying to explain to my mother what I had done. This was when I was a graduate student and I’d had my first idea, proving my first theorem. Despite her willingness to listen and understand, she lost the thread of the ideas as I was going through all the needed definitions—I remember we got lost when we got to “group.” It was impossible for her to understand at the level she wanted to without years of training.
MP: Let’s get back to your undergraduate years at Edinburgh. You said that the students who were doing mathematics there didn’t seem to be interested in what you really regarded as mathematics. Were they especially utilitarian in their outlook?
McDuff: You have to understand that I was, you would say, a dreadful snob. I just didn’t talk to them, so I don’t know what they were interested in. On the other hand, we did have a few recitations together, and if there had been somebody else who was really involved in the classes, then it’s conceivable I would have talked to them.
MP: They didn’t strike you as interesting.
McDuff: They didn’t strike me as interesting, and that says a lot about me as well as a lot about them.
MP: You are referring to students at Edinburgh. What about Cambridge students?
McDuff: I wasn’t at Cambridge for my undergraduate work. I won a scholarship to Cambridge, but I didn’t go there. I went to the University of Edinburgh because that’s where David was.
MP: But then you went on to Cambridge for graduate work.
McDuff: Yes. As a graduate student, I did talk to some of the others, but graduate school in Britain is very different from what it is in America. Much more specialized, students have only three years, though usually with no teaching duties. Because I had distinguished myself at Edinburgh, I was excused from part three of the Cambridge Tripos.
The Tripos is roughly the equivalent of a master’s degree; one takes six lecture courses in different subjects, with final exams at the end of the year. If I had done that, then presumably I would have gotten to know both more mathematics and more students. Instead I immediately started working with my advisor G. A. Reid on my dissertation topic, interacting only with the small group of students in functional analysis, none of whom were working on exactly the same subject as I was. After my first semester in Cambridge, I married David, who then came to join me. My life outside mathematics was with him, which cut me off from the other students.
Also, I still was not talking to people. For example, John Conway was there. He was this idiosyncratic person who wore no shoes, had six kids. . . . I never talked to him. One end of the Common Room was full of his games, which now I think are fun, but in those days I didn’t want to be involved. I was terribly serious, seeing mathematics as a very high, abstract, artistic endeavor; games were not part of it.
An Encounter with I. M. Gelfand
MP: After two years at Cambridge, you went off to Moscow for six months.
McDuff: Right.
MP: You ended up working under Gelfand because his was the only name that was familiar to you before you went there. Is that right?
McDuff: More or less. I hadn’t thought that I had to study with anybody. My advisor had not asked: “Well, do you know what you want to do when you get there?” I just hadn’t thought about it; I was very naive in many ways. But luckily, when they asked me in the student office at Moscow University who I wanted as my advisor, I said Gelfand, and that turned out to be great.
MP: Good choice.
McDuff: His was the first name that came to mind, probably because I had written my undergraduate project on his book on distributions.
MP: Tell us about getting to know him and the impact that he had on you.
McDuff: When I first met him, he asked what I was doing, why I was there, and then said that he was much more interested in the fact that my husband David was writing a thesis about the Russian symbolist poet Innokenty Annensky than that I had solved this problem about von Neumann algebras. Nevertheless, he then tried to figure out what he could teach me.
Gelfand amazed me by talking of mathematics as if it were poetry. He tried to explain to me what von Neumann had been trying to do and what the ideas were behind his work. That was a revelation for me—that one could talk about mathematics that way. It is not just some abstract and beautiful construction but is driven by the attempt to understand certain basic phenomena that one tries to capture in some idea or theory. If you can’t quite express it one way, you try another. If that doesn’t quite work, you try to get further by some completely different approach. There is a whole undercurrent of ideas and questions.
MP: What is the single biggest thing that he gave you? He was clearly a very inspirational person for you.
