Celebratio Mathematica

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.

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.

And in­tu­ition.

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?

Wil­helm Mag­nus.

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.

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 Ral­ph Fox, 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.

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, and Louis Niren­berg said 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].

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.

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.

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.

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.

Yes.

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.

Yes.

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.

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.

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 when Bill Thur­ston made the dis­cov­ery that earned him the Field’s Medal, and the second time was when Vaughan Jones 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 Car­o­lyn Series. 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.