Celebratio Mathematica

Karen Uhlenbeck

Interview with Karen Uhlenbeck

by Allyn Jackson

Born in 1942, Kar­en Keskulla Uh­len­beck re­ceived her Ph.D. in 1966 from Bran­de­is Uni­versity, un­der the dir­ec­tion of Richard Pal­ais. After postdoc­tor­al po­s­i­tions at the Mas­sachu­setts In­sti­tute of Tech­no­logy and the Uni­versity of Cali­for­nia at Berke­ley, she held fac­ulty po­s­i­tions at the Uni­versity of Illinois at Urb­ana-Cham­paign (1971–1976), the Uni­versity of Illinois at Chica­go (1976–1983), and the Uni­versity of Chica­go (1983–1988). In 1988, she was ap­poin­ted to the Sid W. Richard­son Found­a­tion Re­gents Chair in Math­em­at­ics at the Uni­versity of Texas at Aus­tin. Now re­tired from Texas, she is cur­rently a vis­it­or at the In­sti­tute for Ad­vanced Study in Prin­ceton.

Over the past half-cen­tury, Uh­len­beck has been one of the out­stand­ing fig­ures in ana­lys­is and has made es­pe­cially pro­found con­tri­bu­tions in geo­met­ric ana­lys­is. She is prob­ably best known for her work in math­em­at­ic­al gauge the­ory, which laid the found­a­tions for some of the most sig­ni­fic­ant ad­vances in math­em­at­ics in the twen­ti­eth cen­tury. For this achieve­ment, the Amer­ic­an Math­em­at­ic­al So­ci­ety awar­ded Uh­len­beck the 2007 Steele Prize for a Sem­in­al Con­tri­bu­tion to Re­search. Among her oth­er hon­ors are a Ma­cAr­thur Fel­low­ship (1983) and the Na­tion­al Medal of Sci­ence (2000). She was elec­ted to the U.S. Na­tion­al Academy of Sci­ences in 1986 and to the Amer­ic­an Philo­soph­ic­al So­ci­ety in 2012. In 1990, she be­came the second wo­man, after Emmy No­eth­er in 1932, to give a plen­ary ad­dress to the In­ter­na­tion­al Con­gress of Math­em­aticians.

What fol­lows is the ed­ited text of an in­ter­view with Uh­len­beck, con­duc­ted in Prin­ceton by Allyn Jack­son in March 2018.

Interested in everything

As a child I read a lot, and I read everything. I’d go to the library and then stay up all night reading. I used to read under the desk in school […] I read all of the books on science in the library and was frustrated when there was nothing left to read.

Jack­son: You have such a beau­ti­ful maid­en name, Kar­en Keskulla.

Uh­len­beck: Thank you. I still sign my­self Kar­en Keskulla Uh­len­beck. My grand­fath­er was Es­to­ni­an, al­though he was born in Riga, which is in Latvia. “Keskula” was the ori­gin­al spelling. It comes from the Finnish–Es­to­ni­an lan­guage and means “cen­ter of the vil­lage”. If you go to Hel­sinki and drive around, you will see signs for “Keskulla” all over! That’s on my fath­er’s side. My grand­moth­er was Ger­man.

Jack­son: Was your fath­er born in the United States?

Uh­len­beck: Yes. I’m not sure about my grand­moth­er, but I think she was born here too. I do not know much about my back­ground. We are not a his­tory-keep­ing fam­ily.

My moth­er was one of 12 chil­dren of a Meth­od­ist min­is­ter. He died when she was 5 and while her own moth­er was preg­nant. My moth­er had an older broth­er who was able to help the fam­ily through. But she did not grow up with a lot of money.

Jack­son: Where was this?

Uh­len­beck: In Farm­ing­dale, New Jer­sey. There was a lot of sis­terly help too — my moth­er was very at­tached to at least one of her older sis­ters. It wasn’t a ques­tion of their moth­er tak­ing care of 12 chil­dren. I knew my ma­ter­nal grand­moth­er, and she wasn’t one to take care of any­body. She was a re­mark­able wo­man. She must have been at least 5 feet 11 inches [about 1.8 meters] tall, which at that time made her a very tall wo­man. She was al­ways in­ter­ested in everything and very talk­at­ive. I come from a line of strong wo­men on the ma­ter­nal side.

Jack­son: Your moth­er be­came an artist.

Uh­len­beck: She was sup­por­ted by at least one older sis­ter in that. She star­ted col­lege around 1928, at the Art Stu­dent’s League in New York City. After about a year, she real­ized that she wouldn’t get a job as an artist. So she trans­ferred to the City Col­lege of New York and got a de­gree in edu­ca­tion. I think for the next few years of her life, un­til she met my fath­er, she taught art in high school, and maybe grade school too. She al­ways liked teach­ing little kids.

I don’t know how those first years of the De­pres­sion were for her. I think there was a time when she did not get enough to eat. She gradu­ated from col­lege in 1932, as my fath­er did. That was not a good year to gradu­ate.

One of my fath­er’s aunts had a job as a buy­er for Macy’s and was fairly well in­to the up­per middle class. She fin­anced my fath­er to study mech­an­ic­al en­gin­eer­ing at MIT. In 1932 the only job he could find was in the gold mines of Cali­for­nia. After a year or so, he went to Utah and worked in the mines there. He fell in love with the moun­tains out West.

The De­pres­sion men­tal­ity hung over our child­hood. My par­ents were both up­wardly mo­bile, edu­cated, in­tel­lec­tu­al people who at a crit­ic­al time of their lives, when they were start­ing out, had really severe eco­nom­ic prob­lems. There was a back­ground of real worry about money, but also an in­tel­lec­tu­al back­ground.

Some of my moth­er’s sib­lings went to col­lege, and her fath­er was some­how as­so­ci­ated with Drew Uni­versity.

Jack­son: Was your fath­er the first in his fam­ily to go to col­lege?

Uh­len­beck: Yes, I’m pretty sure he was. He grew up in Ossin­ing, New York. After he had been out West, he got a job with the Alu­min­um Com­pany of Amer­ica, which after a while be­came Al­coa. He and my moth­er spent most of their mar­ried life here in New Jer­sey, un­til they re­tired to Santa Fe.

Jack­son: What kind of paint­ings did your moth­er paint?

Uh­len­beck: My moth­er, through all her life, was really in­ter­ested in scenery. But a lot of her early work is very much 1930s work. I trace it when I go to art mu­seums and can see things like my moth­er painted. Through my child­hood she mostly painted scenes around New Jer­sey. She tried paint­ing her kids but that didn’t work very well.

Jack­son: She didn’t paint people so much?