McDuff: He was the first person I had met who saw mathematics in the way that I imagined it. At that point, married to a poet, I was very idealistic. I saw mathematics with its abstract beauty as one way of expressing human thought and creativity. Gelfand also saw mathematics as part of everything else. Whether he was reading books to me, or we were listening to music, for him, that was doing mathematics. At that time in Russia, there were so few outlets for people’s creativity that many people became mathematicians who in the West would probably have done other things. We think that mathematical talent is something very special, possessed by only a few. But many different kinds of minds can contribute to mathematics, and they did so in Russia. That’s one reason it was such a vibrant and broad mathematical culture. In addition he used so much mathematics, while I knew nothing. He gave me his recent paper to read, “Cohomology of the Lie Algebra of Vector Fields on a Manifold.” I had been so narrowly educated that I didn’t know what cohomology was, what a Lie algebra was, what a vector field was, or what a manifold was.
Figure 12.10 At Gelfand’s seminar, Moscow University fall 1969, McDuff talking, Gelfand explaining.
MP: So, he inspired you to learn a lot of new mathematics.
McDuff: Yes, he opened my eyes to many things, but I was only in Russia for six months.
MP: A very important six months. It sounds as if he spent a lot of time with you.
McDuff: Being a young and very inexperienced mathematician from the West, I was a complete novelty. There were very few visitors from the West in those days, and Gelfand was eager to practice his English. He tried to help me find a way forward, suggesting many things for me to read. He once said toward the end of my time in Moscow, when I’d reexpressed one of his results in the language of sheaves that I was just learning: “Oh, you’re quick.” He realized that I could do some mathematics and gave me a letter of recommendation to help me in the future. He gave me a vision and he spent time with me. He obviously thought I was worthwhile, which was very encouraging.
Gelfand cared deeply about education. In Russia talented young Jewish people couldn’t get the education they wanted and were largely shut out of the university system. He found ways to bring mathematics to them, for example through evening schools that he set up.
MP: He also wrote some very elementary and innovative books on algebra and trigonometry.
McDuff: He grew up in the provinces, excluded because he was Jewish and poor. He devoted considerable energy to creating opportunities for talented young people, inviting them to his seminar and trying to make the ideas accessible.
Changing Fields—Not Easy
MP: After those six months in Russia, had your mathematical interests shifted?
McDuff: When I came back to Cambridge, I was working in a completely different field.
MP: That falls into what some people in sports would call the guts ball category—shifting fields. At that point you had done work, very good work, in functional analysis.
McDuff: But I didn’t know where to go with it. If I had gone to France and talked to someone like Dixmier, I might have found out what my work in functional analysis was related to and where it might lead; but my thesis advisor didn’t really know. I had no clue, and Gelfand was interested in other things. So, what was I to do but try another area of mathematics? It wasn’t easy; I was starting again and didn’t really have a framework. Gelfand suggested that I work with Frank Adams, but Adams was just in the process of moving to Cambridge and didn’t know anything about me. He suggested to me that I study algebraic K-theory, which I did, but it didn’t help me get my footing in this new field.
After submitting my PhD upon my return from Russia, I spent two more years in Cambridge as a Science Research Council post-doc, basically learning on my own except that I talked to some topologists. I took a course on four-manifold topology from Casson, went to wonderful lecture courses by Adams on Quillen’s work on homotopy groups of spheres, and read a variety of books: Milnor on Morse theory; Lang and Serre on number theory. At that stage, I was still learning passively, not working on any problems.
Since I had no duties, David and I were able to spend several months each spring in my parents’ place in Tuscany. This was a peasant cottage, built of stone with no running water or electricity, in the middle of an olive grove owned by Aleksander Zyw, a painter friend of the family. We spent idyllic months there. After each day’s work, we would walk up the hill to drink wine with Aleksander and his Scottish wife Leslie and talk about painting, olive trees, and life in general.
Back in Cambridge for the second year of my postdoc, I gave birth to Anna. At the end of that year, the department paid for me to go to a K-theory conference at the Battelle Institute in Seattle, where I met Graeme Segal. We got to know each other and started working together. I did the equivalent of a second PhD with him, finally getting back into research by working on problems he suggested. For a long time, feeling totally inadequate because of my ignorance, I had just been trying to learn things with no sense of where the questions are, what I might contribute. If I’d continued on my own, it’s not clear to me that I would have found a way back.
MP: So after your postdoc at Cambridge, you took a position at the University of York. How did that happen?
