Celebratio Mathematica

Dusa McDuff

My Life

by Dusa McDuff

On two dif­fer­ent oc­ca­sions re­cently, (male) math­em­aticians asked me in all in­no­cence: But you surely nev­er suffered any dis­crim­in­a­tion? This es­say is partly in re­sponse to them and partly amp­li­fies a talk I gave last year to par­ti­cipants in the GROW con­fer­ence about my fam­ily back­ground. I am con­vinced that it is only be­cause there was such a strong aca­dem­ic tra­di­tion in my fam­ily (among both wo­men and men) that I sur­vived as a math­em­atician. One in­dic­a­tion of how rare this was is the fact that no fe­male math­em­aticians were elec­ted as Fel­lows of the Roy­al So­ci­ety of Lon­don for the al­most fifty years between 1947, when Dame Mary Cartwright be­came the first such Fel­low, and 1994, when I was elec­ted.

1: The author’s grandfather, George Rivers Blanco White, studied mathematics at Cambridge and placed second in his year in final exams.
Photo courtesy of Dusa McDuff

My moth­er’s fath­er, George Rivers Blanco White (1883–1966), pic­tured in Fig­ure 1, stud­ied math­em­at­ics at the Uni­versity of Cam­bridge, was a ju­ni­or wran­gler (i.e., he placed second in his year in the fi­nal ex­ams), and chose to fight as a private rather than an of­ficer in the First World War in an ar­til­lery unit in the trenches (cal­cu­lat­ing tra­ject­or­ies). So­cially pro­gress­ive, he be­came a law­yer and even­tu­ally a dis­tin­guished di­vorce court judge. I met him only a few times, but my moth­er had been close to him and told me about him.

My fath­er, Con­rad Hal Wad­ding­ton (1905–1975), pic­tured in Fig­ure 2, was a dis­tin­guished de­vel­op­ment­al bio­lo­gist, ge­net­i­cist, and thinker, whose work is still well known today be­cause he was one of the fore­fath­ers of sys­tems bio­logy. I was closest to him when I was about thir­teen, when he was writ­ing The Eth­ic­al An­im­al, his at­tempt to for­mu­late a sci­entif­ic jus­ti­fic­a­tion for mor­al­ity, and I spent even­ings in his study dis­cuss­ing his ideas. He didn’t have much time for math­em­at­ics, think­ing it too dry and bor­ing.

2: The author’s father, Conrad Hal Waddington, was a developmental biologist, geneticist, thinker, and a forefather of systems biology.
Photo courtesy of Dusa McDuff

My moth­er, Mar­garet Justin Wad­ding­ton, pic­tured in Fig­ure 3, was a trained ar­chi­tect and town plan­ner with a full-time job in the civil ser­vice in Ed­in­burgh, design­ing coun­cil hous­ing and later do­ing re­search in­to the most ef­fi­cient designs for emer­gency rooms in hos­pit­als. It was very un­usu­al for mar­ried wo­men to have pro­fes­sion­al ca­reers in Ed­in­burgh, which was one reas­on why our fam­ily had very few friends there. An­oth­er reas­on was the ef­fect of the war: my fath­er was re­war­ded for his dis­tin­guished war work by the Pro­fess­or­ship in Ed­in­burgh, but al­most all my par­ents’ friends re­mained much closer to Lon­don. Des­pite hav­ing to work in a gov­ern­ment of­fice after the war, in the 1930s she did man­age to build a house in the mod­ern­ist style that is now a \( 2^* \) lis­ted build­ing, an hon­or in Bri­tain.

I was al­ways ex­pec­ted to get an ad­vanced edu­ca­tion, have a ca­reer, and be self-sup­port­ing — again, un­usu­al for the time and very help­ful, be­cause that was one battle I did not have to fight. Also, I hardly had to fight for the de­cision to study math­em­at­ics. I had al­ways ex­celled at it, and my school had a won­der­ful math­em­at­ics teach­er who showed me the beauty of Eu­c­lidean geo­metry and the el­eg­ance of cal­cu­lus.