Uh­len­beck: Right. About the time I went away to col­lege, she switched from oil paint­ing to wa­ter col­or, I sus­pect be­cause of the chem­ic­als in­volved.

Jack­son: You were one of four kids in your fam­ily. What did the oth­er three end up do­ing?

Uh­len­beck: Well, they had dif­fi­culties. My fath­er in­sisted my broth­er try to get a de­gree in en­gin­eer­ing, and he did study en­gin­eer­ing, but then he went to di­vin­ity school. He has not had an easy life. My young­est sis­ter was badly han­di­capped, and she died more than 20 years ago. My middle sis­ter is a poet. She lives with her hus­band in Carl­isle, Mas­sachu­setts. They are very in­ter­ested in wild­life and eco­logy.

Jack­son: You were the old­est?

Uh­len­beck: Yes.

Jack­son: Were there im­port­ant in­flu­ences in your youth in math­em­at­ics and sci­ence? Was your fath­er an in­flu­ence?

Uh­len­beck: Cer­tainly there was in­dir­ectly an in­flu­ence from my fath­er. We were great read­ers, and my fath­er took out from the lib­rary books by people like Fred Hoyle and George Gamow, which I ended up read­ing. He was more wor­ried about my broth­er be­ing the sci­ent­ist than he was about me. My moth­er’s friends were paint­ers, and they did not have your av­er­age, nor­mal life­style. So when I was young, I was in­tro­duced to a lot of people who did not live nor­mal, middle-class lives. I have very strik­ing memor­ies of that.

Jack­son: Your fath­er wanted your broth­er to be­come in­ter­ested in sci­ence be­cause he was a boy?

Uh­len­beck: Yes.

Jack­son: Did your broth­er ex­hib­it much in­terest in sci­ence?

Uh­len­beck: No. His in­terests were al­ways some­what mys­ter­i­ous at the time, but he cer­tainly hated sci­ence. It was kind of a miser­able up­bring­ing for him. He is now a ser­i­ous mu­si­cian.

Jack­son: So you were freer to do what you were in­ter­ested in.

Uh­len­beck: Yes. We lived in West Mil­ling­ton, near Bask­ing Ridge. West Mil­ling­ton wasn’t even a small town. Mil­ling­ton was a town, and West Mil­ling­ton was the name for a few houses. It was out in the coun­try. That was my moth­er’s in­flu­ence. She al­ways liked the scenery. My par­ents did not in­ter­act much with the loc­al people. My fath­er had some friends through his pro­fes­sion, and my moth­er had her artist friends and her ex­ten­ded fam­ily of el­ev­en broth­ers and sis­ters. One was in Michigan, and one was in Alaska, but the rest lived in New Jer­sey.

Jack­son: In high school, were there teach­ers who en­cour­aged you to study math­em­at­ics?

Uh­len­beck: Not really. I was a good stu­dent, so my teach­ers en­cour­aged me. My most vivid memor­ies are of my Lat­in teach­er. A good num­ber of stu­dents at that time were tak­ing Lat­in, and I took four years of it. My in­ter­pret­a­tion of that now is that Lat­in was the only hard course I had. You couldn’t just do it, you had to work at it!

Jack­son: You didn’t have to work at math classes?

Uh­len­beck: No. Be­cause of my sched­ule I couldn’t take the “hon­ors” math­em­at­ics class. So the hon­ors math teach­er had me in a couple of times after school and taught me the rudi­ments of cal­cu­lus. This helped tre­mend­ously when I took en­trance ex­ams for the hon­ors col­lege at the Uni­versity of Michigan.

I once had a date with some really weird kid who was the son of friends of my par­ents. He lent me a cal­cu­lus book, and I made my way through it. My fath­er had some text­books from MIT, in­clud­ing one on op­tics that I went through without really com­pre­hend­ing it. I was in­ter­ested in everything. I don’t think I was very a com­fort­able kid to be around.

Jack­son: Why?

Uh­len­beck: I was al­ways tense, want­ing to know what was go­ing on and ask­ing ques­tions. I was not prop­erly so­cial­ized.

I didn’t have to learn it, I could create it myself

I learned about mathematics in my freshman honors course at the University of Michigan. I still remember the thrill of taking limits to compute derivatives, and the little boxes used in proving the Heine–Borel theorem. The structure, elegance and beauty of mathematics struck me immediately, and I lost my heart to it.

Uh­len­beck: When it came time to go to col­lege, there was no ques­tion of go­ing to col­lege in New Jer­sey. Rut­gers and Prin­ceton were all-male, and Douglas was a wo­man’s col­lege. My moth­er did not want me to go to a wo­man’s col­lege. I would have liked to ask her why; it just nev­er came up.

So I picked out three places. We had cous­ins in Michigan, so Uni­versity of Michigan was one of the schools. Ac­tu­ally large num­bers of stu­dents from the met­ro­pol­it­an area of New Jer­sey and New York went else­where to col­lege, in­clud­ing the Uni­versity of Michigan. My fath­er had gone to MIT, so I ap­plied there. And for some reas­on I picked out Cor­nell; it soun­ded like a good place. I got in every­where. I knew that my par­ents were wor­ried about money. They didn’t put pres­sure on me, or if they did it was subtle, but I chose Uni­versity of Michigan be­cause it was a lot cheap­er than the oth­ers. It was a very good choice for me. About ten years later, I taught at MIT, and the wo­men there were miser­able! Two wo­men in every cal­cu­lus class, kind of hud­dling to­geth­er. Uni­versity of Michigan was in­tel­lec­tu­ally great, but also so­cially great be­cause the girls from the Mid­w­est were much more open and lively than the ones in New Jer­sey. In New Jer­sey it was very clique-ish, and I was a “brain”.

I don’t know any­body who likes high school, so I don’t know wheth­er it’s worth dwell­ing on the fact that high school wasn’t much fun! I was in the band, and I played cla­ri­net. I was very act­ive in sports, though girl’s sports did not ex­ist then as they do now. Since we lived out in the coun­try, we had trans­port­a­tion prob­lems if I wanted to stay after school.

I had two or three ses­sions with that hon­ors math teach­er in high school, and what he taught me was very form­al and rote. I re­mem­ber the mo­ment at which I really grasped something about cal­cu­lus, which was in col­lege.

Jack­son: Can you tell me about that mo­ment?

Uh­len­beck: I had a very ad­vanced course, sim­il­ar to an un­der­gradu­ate real ana­lys­is course. We had done lim­its. I went to a help ses­sion, and the teach­ing as­sist­ant showed how to take a de­riv­at­ive. I re­mem­ber get­ting up and look­ing at the guy next to me and say­ing, “Are we al­lowed to do that?” It was a mo­ment where sud­denly I real­ized there were all sorts of things in math­em­at­ics that you “were al­lowed to do”.