McDuff: There were very few jobs in the U.K. that year (1972), just four I believe, and I was lucky enough to get one. Some years later, my Cambridge advisor said, “What a pity you didn’t apply for that lectureship at Cambridge. . . .” Why didn’t he suggest it at the time?
Figure 12.11 McDuff and daughter Anna near Kenilworth, Warks, 1978.
I left Cambridge because there were no lectureships that year, and the only possible fellowship was at Girton, where the salary wasn’t enough to support a husband and child. I wasn’t eligible for one of the better-paying fellowships, say at Trinity, because they were reserved for men.
Although there wasn’t overt prejudice at Cambridge when I was a graduate student, the Cambridge system wasn’t set up for women to have a career. I was somewhat anomalous, supporting a child and husband, and I couldn’t survive on a very small fellowship from Girton. So I left.
MP: So you went off to York.
McDuff: York, yes. At that time, I had a nine-month-old child, my first teaching job, and a husband who refused to do anything around the house.
MP: You were busy coping with lots of responsibilities.
McDuff: And I was trying to do mathematics, so I was very busy.
MP: Well, I think a nine-month-old child would be more than enough to occupy your time.
McDuff: David did look after Anna, I have to say, but he wouldn’t change her nappies; I had to drive home five miles every lunchtime for that. He said they were too geometric (they did have to be folded). We also couldn’t afford good disposable nappies. I washed them in a machine, and hung them out to dry in the garden.
MP: My wife and I are greatly enjoying our first grandchild, and I have gotten real insights into how much effort goes into caring for a baby.
McDuff: It really is a lot of work. It’s also hard to do mathematics when your time is chopped up into little pieces.
MIT and Graeme Segal—A Turning Point
MP: So, after your Russian experience, you came to feel you could do research again.
McDuff: Slowly, very slowly. I very much enjoyed being at York, teaching for the first time. Feeling less intimidated once out of Cambridge, I organized seminars with other young faculty and learnt with them. We were allowed to carry through some ideas about changing the structure of the undergraduate program, introducing a choice of courses and student projects into the last year. I continued working closely with Segal, mostly via letters, completing a paper on configuration spaces of positive and negative particles.
In 1974–75, I was invited to MIT as a visiting assistant professor. While there I realized how far away I was from being the mathematician I wanted to be, but I also realized that I could do something about it. I became more aware of the relevance of feminist ideas. Before I’d thought that I was beyond all that since I already earned the living, but I’ve slowly learnt that these matters go much deeper.
In those days, I was still a follower, not interacting with anyone on a basis of equality. I had most inappropriate role models, either sirens such as Lou Andreas Salomé or sufferers such as the Russian poet’s wife Nadezhda Mandelstam—neither of much help in becoming a creative mathematician. It was also harder than it is today for a young woman to interact with other (male) mathematicians in a purely professional way as a student or colleague; there were too few of us.
Figure 12.12 McDuff in Toronto, 1974.
That year, for the first time, I met other female students of mathematics to whom I could relate. I also had a mathematical idea again, the first real idea since my thesis, which grew into a joint paper with Segal on the Group Completion Theorem. (Again we collaborated by mail, since he was in Moscow and Oxford.) The year at MIT was crucial in building up the research side of my career. I woke up and realized that I could affect my life.
MP: It seems that the influence of Gelfand, Segal, and your MIT experience were key factors in your development as a research mathematician.
McDuff: Because I had a child, was very busy, and had changed fields, it would have been very easy for me not to succeed as a research mathematician. I read a very interesting book* 20 years ago about women in academia that showed how small but key things determined whether women remained in the academy or were, as they said, “deflected” by advice they received, inconveniently timed moves they had to make to accommodate their husband’s career, or an illness. If you’re an outsider, it is almost impossible to function. For marginalized people, as women were at that time, it’s not only a question of merit but also one of luck. I had talent and perseverance, but I was also lucky.
Broadening Horizons
MP: Never underestimate the value of luck, but it takes more than that to succeed in research.
McDuff: For me it’s been a long steady haul.
MP: So, in what was to be year three at York, you went to MIT.
McDuff: Right.
MP: Did you simply learn of this appointment that was reserved for women and decide to apply?
McDuff: At that stage I would never have applied; I was invited to come. MIT was looking for women, and I imagine that I. M. Singer had heard about me from Gelfand.