3: The author’s mother, Margaret Justin Waddington, was a trained architect and town planner, designed council housing in Edinburgh and researched efficient designs for emergency rooms.
Photo courtesy of Dusa McDuff

However, I did have to fight the battle of how to be both a wo­man and a math­em­atician. Al­though as a teen­ager I had al­most no idea what either of those iden­tit­ies might be, it was cer­tainly not to be a “blue­stock­ing,” i.e., a ser­i­ous, es­sen­tially sex­less old maid (you see all the pre­ju­dices). By the time I was fif­teen or so, I also had quite a bit of scorn for wo­men. I went to an all girls’ school, which, in ret­ro­spect, had many ad­vant­ages, but the teach­ing was not as rig­or­ous as that in boys’ schools, and a sig­ni­fic­ant co­hort of the girls were there to learn to be young ladies and fu­ture wives. Even to go on to uni­versity rather than Atholl Cres­cent (a fin­ish­ing school in Ed­in­burgh where one learnt to be a wife) was quite a step.

I know very little about my great-great-grand­moth­er, Mrs. Robison. The reas­on for her rather resigned ex­pres­sion in Fig­ure 5 will be­come clear, as will that for my great-grand­moth­er Maud’s sour­ness. Maud (1865–1953) was re­mark­able. She was part of a group of wo­men who won Votes for Wo­men in New Zea­l­and in 1893 — the first coun­try in the world to achieve this. Com­ing to Lon­don in the 1890s, she joined the Fa­bi­an so­cial­ists, foun­ded their wo­men’s group, and as part of a pro­ject with them wrote Round About a Pound a Week, a book about the Lon­don work­ing classes in 1912 that is still in print today and was re­cently used as a so­ci­ology text in Stony Brook Uni­versity. (In­cid­ent­ally, I still get more roy­al­ties from that book than I do for my own.) I nev­er met Maud: my moth­er had deeply of­fen­ded her and there was no com­mu­nic­a­tion.

4: Dusa Waddington in 1960–1961.
Photo courtesy of Dusa McDuff

The cent­ral fig­ures in Fig­ure 5 are my grand­moth­er Am­ber and her daugh­ter Anna-Jane. There was a big scan­dal in Lon­don about Am­ber’s preg­nancy and Anna-Jane’s birth be­cause the fath­er was H. G. Wells. Though HG was firmly mar­ried and with chil­dren, he and Am­ber had fallen in love a few years be­fore.1 He wrote sev­er­al books about her, not­ably Ann Veron­ica, a nov­el fea­tur­ing the “new wo­man.” Though this new wo­man is very ap­peal­ing and in­tel­lec­tu­ally alive, in the end Ann Veron­ica mar­ries the teach­er she so in­spired and then es­sen­tially ceases to have an in­de­pend­ent life — at least Wells goes no fur­ther with the story.

5: Four generations of women in the author’s family: her great-grandmother, Maud Pember Reeves; her half-aunt Anna-Jane; her grandmother, Amber Pember Reeves (Dusa) holding Anna-Jane; and her great-great-grandmother, Mrs. Robison.
Photo courtesy of Dusa McDuff

My grand­moth­er, of course, did go fur­ther with her story. My grand­fath­er, who had loved her in their stu­dent days in Cam­bridge, mar­ried her while she was preg­nant, for which she was al­ways very grate­ful. Be­sides hav­ing three chil­dren, she wrote some nov­els (A Lady and Her Hus­band was re­cently re­is­sued), and oth­er books such as one about eco­nom­ics for the Left Book Club, Worry in Wo­men, and Eth­ics for Un­be­liev­ers in­spired by Con­fucius’s philo­sophy. After the Second World War she taught philo­sophy at Mor­ley Col­lege (now part of the Uni­versity of Lon­don). However, she was con­sidered by both my par­ents to have been without a real ca­reer and to have largely wasted her tal­ents. My fath­er once re­marked that she could have had a won­der­ful life in Lon­don: she was very at­tract­ive, knew every­one, and could have been the mis­tress of the likes of Ber­trand Rus­sell — “What a wasted op­por­tun­ity!” he said. But she al­ways thought that HG was the most bril­liant man she had ever met.