Years later, I met the teach­er in this course, Max­well Reade, and he said that I used to sit in the last row and read the New York Times. I hon­estly don’t re­mem­ber that! But I do re­mem­ber find­ing the course bor­ing be­cause he would do something one day, and then come back the next time and do the same thing again. As a pro­fess­or and as a teach­er, I know that’s what you’re sup­posed to do. But as a stu­dent who was fas­cin­ated by the math­em­at­ics, I just saw this as re­pe­ti­tion. It was an hon­ors math class and star­ted out with 80 stu­dents. By the end of the second year, there were about eight left.

Jack­son: How many wo­men were in the class?

Uh­len­beck: I don’t think there were many wo­men in the ori­gin­al class. I think there were three wo­men among the eight at the end. Both of the oth­er two also have Ph.D.s in math. I’m not sure I can re­mem­ber their names.

Jack­son: That’s pretty amaz­ing, three out of eight.

Uh­len­beck: Oh, yeah. It was a great en­vir­on­ment. Of course it was a com­pletely male at­mo­sphere. The lib­rar­ies in mid­west­ern uni­versit­ies have a cer­tain feel. An oak table, a lot of thick volumes — it’s some­how a male at­mo­sphere. That was the way it was, and I ac­cep­ted it, in the way young people ac­cept most things. I was a suf­fi­ciently good stu­dent that I was very much en­cour­aged. I took gradu­ate courses as an un­der­gradu­ate and had a lot of en­cour­age­ment to ap­ply to gradu­ate school.

I re­mem­ber an­oth­er “aha” mo­ment. I was tak­ing a course from a rather fam­ous guy named Ray­mond Wilder,1 who had been a stu­dent of R. L. Moore.2 There were fifty stu­dents in the class, but Wilder was still teach­ing like R. L. Moore.3 In this class I had an “aha” mo­ment when I real­ized I could do what was in the books. I didn’t have to learn it, so to speak — I could cre­ate it my­self. Those kinds of courses are tre­mend­ously valu­able, for prob­ably all kinds of stu­dents, be­cause they make you real­ize that you can think your way through something.

Jack­son: Al­though you star­ted as a phys­ics ma­jor, you ended up ma­jor­ing in math.

Uh­len­beck: Yes. I did my ju­ni­or year abroad. I had a boy­friend who was a math gradu­ate stu­dent and was Pol­ish. He thought I needed more of an edu­ca­tion — I was too Amer­ic­an or something! I ap­plied for a ju­ni­or year abroad pro­gram through Wayne State Uni­versity. The pro­gram had stu­dents from all over the United States — from Yale, Prin­ceton, Stan­ford, Wheaton Col­lege in Mas­sachu­setts. I went to Mu­nich for a year, and it was a fas­cin­at­ing ex­per­i­ence. I learned to speak Ger­man reas­on­ably well. And I sat through Ger­man math lec­tures, which was an eye-open­er. The lec­tures were com­pletely pol­ished. They were so easy to fol­low be­cause everything was worked out so clearly. I’d been used to pro­fess­ors who pre­pare for fif­teen minutes be­fore lec­tur­ing. I learned to ski, I learned to like op­era, and I was in­tro­duced to the world of the theat­er. It was a won­der­ful ex­per­i­ence.

Jack­son: Do you re­mem­ber any of the pro­fess­ors in Mu­nich?

Uh­len­beck: I took some al­geb­ra­ic to­po­logy from Rieger,4 and I took dif­fer­en­tial equa­tions and com­plex ana­lys­is from Stein.

Jack­son: This was Karl Stein?5

Uh­len­beck: I think it was him. I might know his name from math­em­at­ics, from Stein man­i­folds. Of course at that time I didn’t know that, I wasn’t far enough along. His lec­tures were beau­ti­ful.

There were also ex­er­cise ses­sions, Übun­gen, and I re­mem­ber that the oth­er math stu­dents in our pro­gram were not as ad­vanced as I was. They were from places like Yale and Prin­ceton, so it was an en­cour­aging dis­cov­ery that the edu­ca­tion I was get­ting in the hon­ors pro­gram at the Uni­versity of Michigan was really good. I owe a lot to the Uni­versity of Michigan.

Jack­son: Do you re­mem­ber pro­fess­ors from Michigan oth­er than Wilder who were im­port­ant to you at that time?

Uh­len­beck: I had a course in al­geb­ra­ic to­po­logy from Frank Ray­mond. I did not take a course from Paul Hal­mos al­though he was at Michigan at the time.

Jack­son: There were no wo­men pro­fess­ors there at the time?

Uh­len­beck: No. There were oth­er wo­men in the classes, and I know sev­er­al of them now, for ex­ample Har­riet Pol­lat­sek and Martha K. Smith. There is a hand­ful of wo­men who were in the hon­ors pro­gram at Michigan and went on to get Ph.D.s in math. One of the the­or­ies was that fath­ers would not send their tal­en­ted daugh­ters to elite schools; if they’d been male they would have been sent else­where. So a group of very tal­en­ted wo­men ended up in the hon­ors math pro­gram at Michigan. I’m not sure I ever bought that the­ory.

The value of the imperfect role model

There were a handful of women in my graduate program [at Brandeis], although I was not close friends with any of them. It was self-evident that you wouldn’t get ahead in mathematics if you hung around with women. We were told that we couldn’t do math because we were women. So, if anything, there was a tendency to not be friendly with other women. There was a lot of blatant, overt discouragement, but there was also subtle encouragement. There were a lot of people who appreciated good students, male or female, and I was a very good student. I liked doing what I wasn’t supposed to do, it was a sort of legitimate rebellion. There were no expectations because we were women, so anything we did well was considered successful.

Jack­son: You got your bach­el­or’s de­gree from Michigan in 1964. When did you start think­ing about gradu­ate school in math­em­at­ics?

Uh­len­beck: I just fell in­to it. When I was an un­der­gradu­ate, I ap­plied for some in­tern­ships at places like IBM and Bell Labs. I nev­er got a po­s­i­tion, and I al­ways think my life might have been very dif­fer­ent if I had. I was not in­ter­ested in teach­ing — I thought people were way too dif­fi­cult to deal with! So I fell in­to math­em­at­ics as something that I could do. My gradu­ate stu­dent boy­friend had trans­ferred to NYU. I don’t re­mem­ber where I ap­plied but I ended up go­ing to NYU. This was partly be­cause my moth­er had gone to col­lege in New York and partly be­cause NYU had a good repu­ta­tion for wo­men at that time.