While there, I realized that instead of being envious of other people’s opportunities, I could arrange them for myself. I applied to the Institute for Advanced Study and got in. There was a job coming up in Warwick. I didn’t have to be at York; I could apply to Warwick. It was that kind of thing.
MP: It sounds like something of an awakening. So, you went to Warwick for two years.
McDuff: Yes. I was very happy there.
Figure 12.13 McDuff, about 1985.
MP: Warwick was a very young university at that point.
McDuff: York was also a “new university,” as they were called. But Warwick had more international connections and possibilities because of the Mathematical Institute set up by Zeeman.
MP: In the early 1970s, were all Cambridge colleges still single sex?
McDuff: I felt excluded from Cambridge for that reason. There were one or two other women floating around the department, but they were pretty marginal. The few female graduate students typically got married and left the field after their PhD. There was a senior woman in statistics, but I did not know her. Girton had women mathematicians, Dame Mary Cartwright for example. But although I was formally a member of that college, it had no provision for married students, and I never went there. My one interaction with Mary Cartwright was when I was a schoolgirl applying to Cambridge; I had tea with the Mistress of Girton College, who just happened to be the distinguished mathematician Mary Cartwright. I don’t remember knowing that she was a mathematician. She handed me a cup of tea in a delicate cup. It was a formal occasion, not a mathematical one.
MP: You said that, during this period, your husband David was something of a house dad.
McDuff: After finishing his PhD, he didn’t want a regular job, instead translating poetry from many languages. He was not at all domestic.
MP: So in 1975, after being at MIT, you went back to York, and that coincided, roughly, with your separation from David.
McDuff: Right.
MP: And then on to Warwick in 1976.
McDuff: That was a very good place for me. I could well have stayed there for the long haul, but I moved to Stony Brook in 1978 for personal reasons I’ll talk about later. The two institutions are alike in many ways, not quite at the center of things but rich with opportunities.
Isolation
MP: You frequently speak of being isolated in your writing.
McDuff: Right.
MP: Many people continue to believe that mathematics is a single-person sport and not a team sport. You seem to be suggesting very strongly that interactions with other mathematicians are very important.
McDuff: I think they are. I talked about this with my husband, Jack Milnor, recently, because he is somebody who almost always works alone; he does talk to people, but not that much. What he said was that when you’re learning a subject, it is vital to talk to others. You have to grow up in a community, know where the subject is at, and what the interesting problems are. Once you have a general framework, you can fruitfully work on your own—though even then it’s often good to talk to others.
A large part of the problem was my attitude. If I’d had role models of women challenging authority I might have done better—to do research you have to ask questions. I did not know anyone who was attempting to live a similar life, and so it took a long time for things to come together. The late sixties was the time of the “Free University” in Cambridge, lots of far-left politics, very antiauthority. There were very few adults whom I was willing to talk to—Gelfand and Aleksander Zyw—but not my parents or anyone who might have given me sensible advice.
I kept myself apart from most other women since they didn’t seem to share my ambitions. I also isolated myself because of my life with David; he was not at all sociable either.
MP: You said that Gelfand opened a number of fields to you. That might have helped with the isolation problem.
McDuff: Eventually it did.
MP: He introduced you to subject matter that he thought you should know and revealed the interconnectedness of many of these ideas. A mathematician I talked to recently said that underscores a defect of American education in mathematics. He was talking about it at the undergraduate level as well as the doctoral level. He claims that students get pushed through a number of required courses and very often finish a bachelor’s degree or an advanced degree with only a weak idea of how mathematical subjects connect. For that reason he recommends developing new sorts of modern capstone experiences that bring the ideas together. I don’t know if that’s particularly easy. I think it’s easier to do it at the undergraduate level than at the graduate level. Maybe that’s part and parcel of education in Russia.
McDuff: Well, I don’t know what education in Russia is like now, but the Gelfand seminar reflected his very broad interests. His attitude was that everything was one.
You can’t learn all of mathematics, but the education that I got as a graduate student in Britain was very narrow. Education there is still rather narrow because, at least until very recently, PhD students were allowed only three years. I think it’s much better in the States than in Britain. Instead of starting work immediately on some thesis topic, students have one or two years of general courses and can then decide whom to work with. Even though people may still be specialized, they have certainly seen more mathematics when they finish than I had.