Look­ing back now, I think she did re­mark­ably well, con­sid­er­ing that between the wars, when there was so much un­em­ploy­ment, it was not con­sidered prop­er (and of­ten simply not al­lowed) for mar­ried wo­men to have full-time jobs. However, she (as well as my moth­er and my young­er self) thought that boys were bet­ter than girls. There was no fam­ily tra­di­tion that wo­men could be as good as men. My moth­er’s solu­tion to the prob­lem was to ig­nore the fact that I was fe­male, a fine ap­proach when I was young, but not much help to me in nav­ig­at­ing ad­oles­cence.

Dusa was the name HG used for Am­ber, be­cause she had long black snaky hair. (I later learnt that HG had a dif­fer­ent ver­sion: in his es­say On Loves and the Lov­er Shad­ow about his love life, he said that Am­ber had chosen it for her­self as a teen­ager, be­cause she iden­ti­fied with Me­dusa’s head held up by Perseus in Bern­ini’s statue.) Since I was called after her, I was very af­fected by her story. Identi­fy­ing with the mon­strous Gor­gon2 made me feel unique, which was both good and bad. I could cer­tainly nev­er settle for something I con­sidered or­din­ary, a great im­petus for me to con­tin­ue to try to find my way in math­em­at­ics. On the oth­er hand, I had very un­real­ist­ic ideas about what kind of wo­man I wanted to be (or could be), and I also had to find a geni­us equi­val­ent to HG….

I did find my geni­us (a fledgling poet). He had the good qual­ity of be­ing com­pletely amathem­at­ic­al (be­cause he couldn’t pass the arith­met­ic ex­am to get in­to Ed­in­burgh Uni­versity, he had to do a pa­per in bot­any), which meant that I was com­pletely free in that re­spect. In the course of his PhD in Rus­si­an sym­bol­ist po­etry, we man­aged to go to Mo­scow to­geth­er for six months in 1969–70, where I stud­ied with the great I. M. Gel’fand. This had an enorm­ous in­flu­ence on my math­em­at­ic­al de­vel­op­ment. So des­pite the many dif­fi­culties, liv­ing with a poet did in the end spur my ca­reer.

In 1967 I went as a gradu­ate stu­dent to Cam­bridge, without real­iz­ing that the de­cision of where to study and whom to study with was at all con­sequen­tial. Start­ing out in func­tion­al ana­lys­is, I stud­ied von Neu­mann al­geb­ras with George Re­id, who had re­cently re­turned from a year at Tu­lane. Dur­ing my second year, he showed me a re­cent pa­per by an­oth­er gradu­ate stu­dent who had just dis­covered a third \( \mathrm{II}_1 \) factor.3 Quite quickly, I man­aged to ex­tend these new ideas. Dur­ing a meet­ing where I showed sev­en­teen or so new factors to George, he re­marked, “If only you could it­er­ate one of these con­struc­tions….” That seed was enough for me to find an it­er­a­tion pro­cess that al­lowed the con­struc­tion first of count­ably many and then of un­count­ably many dif­fer­ent \( \mathrm{II}_1 \)-factors. This work was pub­lished in the An­nals of Math­em­at­ics; its ex­ist­ence helped me to get jobs and was a great mor­al sup­port for many years.