Jack­son: Cath­leen Mor­awetz was there.

Uh­len­beck: She was the only wo­man I ever took, or even audited, a course from. I want to tell this care­fully, as it is im­port­ant to me. At the time, I was not very im­pressed by her. I was crit­ic­al of her hair and clothes, her lec­tur­ing style, and even of her math­em­at­ics. I had no con­tact what­so­ever with ap­plied math­em­at­ics and was not in­ter­ested. So I re­solved to be bet­ter than my one role mod­el, which serves some pur­pose. Many years later, I fell on this memory as a lifesaver. I was go­ing through a rocky peri­od, and the memory of less-than-per­fect lec­tures by a highly re­spec­ted math­em­atician, who I had grown to ad­mire, kept me go­ing. One does not have to be per­fect! One can be hu­man. Role mod­els serve many pur­poses.

Joan Birman and I were gradu­ate stu­dents to­geth­er, and we got to know each oth­er. She was a lot older than I was and already had three kids. She had a house north of New York City, and I went to lunch there. I had a room­mate, Mary An­der­son, who was mar­ried to some­body who went to the IAS for a year. Mary would come up in the middle of the week and go to her classes at NYU and go back to Prin­ceton on the week­ends. She did get her Ph.D. in math, and she lived a fas­cin­at­ing life. We are still in touch, and we have very dif­fer­ent memor­ies of NYU.

After a year at NYU, I mar­ried the boy­friend I had met in Michigan, who is a bio­chem­ist, and I took the name Uh­len­beck.

Jack­son: He is the son of the phys­i­cist George Uh­len­beck. And then?

Uh­len­beck: We moved to Bo­ston be­cause my hus­band went to Har­vard. I had an NSF Gradu­ate Fel­low­ship, and that helped me move. This was around 1967, not long after Sput­nik. Throughout my col­lege ca­reer, there were many pro­grams en­cour­aging stu­dents to go in­to math and sci­ence. They were en­cour­aging every­body, and wo­men coun­ted.

I trans­ferred to the math gradu­ate pro­gram at Bran­de­is Uni­versity. I did not even ap­ply to MIT and Har­vard. I have no idea where this kind of gut wis­dom came from, ex­cept that I knew that I wasn’t in­ter­ested in the wo­man ques­tion. I did not want to be an ex­cep­tion­al wo­man. It was a gut feel­ing; I can’t cite any ter­rible ex­per­i­ences I’d had. I just knew I would be isol­ated at those places. Bran­de­is was a smal­ler school, and I felt that I would be safer there. And it was true. I don’t think the pro­fess­ors were all that eager to see me. A lot of them didn’t want or ap­prove of wo­men stu­dents. But I was a suf­fi­ciently good stu­dent that whatever their feel­ings had been about wheth­er wo­men should do math, I was able to over­come them without any trauma, so to speak.

But I nev­er did get along with the sec­ret­ar­ies! They were much harder. I re­mem­ber the early parts of my ca­reer be­ing made dif­fi­cult by hav­ing to get along with sec­ret­ar­ies. By the way, in the later part of my ca­reer, es­pe­cially at the Uni­versity of Texas and the IAS, I got ter­rif­ic sup­port from the staff. Things have really changed.

Jack­son: Your ad­viser at Bran­de­is was Dick Pal­ais. How did it come about that he be­came your ad­viser?

Uh­len­beck: I chose him very con­sciously be­cause he was in­volved in a new sub­ject called glob­al ana­lys­is — us­ing to­po­logy to solve prob­lems in ana­lys­is, or ana­lys­is to solve prob­lems in to­po­logy. Do­ing dif­fer­en­tial to­po­logy in in­fin­ite di­men­sions was a big thing the second year I was in gradu­ate school. There was a sem­in­ar every Fri­day night at MIT that I went to on this new stuff, and I was very taken with it. Dick Pal­ais had been at the IAS and ran a sem­in­ar there, and he had pub­lished a book about the Atiyah–Sing­er in­dex the­or­em.6 I con­sciously de­cided that I wanted to do this new math­em­at­ics rather than, say, prove a small change in some bound­ary value prob­lem.

It was a good fit for me be­cause I am by nature an ana­lyst, while my thes­is ad­viser was a dif­fer­en­tial to­po­lo­gist. He framed the prob­lems for me and taught me the ab­stract think­ing, which I am not so good at. He is a beau­ti­ful ex­pos­it­or and really filled in some part of my know­ledge that I nev­er could have got­ten my­self. And I was able to go to the lit­er­at­ure and work out the tech­nic­al de­tails of the ana­lys­is, which was not something he knew how to do.

Jack­son: Can you talk about the con­nec­tion of this to the work of Eells and Sampson?

Uh­len­beck: Their work is found­a­tion­al. There was a pa­per by Eells and Sampson that ac­tu­ally con­struc­ted har­mon­ic maps in­to man­i­folds of neg­at­ive curvature us­ing a heat flow.7 That was one of the first times that any­one had ac­tu­ally used to­po­lo­gic­al con­di­tions in such a set­ting. The neg­at­ive curvature is re­lated to a to­po­lo­gic­al con­di­tion. Their pa­per showed that you could get solu­tions in any ho­mo­topy class, and in many cases, if you had a little bit more neg­at­ive curvature, they were unique.

That was one of the first pa­pers that con­nec­ted ana­lys­is on man­i­folds and to­po­logy. In sur­vey talks where this ana­lys­is and to­po­logy con­nec­tion comes in, I would al­ways also cred­it the Bott peri­od­icity the­or­em, which was proved in the late 1950s. That’s a case where the to­po­logy was in­vest­ig­ated by look­ing at geodesics, and it def­in­itely uses ana­lys­is. An­oth­er is the Atiyah–Sing­er in­dex the­or­em, which con­nects the di­men­sion of the solu­tion space with to­po­logy.

Also fun­da­ment­al was the Pal­ais–Smale con­di­tion. Steve Smale and my ad­viser Dick Pal­ais dis­covered it pretty much in­de­pend­ently. It shows that the Morse the­ory that was de­veloped by Mar­ston Morse could be car­ried out in in­fin­ite di­men­sions. This is the point at which I began gradu­ate school, in 1964. This area was open­ing up, con­nect­ing ana­lys­is and to­po­logy and geo­metry. I was at­trac­ted to that — the area in between things. It was like jump­ing off a deck where you didn’t know what was go­ing to hap­pen.

Jack­son: In your re­sponse to get­ting the Steele Prize, you said that you were at Bran­de­is in the thick of glob­al ana­lys­is and that the area fell in­to dis­fa­vor later. What happened?