MP: So most new PhDs in Britain are about 24 years old when they complete their degrees.
McDuff: I was 24 when I’d done mine.
MP: In the United States, the average is closer to 27, a dramatic difference.
McDuff: It used to be the case in England that you specialized more as an undergraduate, so you had a bit more mathematics completed before starting graduate school, but not in Edinburgh because the Scottish university system was more like the American. Of course, if you are in a department with broad enough interests, you can continue developing and growing.
MP: Do you still feel isolated?
McDuff: No, I don’t feel isolated anymore, but that’s fairly recent—really starting when I became interested in symplectic geometry in the mid-1980s.
Symplectic Geometry
MP: I have to compliment you on your article on an introduction to symplectic geometry. It’s the first one that really began to make sense to me. Your exposition is a gift.
McDuff: Which introduction was that?
MP: The one you gave at the European Women in Mathematics Conference.
McDuff: Oh, that recent one.
MP: As I say, it was a gift.
McDuff: At first I thought you were talking about the book I wrote with Dietmar Salamon, called Introduction to Symplectic Geometry, whose first chapter is “From Classical to Modern.” One of the things I did learn in Edinburgh was very old-fashioned classical mechanics, spinning tops, canonical transformations, and all that. Those are the roots of symplectic geometry, so this chapter started off talking about classical mechanics and then discussed Gromov’s modern approach to the geometry of Euclidean space.
Jack Milnor
MP: Somewhere along the line, you encountered Jack Milnor.
McDuff: I met him when I was at the Institute for Advanced Study in Spring 1976.
MP: Apparently you and Jack do discuss mathematics.
McDuff: To some extent. I’ve never collaborated with him. But we do talk about mathematics, and he occasionally reads things that I write. For example, he read the article of mine you liked, helping me make it understandable to somebody who doesn’t know the subject.
MP: I’ve heard more than one mathematician say, “Never marry a mathematician in the same field, because that can only lead to conflict.” That was one of the reasons for asking that question, but your fields are sufficiently different that mathematical conflicts are less likely to occur.
McDuff: Well, he’s moved out of topology into dynamical systems. There are relations between symplectic geometry and dynamical systems, but we don’t talk much about them. He’s wonderful when I want to ask questions about topology. We do interact about mathematics, somewhat more when we met than now, but I’ve always felt that it’s better for me to be independent.
MP: I could have guessed that by now. So, as of the moment, you’re continuing to work vigorously in symplectic geometry, not anticipating a move into some other field.
McDuff: Symplectic geometry is incredibly rich. Many brilliant young people have come into the field. It now encompasses a huge amount of mathematics and relates to many other areas, so I am not tempted to do something completely different. The homotopy theory I studied as a postdoc is relevant, as are many other things I learnt along the way. For example, I first learnt about continued fractions in order to teach a workshop for the Stony Brook undergraduate Women in Science program, but it was a crucial ingredient of my latest work about embedding ellipsoids. Of course, there are many other topics like mirror symmetry and algebraic geometry that I don’t know enough about, but I am more confident that I can learn them as needed.
Figure 12.14 McDuff and Jack Milnor, about 1978.
MP: There are all kinds of opportunities, even full-blown journals in symplectic geometry.
McDuff: The Journal of Symplectic Geometry is fairly recent, I would say, about five years old or so.
MP: That’s young for a journal.
McDuff: The field has influence. The ideas of Floer theory, first expressed in symplectic geometry, have now permeated low-dimensional topology, with close connections to gauge theory and homological algebra. Via mirror symmetry, we now understand that symplectic geometry is in some sense the twin of complex geometry.
Dusa McDuff as Role Model
MP: In 1978 you moved to Stony Brook to be closer to Milnor, who was at the Institute in Princeton, and in 1984 you were married. At Stony Brook you quickly became a full professor and department head. As department head you worked to improve the curriculum. Tell us about that.