6: Gel’fand’s seminar opened the author’s eyes to many areas of mathematics, and he advised her to become a topologist.
Photo courtesy of Dusa McDuff

Fig­ure 6 shows a pho­to­graph of me in Mo­scow, Fall 1969, at Gel’fand’s sem­in­ar. When I got there, I knew very little math­em­at­ics. Gel’fand opened my eyes to num­ber the­ory, man­i­folds, to­po­logy, ho­mo­topy the­ory, ho­mo­lo­gic­al al­gebra, Lie groups… 4 After Mo­scow, I came back to Cam­bridge for two years, tak­ing Gel’fand’s ad­vice to be­come a to­po­lo­gist. It was a time of learn­ing, both to be a to­po­lo­gist and a moth­er. I felt very isol­ated and ig­nor­ant: it would have been very help­ful to have a ment­or at this time, someone to en­cour­age me and sug­gest prob­lems for me to work on. Be­cause I had totally switched fields, no one felt re­spons­ible for look­ing out for me, though, to be fair, some people did help me at a few cru­cial mo­ments in the next few years: sup­ply­ing money to go to a con­fer­ence in the US, giv­ing me a job at the Uni­versity of York, and later in­vit­ing me to fill a vis­it­ing po­s­i­tion at MIT.

Was I ever dis­crim­in­ated against? There are two kinds of dis­crim­in­a­tion: ex­pli­cit and im­pli­cit. For the most part, ex­pli­cit dis­crim­in­a­tion did not af­fect me much. However, in ret­ro­spect, im­pli­cit dis­crim­in­a­tion — for ex­ample, the fact that I was so isol­ated as a postdoc be­cause I could not share in col­lege life — as well as my own in­tern­al­ized miso­gyny, did have a sig­ni­fic­ant ef­fect, though I hardly no­ticed this at the time. An­oth­er im­port­ant factor, and one that I was aware of, was per­vas­ive but not overt: it was very rare that wo­men be­came pro­fes­sion­al sci­ent­ists in Bri­tain at the time, largely be­cause sci­ence (and par­tic­u­larly “hard” as op­posed to “life” sci­ence) was con­sidered such a very un­fem­in­ine thing to do. Even I thought it was un­fem­in­ine, and I had there­fore to show my fem­in­in­ity in oth­er ways: for ex­ample, my moth­er nev­er did any house­work and kept her maid­en name pro­fes­sion­ally, while I, ever the con­trary teen­ager, glor­ied in cook­ing for my boy­friend and changed my name when I got mar­ried. I al­ways felt alone, had no co­hort of friends for mu­tu­al en­cour­age­ment. Even when the second-wave wo­men’s move­ment got star­ted in Bri­tain, I thought it was ir­rel­ev­ant to me, since they were fight­ing the battle of tak­ing their pro­fes­sion­al lives ser­i­ously, while I was already re­spons­ible for sup­port­ing my fam­ily. These days, when most of the ob­vi­ous bar­ri­ers to wo­men’s par­ti­cip­a­tion in math­em­at­ics have been re­moved, there still re­main very strong and in­si­di­ous in­tern­al bar­ri­ers, shown in such phe­nom­ena as ste­reo­type threat or im­poster syn­drome. The pre­ju­dices that lead to people ac­cept­ing as com­pletely nor­mal that wo­men should not get de­grees at Cam­bridge (they first could get Cam­bridge de­grees in 1948) are very strong and do not dis­ap­pear im­me­di­ately when the ex­tern­al bar­ri­er is re­moved.