Uh­len­beck: It was a sub­ject that was sud­denly very big for a year or so, but then there were no big pay­offs. About ten years later, it came back as geo­met­ric ana­lys­is, with people like S.-T. Yau prov­ing the­or­ems. It was not a smooth tra­ject­ory. What was in the minds of the people who were work­ing in it in the 1960s ac­tu­ally came to fruition in the 1970s and 1980s. It wasn’t very pop­u­lar in between.

I should add something else about that peri­od of my life. My in-laws lived in New York, and we vis­ited them of­ten. My fath­er-in-law was a phys­i­cist who had moved from the Uni­versity of Michigan to Rock­e­feller Uni­versity. He was a com­bin­a­tion of amused by me and en­cour­aging to me. It was in­ter­est­ing and use­ful to get the per­spect­ive of an aca­dem­ic fam­ily. I val­ued them be­cause I had come from out­side aca­demia and from out­side this lar­ger in­tel­lec­tu­al world. My in-laws knew a wide range of people in aca­demia from vari­ous places. My moth­er-in-law taught me to cook. I did know how to cook ba­sic stuff from my moth­er, but my moth­er didn’t think of cook­ing as an art. So my in-laws were very im­port­ant to me.

Jack­son: You said your fath­er-in-law was amused but en­cour­aging. Amused, mean­ing…?

Uh­len­beck: I think he just en­joyed the fact that I was dif­fer­ent and wanted to do things in a way that he was very un­fa­mil­i­ar with.

Jack­son: Some men would have been put off by that.

Uh­len­beck: That’s right. He was not put off.

Jack­son: His wife was not an aca­dem­ic?

Uh­len­beck: No. They were Dutch and had both grown up in In­done­sia. They’d seen a lot of the world. It was a real eye-open­er to meet people from such a dif­fer­ent back­ground.

Jack­son: After your Ph.D. you taught at MIT and then you spent two years at Berke­ley, 1969 to 1971. What are your im­pres­sions of Berke­ley from that time?

Uh­len­beck: Polit­ics. The Vi­et­nam War. I also learned a lot of math­em­at­ics.

Jack­son: Who do you re­mem­ber in the math de­part­ment from that time?

Uh­len­beck: In­ter­est­ingly enough, an old timer named Abe Taub.8 He was at that time a math­em­at­ic­al phys­i­cist, and I took some quantum mech­an­ics from him and some gen­er­al re­lativ­ity from his col­league Ray Sachs. I also learned a little bit of quantum field the­ory, but that was in­ter­rup­ted when the uni­versity closed down be­cause of demon­stra­tions. Abe Taub im­pressed me quite a bit. He had worked on shock waves and flu­id mech­an­ics and had also been in­stru­ment­al in the con­struc­tion of one of the first com­puters, at the Uni­versity of Illinois. He later moved to Berke­ley. I was in­ter­ested in how you could use very ana­lyt­ic­al means to study things like shock waves, and I was im­pressed by his breadth.

At Berke­ley I didn’t have many people to work with. I got in­volved in prov­ing some things about ei­gen­val­ues and ei­gen­func­tions, and I worked on Lorentz geo­metry, which is geo­metry in space­time. These things were not ac­tu­ally in­spired by any­body; I just picked them up. I was a little bit frus­trated that I couldn’t get any­body else in­ter­ested in them! But I’m not be­ing crit­ic­al; it surely was part of the at­mo­sphere. I listened to a semester of ce­les­ti­al mech­an­ics from Steve Smale in my last term there. By the time I fin­ished the course I had learned mech­an­ics very well.

Jack­son: After Berke­ley you moved to the Uni­versity of Illinois in Urb­ana-Cham­paign in 1971. Both you and your hus­band got po­s­i­tions there.

Uh­len­beck: We thought it would be a little bit like the Uni­versity of Michigan, where we were both un­der­gradu­ates and which we had en­joyed. Some­how or oth­er there was a fuss about my job. I don’t know what it was — people seemed to en­joy telling you that you didn’t really get the job.

Jack­son: That you wer­en’t really qual­i­fied?

Uh­len­beck: Yes. They said they wanted to make a lot of of­fers, and they could make only one, the im­plic­a­tion be­ing that some­how the ad­min­is­tra­tion had something to do with it — which could have been true.

Four or five wives of pro­fess­ors taught in the math de­part­ment, but they were not part of the sem­inars. And there was a wo­man pro­fess­or, M.-E. Ham­strom.9

Jack­son: You wer­en’t happy at Urb­ana.

Uh­len­beck: It’s hard to know wheth­er I was un­happy, or my mar­riage wasn’t work­ing out, or whatever. I didn’t feel at home math­em­at­ic­ally, for sure. It was a big change from Berke­ley. I don’t re­mem­ber that I talked math­em­at­ics so much to people in Berke­ley, but some­how I had a lot more people to talk to. It might even have been just age. At Illinois there wer­en’t so many people in my age group, and none with my back­ground.

The wives did all the en­ter­tain­ing in the math de­part­ment. Years later, when I went back to vis­it, there were no parties be­cause the wives were all work­ing! But dur­ing my time at Illinois, there was really a di­vi­sion between wo­men and men in the de­part­ment. The at­mo­sphere was stifling to me. Up to that time, my ex­per­i­ence in aca­demia had been an es­cape from feel­ing con­fined.

From regularity of maps to minimal surfaces

[Lesley Sibner is] a typical example of a woman who is a very good research mathematician but who is not recognized. She is at a technical institute. The only good thing about the job is that she periodically gets good graduate students. She is underpaid and teaches a lot […]. [T]o be really successful you have to be protected, and there is no way to do it any other way. I think about this all the time.

Uh­len­beck: I must have traveled quite a bit when I was at Urb­ana — I don’t know how I did it, but I must have. I be­came very friendly with Les­ley Sib­n­er,10 who was at Poly­tech­nic Uni­versity in Brook­lyn. I first met her in Trieste at a con­fer­ence in 1971 or 1972. I went to a talk on a sub­ject I thought I would know something about. I was ex­pect­ing some guy to get up and speak, be­cause I didn’t know who Les­ley Sib­n­er was. And this wo­man in a purple suede pant­suit and high heels came char­ging in and gave a beau­ti­ful talk on stuff that I knew about and was in­ter­ested in. I was so ex­cited to meet her. We be­came friends, and I made quite a few trips to New York and would al­ways see her and her hus­band Robert Sib­n­er, who was a math­em­atician at the City Uni­versity of New York. Les­ley was an ex­cep­tion­al wo­man. She had ac­tu­ally star­ted out study­ing act­ing and came to math­em­at­ics a little bit late. She was a very good math­em­atician and a won­der­ful per­son.