McDuff: I made efforts to improve the undergraduate curriculum at Stony Brook by introducing new classes and working to attract more majors. I like teaching. I was involved in calculus reform, teaching calculus with computers, and experimenting with new curricula. I haven’t been able to work consistently at that because it’s just too much to spend a lot of time doing that and have a family and do research. (My son, Thomas, was born in 1984.) At Stony Brook I worked with others to improve the undergraduate program, but there is always a huge amount left to do.
Figure 12.15 McDuff with son Thomas, 1985.
MP: Like it or not, you’ve become a role model for women in mathematics. How do you handle that?
McDuff: At some point I had to decide whether I’d put more effort into doing administration or into improving teaching—getting involved in a major way and doing it properly, or whether I wanted to concentrate on research. There are many prominent women scientists who’ve recently become deans or university presidents. I decided I didn’t want to take that route, since what I really cared about was doing research. That’s also important for women, to see that women can excel at research. I recently moved to Barnard partly for that reason, since it gives me a visible platform on which to be both a woman and a research mathematician.
I always try to encourage young women, indeed women at all stages. I had very little advice, and talked to very few people early in my career. Now I talk to people a lot; it’s good to encourage dialogue. There are many ways of being a successful woman mathematician. When I gave my acceptance speech after winning the Satter Prize, I said that it would be good for other women to talk about how and why they became mathematicians, for my story is certainly not the only one. A few months later, the Notices of the American Mathematical Society published fascinating autobiographical accounts by six other women. That was very valuable because it showed women from different backgrounds, with different motivations and interests, and different ways of contributing.
MP: Such articles actually do lots of good for mathematics in that they provide a lot of information, counseling as it were, of what it means to be a woman in mathematics and what they’re up against.
McDuff: Now there are several very successful programs designed for women. People have been trying various approaches, some of which have really worked. For example, the Nebraska Conference for Undergraduate Women and the program at the Institute for Advanced Study are both excellent. There are still not enough women in the profession for us to take their presence for granted. Anything that can bring women together and help them meet others, who are possibly quite different but facing some of the same issues, is to my mind very helpful.
MP: Some women mathematicians I have spoken with believe that employment opportunities for women mathematicians have actually declined in recent years.
McDuff: It’s still very spotty. There are some really brilliant young women mathematicians now; it is much more accepted that women can be good mathematicians.
There’s still a lot of hidden prejudice, not just about women. There are so many attributes—accent, color, or background—that make one think a person couldn’t possibly be a mathematician. It takes a long time for the academic culture to change.
Some good departments have made serious efforts to overcome this problem. In those that haven’t, there aren’t many women. In general, I think the situation is much better than it was. There are many more opportunities. Every year there are some women everybody wants to hire. Coming here to MSRI is great fun; there are a lot of young women in the program, which improves the atmosphere for everyone.
One can’t be complacent. Everybody should be encouraged, not just women. The old attitude used to be that if people are any good they’ll survive, and if they don’t, there are always others. I disagree; we have to care about everyone.
Figure 12.16 McDuff, about 2004.
Career, Family, Husband—A Lot of Juggling
MP: In one of the articles you have written you say that you only survived because of the confidence that was instilled in you by the success of your work on von Neumann algebras.
McDuff: My upbringing also gave me a lot of confidence. I felt I could do everything, and I tried to do everything—to have a career and a family and a brilliant husband.
But it’s not possible to keep all those balls up in the air at the same time.
MP: It’s difficult. I don’t think it’s gotten easier.
McDuff: It’s not gotten much easier, except that attitudes toward gender issues have improved. Marriage is now more of an equal partnership, while in the past women and men were expected to play very different roles. It is less of an anomaly for a young woman to have the ambition to succeed as a mathematician. In my day it was considered so unfeminine that I had to spend a lot of energy proving I was a woman.
At the beginning, before I had found my way as a mathematician, the fact that I had written a good PhD thesis gave me belief in myself. The visibility of my thesis also enabled me to get jobs. No doubt I was known because I was one of the very few women doing research mathematics at that time, but I had also shown that there are infinitely many type II1 factors, a question left open since the foundational papers of Murray and von Neumann in the 1940s. So that gave me a firm basis on which to build a life.
*Nadya Aisenberg and Mona Harrington, Women of Academe: Outsiders in the Sacred Grove (Amherst: University of Massachusetts Press, 1988).