In the 1960s there were, of course, very vis­ible mani­fest­a­tions of the idea that aca­dem­ic life is not for wo­men. At the time, most Ivy League uni­versit­ies in the States did not ad­mit wo­men, and in Bri­tain al­most all the col­leges at the most pres­ti­gi­ous uni­versit­ies (Ox­ford and Cam­bridge) were single sex. As a gradu­ate stu­dent, I was a mem­ber of Gir­ton Col­lege, Cam­bridge, but I nev­er went there for sev­er­al reas­ons: it was sev­er­al miles away from the town cen­ter, re­search was done in the de­part­ment rather than in col­leges, I was too pre­ju­diced against wo­men to think it worth ex­plor­ing Gir­ton, and, like all oth­er col­leges at the time, Gir­ton had no pro­vi­sions for mar­ried stu­dents. The male stu­dents, of course, did dine in their col­leges, which were also in the town cen­ter. In the early 1970s when some male col­leges began to ad­mit wo­men, my thes­is ad­visor, George Re­id, told me that as far as he was con­cerned this would only hap­pen at his col­lege over his dead body. So I did feel ex­cluded. Even in the math­em­at­ics de­part­ment, the few wo­men were either go­ing to get mar­ried when they gradu­ated or were go­ing back to their home abroad or were very mar­gin­al­ized. There were no role mod­els to show the way or even an in­cip­i­ent wo­men’s move­ment with which to de­vel­op aware­ness and plot res­ist­ance.

The most dis­crim­in­at­ory situ­ation I en­countered was when I was try­ing to get a job in Cam­bridge after my postdoc. I was mar­ried with a hus­band and baby to sup­port, but the bet­ter pay­ing fel­low­ships were at men’s col­leges so that I could not ap­ply, while Gir­ton was too poor to pay enough for a fam­ily to live on. There had been an open uni­versity lec­ture­ship the pre­vi­ous year, which I was eli­gible for. But no one told me about it, and I was com­pletely out of the loop. In fact, the way I heard about this was when George Re­id said to me at one point: “What a pity you didn’t ap­ply for that job last year….”

So I had to leave Cam­bridge, pos­sibly a good thing in the end. Ob­tain­ing one of the few lec­ture­ships avail­able in the UK that year, I en­joyed my time as a young fac­ulty mem­ber in York, start­ing to work with col­leagues, mak­ing friends, and lead­ing a some­what more nor­mal life. In the sum­mer be­fore go­ing to York, I went to a con­fer­ence on K-the­ory at the Bat­telle In­sti­tute in Seattle (in­cid­ent­ally, my first vis­it to the US), where I met Graeme Segal. We star­ted a col­lab­or­a­tion, in which I first func­tioned much as a gradu­ate stu­dent, but gradu­ally be­came a some­what more equal part­ner. As an in­dic­a­tion of my state of mind then, here is the story of my meet­ing with Ar­mand Borel in Seattle. On the first morn­ing of the meet­ing, I was up very early with jet lag and had a very pleas­ant break­fast con­ver­sa­tion about jazz clubs in New York with an in­tel­li­gent and friendly man. When I dis­covered later that my break­fast com­pan­ion was Borel, I said not one more word to him. Even later, when I vis­ited the In­sti­tute for Ad­vanced Study I nev­er talked to him: I simply felt I knew too little to face him.

I could well have stayed in York (and as a res­ult per­haps done rather less math­em­at­ic­al re­search) but didn’t, be­cause, no doubt on Gel’fand’s re­com­mend­a­tion, I was offered the won­der­ful op­por­tun­ity of a year as a vis­it­ing as­sist­ant pro­fess­or at MIT in 1974–1975. One of the first in­sti­tu­tions to act­ively pro­mote wo­men in sci­ence, MIT had re­served this po­s­i­tion for a wo­man. In my case, this in­ter­ven­tion was very suc­cess­ful, since it gave me a chance to be­come part of a top-level re­search com­munity. After that I fi­nally began to func­tion more in­de­pend­ently, cre­at­ing op­por­tun­it­ies for my­self, hav­ing new math­em­at­ic­al ideas, and slowly re­build­ing con­fid­ence that I really could do math­em­at­ics. Fi­nally I got to the fol­low­ing stage: I was talk­ing to Raoul Bott for the first time, ex­plain­ing something that I was work­ing on with Graeme Segal, and I said, “I thought that…,” which he in­ter­rup­ted with “You thought?” I con­fid­ently replied, “Yes!” and con­tin­ued with my ex­plan­a­tion.