Jack­son: Did you write any pa­pers with her?

Uh­len­beck: We wrote a pa­per11 about ten years after we had met. I was by then at the Uni­versity of Texas at Aus­tin.

Jack­son: You left your hus­band and, in 1976, took a po­s­i­tion at the Uni­versity of Illinois at Chica­go. You liked the at­mo­sphere there much bet­ter than Urb­ana.

Uh­len­beck: Right. I was very com­fort­able there. I also learned some Teichmüller the­ory from How­ie Mas­ur, who had the of­fice next to mine. This proved use­ful when I tried to un­der­stand Thur­ston’s work.

Jack­son: It was when you were at the Uni­versity of Illinois at Chica­go that you wrote a pa­per with Jonath­an Sacks.12

Uh­len­beck: He came to Urb­ana as a postdoc and star­ted talk­ing to me there. I didn’t know very much about min­im­al sur­faces, but we talked about them and worked to­geth­er. He brought the know­ledge of the sub­ject of min­im­al sur­faces, and I brought the main idea, which was really a fairly simple idea from com­plex ana­lys­is. We wrote a couple of pa­pers to­geth­er. I was very close to him for a num­ber of years, but then I lost track of him.

Jack­son: You said you brought the main idea from com­plex ana­lys­is. What was that idea?

Uh­len­beck: It was the idea of res­cal­ing the prob­lem. If you have something that doesn’t de­pend on scale, you can blow it up and look at it at a dif­fer­ent scale.

Jack­son: This res­cal­ing idea is what you called “bub­bling” in the pa­per?

Uh­len­beck: That’s right.

Jack­son: Why did you call it bub­bling?

Uh­len­beck: You have a prob­lem in \( \mathbb{R}^2 \), and something is hap­pen­ing at a par­tic­u­lar point. You ex­pand tiny discs to large ones, and what you find hid­den at that point is a map to a sphere. So it’s like a bubble. Oth­er kinds of prob­lems can bubble too, but they usu­ally have to have some scale-in­vari­ance in them.

Jack­son: Later you used this idea of bub­bling in the con­text of Yang–Mills equa­tions, is that right?

Uh­len­beck: Yes. Scale-in­vari­ant prob­lems all be­have very sim­il­arly. An­oth­er prob­lem that I didn’t even know about at the time was the Yamabe prob­lem, which is also scale-in­vari­ant. A lot of prob­lems that are totally geo­met­ric are scale-in­vari­ant. That is, if you have a prob­lem where you are look­ing for a met­ric, and some­how or oth­er, there is no in­trins­ic ex­tern­al scale on it, you can ac­tu­ally change the scale that you are look­ing at.

Jack­son: Can you tell me about the prob­lem you were work­ing on with Sacks?

Uh­len­beck: Con­cep­tu­ally, it was a prob­lem of find­ing min­im­al sur­faces. I nev­er learned and still don’t really know the Fe­der­er–Flem­ing ap­proach us­ing geo­met­ric meas­ure the­ory. The al­tern­at­ive ap­proach was to find them with con­form­al para­met­er­iz­a­tions. The prob­lem is, these con­form­al para­met­er­iz­a­tions don’t see scale. That is, if you write down an in­teg­ral, over a piece of \( \mathbb{R}^2 \), that rep­res­ents a piece of the min­im­al sur­face, then you can ac­tu­ally get the same an­swer by chan­ging the scale of the piece in \( \mathbb{R}^2 \). It’s a cal­cu­lus of vari­ations prob­lem, and I had learned a lot about the cal­cu­lus of vari­ations as a gradu­ate stu­dent. The idea was to write down the en­ergy func­tion­al that would pro­duce the min­im­al sur­face and make the small per­turb­a­tion that would ac­tu­ally give it scale. And then you could prove it sat­is­fied the Pal­ais–Smale con­di­tion, so that there was a Morse the­ory. There were lots of crit­ic­al points in that case. You let the per­turb­a­tion scale go to 0, and then, al­though there are places where you would not get con­ver­gence, you just res­cale around those places un­til you find something.

What you get when you res­cale is a map from \( \mathbb{R}^2 \) in­to the man­i­fold where you are look­ing for the min­im­al sur­face. The point is that, be­cause this in­teg­ral doesn’t see any­thing but angles, it can’t tell the dif­fer­ence between \( \mathbb{R}^2 \) and \( \mathbb{S}^2 \). You get a map from \( \mathbb{R}^2 \), but you can also think of it as a map from \( \mathbb{S}^2 \) with a sin­gu­lar point. Then there is a reg­u­lar­ity the­or­em — which is also something that I worked on in my thes­is — where if you have a map from a sphere to a man­i­fold that sat­is­fies the har­mon­ic map con­di­tion, you can ac­tu­ally fill in the point where there is a sin­gu­lar­ity and prove that it’s ac­tu­ally reg­u­lar over the sin­gu­lar point. This was forty years ago, and nowadays the tech­nique would be thought of as very stand­ard.

At the time I had not had con­tact with a large num­ber of math­em­aticians. I had got­ten my Ph.D. at Bran­de­is, not from Har­vard or MIT or Berke­ley. And when I was at Berke­ley and MIT, I didn’t really work with any­one else. That was valu­able in some sense, be­cause I didn’t know what people were think­ing about min­im­al sur­faces. So when Jonath­an Sacks taught me the prob­lem, I didn’t have any built-in ma­chinery to think about it. So I was able to think about it on my own.

Jack­son: That’s an ad­vant­age.

Uh­len­beck: It’s def­in­itely an ad­vant­age. With sev­er­al prob­lems, I worked in a con­text where I didn’t know oth­er people who were de­vel­op­ing ma­chinery about the prob­lem.

Jack­son: Was the work with Jonath­an Sacks more geo­met­ric than you had done be­fore?

Uh­len­beck: Yes, more geo­met­ric. I owe that to Jonath­an.

There is something else I should tell you about. When I was a gradu­ate stu­dent, I ran across a very tech­nic­al prob­lem about prov­ing reg­u­lar­ity of maps that have a bad non­lin­ear­ity in the top-or­der term. It came up in my thes­is. Rather than look­ing at the in­teg­ral that would, say, give the min­im­al sur­face equa­tion, I was look­ing at in­teg­rals that had a slightly dif­fer­ent in­teg­rand. You could prove that there was a Morse the­ory for them. They are called \( p \)-har­mon­ic maps now. The prob­lem was, they were weak solu­tions only in a Banach space; they were not smooth. I worked on this prob­lem for sev­er­al years by my­self, pretty much. I read some pa­pers on el­lipt­ic PDE, and dur­ing one of the first con­fer­ences I went to, I met Jürgen Moser.13 He gave me sev­er­al re­prints of his pa­pers on what’s now known as the Moser it­er­a­tion scheme. The work I did on that prob­lem came out of my meet­ing Moser and read­ing the pa­pers he gave me. That’s the hard­est math­em­at­ics I ever did.14

S.-T. Yau no­ticed that pa­per, and he in­vited me to vis­it him. He was at Stan­ford at that time. I was ab­so­lutely startled be­cause no one had ever asked me about that work. I had nev­er talked to any­one about it! I went to vis­it and talked with Yau and Le­on Si­mon, and met Rick Schoen, who was their gradu­ate stu­dent at Stan­ford. I’m not even sure I could read that pa­per nowadays. It was a very hard, very tech­nic­al pa­per on a prob­lem that I be­came con­sumed with. I worked on it for sev­er­al years — one year as a lec­turer at MIT, and two years at Berke­ley. This was the start of a great re­la­tion­ship with S.-T. Yau.

Jack­son: Was there a glob­al ana­lys­is per­spect­ive in that pa­per?

Uh­len­beck: No, this was just straight PDE, but it was mo­tiv­ated by glob­al ana­lys­is. I had some func­tion­als that sat­is­fied a Morse the­ory, and I wanted to show that their crit­ic­al points were ac­tu­ally reg­u­lar maps, that they wer­en’t very dis­con­tinu­ous. And in a way that’s what grew in­to the min­im­al sur­faces pa­pers. I was look­ing for 2-di­men­sion­al min­im­al sur­faces, and the Di­rich­let in­teg­ral, which is what you usu­ally used to min­im­ize, al­most sat­is­fied the Pal­ais–Smale con­di­tion. So if you add a little ep­si­lon term, it would sat­is­fy the Pal­ais–Smale con­di­tion. The idea was to make this small per­turb­a­tion, and then see what happened as the per­turb­a­tion went to zero.

So the min­im­al sur­faces work grew out of this very tech­nic­al PDE prob­lem that I had got­ten sucked in­to when I was work­ing on glob­al ana­lys­is. Sev­er­al oth­er stu­dents who were also do­ing glob­al ana­lys­is went on to do much more ab­stract things. And I went on to do very tech­nic­al things. But since I was trained in a to­po­lo­gic­al-geo­met­ric view­point, I’ve been able to handle to­po­logy. Al­though now I’m al­ways lost whenev­er I go to a talk in to­po­logy!

Changing what geometry is

The next major development in the story of the so-far-only-physicist’s Yang–Mills equations was one by mathematicians! The Atiyah–Singer index theorem said something about the dimension of the space of solutions to the Yang–Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics communities began to change. It is here that the cooperation and cross-fertilization really began.

I don’t know how to describe the tremendous difference this made. The entire position that mathematics held within the scientific community changed. The mathematicians actually had something to say!

Jack­son: What kinds of prob­lems ap­peal to you? What does a prob­lem have to have to grab your at­ten­tion?

Uh­len­beck: It’s a com­bin­a­tion of be­ing fairly con­crete — so one can un­der­stand con­cretely ex­amples — and also con­nect­ing with a lot of oth­er ideas. For ex­ample, you see the ana­lys­is in a min­im­al sur­face equa­tion, but then you also real­ize it has con­nec­tions with oth­er geo­met­ric ques­tions that are not just ana­lys­is. I am def­in­itely very at­trac­ted to the idea that there are a lot of dif­fer­ent fa­cets in math­em­at­ics and see­ing the con­nec­tions.

Jack­son: A lot of ana­lys­is comes from phys­ics.

Uh­len­beck: I’m im­pressed by that. It seems to me that math­em­aticians in some ways are not very in­vent­ive thinkers. They get an idea, and then they sort of beat it to death, push­ing it for­ward. The idea be­hind it is of­ten lost to most of the people work­ing on the prob­lem.

Jack­son: Are phys­i­cists more in­vent­ive?

Uh­len­beck: They def­in­itely think dif­fer­ently, but I’ve nev­er been able to un­der­stand the dif­fer­ent way they think about prob­lems. Math­em­at­ics, es­pe­cially in the last fifty years, has be­nefited tre­mend­ously from ideas that come in from out­side. Some­how it doesn’t grow in­tern­ally on its own.

I take that back. Sep­ar­ate areas with­in math­em­at­ics can in­flu­ence each oth­er. Ex­amples of this would be the Lang­lands Pro­gram and the use of prob­ab­il­ist­ic meth­ods in PDE. The point is, math­em­at­ics is not a ma­chine where you write down defin­i­tions, the­or­ems, proofs, and tack on an ex­ample. It is much more in­ter­est­ing than that.

If I were young­er, I would be very in­ter­ested in learn­ing a lot more about things that are not core math­em­at­ics, but are math­em­at­ics con­nec­ted with oth­er areas.

Jack­son: What do you have in mind there?

Uh­len­beck: All sorts of stuff in com­puter sci­ence, in bio­logy.

Jack­son: Phys­ics as well?

Uh­len­beck: I tried that. I’d go in a dif­fer­ent dir­ec­tion. It is in­ter­est­ing that some­how or oth­er the ideas in phys­ics lead to ideas in math­em­at­ics, which tend to get stuck. Phys­i­cists have a looser idea of con­struc­tions.

Jack­son: You went to a lec­ture in the late 1970s by Mi­chael Atiyah at the Uni­versity of Chica­go.

Uh­len­beck: He gave a series of four lec­tures at the Uni­versity of Chica­go on Yang–Mills and gauge the­ory.

Jack­son: Is that when you got in­ter­ested in gauge the­ory?

Uh­len­beck: Yes. Most math­em­aticians didn’t know any­thing about it. There were two text­books, one by Steen­rod and one by Huse­moller, on fiber bundles.15 They were very ab­stract. The phys­i­cists would al­ways come to us and ask what they could learn from. And we could only give them these very ab­stract text­books. The lan­guage was surely dif­fi­cult for them — it was dif­fi­cult for the math­em­aticians who tried to learn it, and even harder for the phys­i­cists.

Jack­son: It took the phys­i­cists to come up with the Yang–Mills equa­tions. That wasn’t something math­em­aticians would have stud­ied on their own.

Uh­len­beck: They could have, but they didn’t. They had all the tools, but it wasn’t one of the things they were think­ing about.

Jack­son: Why not?

Uh­len­beck: I wish I knew ex­actly why and what makes the dif­fer­ence between math­em­aticians and phys­i­cists see­ing things in a slightly dif­fer­ent con­text.

I had been work­ing on vari­ation­al prob­lems. With the Yang–Mills equa­tions, there’s a second-or­der equa­tion that is the solu­tion of a vari­ation­al prob­lem, which is very nat­ur­al, and you could eas­ily come up with it. But then there are the first-or­der equa­tions, which need a little bit more to­po­logy to un­der­stand. Most math­em­aticians, when this was ex­plained to them, could un­der­stand it im­me­di­ately. It wasn’t totally mys­ter­i­ous.

I re­mem­ber my thes­is ad­viser Dick Pal­ais en­cour­aging me to get in­ter­ested in Yang–Mills — this was after my Ph.D. I had no ba­sic know­ledge of fiber bundles, and I had to learn what con­nec­tions and curvature are. But I got in­ter­ested and worked it out. I could see what the prob­lems were in the ana­lys­is. I just worked on what the ob­vi­ous prob­lem was, which was to make gauge changes so you could con­trol the ana­lys­is.

In 1979–80, I was at the IAS for a geo­metry pro­gram or­gan­ized by Yau. Cliff Taubes was then a gradu­ate stu­dent at Har­vard. He had a girl­friend who was in Prin­ceton at the time and later be­came his wife. He came down to Prin­ceton for a while and talked to me. The first thing he said was, “Tell me about the cal­cu­lus of vari­ations.” And the first thing I said to him was, “The cal­cu­lus of vari­ations isn’t go­ing to do this for you!” That’s the kind of good ad­vice an ex­pert can give to a novice: No, this is not the right thing for what you are try­ing to do! That was the be­gin­ning of a long friend­ship with Cliff Taubes.

Jack­son: You said that when you learned about gauge the­ory, you asked the “ob­vi­ous ques­tion” about the ana­lys­is. Can you tell me about that ques­tion?

Uh­len­beck: I re­mem­ber ask­ing people what gauge the­ory was. The an­swer was that, for the abeli­an bundle, it’s just a case of Hodge the­ory. The con­nec­tions are defined only up to some am­bi­gu­ity, which you are al­lowed to change. It’s very much like the defin­i­tion of a man­i­fold. You have patches on the man­i­fold, and you define how to go from one to the oth­er on the man­i­fold. In gen­er­al re­lativ­ity, you have a tre­mend­ous amount of am­bi­gu­ity in how to write down the met­ric. You write it down in dif­fer­ent co­ordin­ate patches, and you trans­form one to the oth­er. This trans­form­a­tion is more com­plic­ated than for gauge the­ory. So there is no ques­tion about gauge the­ory be­ing a sim­pler prob­lem than gen­er­al re­lativ­ity and equa­tions about Rieman­ni­an man­i­folds.

At first geo­met­ers were think­ing about these prob­lems in gauge the­ory, but people work­ing in PDE just did not think about them as an as­pect of par­tial dif­fer­en­tial equa­tions. Then came the solu­tion of the Calabi con­jec­ture, where you ask the ques­tion of wheth­er a man­i­fold with the right to­po­lo­gic­al con­di­tion can have a met­ric con­di­tion to pair with it. That was solved by Yau in 1977.16 There was a slow awaken­ing to the fact that there were par­tial dif­fer­en­tial equa­tions on man­i­folds that one could ac­tu­ally solve.

A man­i­fold is a bunch of points, and you have co­ordin­ates, but you can change them. So if you write down a par­tial dif­fer­en­tial equa­tion on a man­i­fold, you have to un­der­stand how it can be moved from one co­ordin­ate to an­oth­er. The work of Eells and Sampson is al­ways con­sidered one of the ba­sic, fun­da­ment­al break­throughs be­cause they ac­tu­ally solved a geo­met­ric equa­tion on a man­i­fold, and they did not use co­ordin­ates. They used the heat equa­tion, which is very sim­il­ar to the el­lipt­ic equa­tions that we all learned to use, but theirs was just a new idea of math­em­at­ics that you could do. There was a tre­mend­ous change in the per­spect­ive of what geo­metry was. And it was cer­tainly en­cour­aged by the fact that phys­i­cists came up with ques­tions that needed math­em­at­ic­al solu­tions.

Jack­son: You men­tioned the spe­cial geo­metry year at the IAS, or­gan­ized by S.-T. Yau. He was an im­port­ant fig­ure for you at this time.

Uh­len­beck: I could tell that he ad­mired my math­em­at­ics. He un­der­stood what I was do­ing, and he hon­estly thought I was a good math­em­atician. I’d been en­cour­aged in a kind of fath­erly way by oth­er math­em­aticians, but Yau was dif­fer­ent. I feel I owe a lot to him, be­cause he made me feel like a real math­em­atician. Of course by then he’d pro­duced a series of fant­ast­ic pa­pers, with new ideas in all of them. I felt he was a fant­ast­ic math­em­atician.

Jack­son: So when he thought you were good, it meant a lot to you.

Uh­len­beck: Yes, it meant a lot to me. It wasn’t just be­ing en­cour­aged be­cause I was smart.

Gauge theory: An adopted child

How did gauge theory appear and become successful in mathematics in the space of a few years? The fundamental mathematical ingredients were in place [fiber and vector bundles, connections, Chern–Weil theory, De Rham cohomology, Hodge theory]. In hindsight, the Yang–Mills equations were waiting to be discovered. Yet mathematicians were in themselves unable to create them. Gauge field theory is an adopted child.

Jack­son: Your work in gauge the­ory laid the ana­lyt­ic found­a­tions for a lot of sub­sequent de­vel­op­ments, in­clud­ing Don­ald­son’s work.

Uh­len­beck: That’s right. In a small neigh­bor­hood you can of­ten choose a gauge, which is a choice of co­ordin­ates in the bundle, in which things work very well. It was clear from the PDE point of view that you had to im­pose an ex­tra equa­tion, and you got the equa­tion to be true by choos­ing the right co­ordin­ates. A prob­lem I struggled with for a long time was that, if you bound the curvature of a con­nec­tion, you would like to get a bound on the con­nec­tion in some co­ordin­ates. The pa­per in which I showed how to do this is still one of my more in­flu­en­tial pa­pers.17 Ac­tu­ally, it isn’t a really hard pa­per, but it was very hard to think of the idea. It took me sev­er­al years to think of how to prove that this was true.

Jack­son: How did you come up with the idea?

Uh­len­beck: I just banged my head against the wall. I think it might vaguely have been in­flu­enced by Yau’s solu­tion of the Calabi con­jec­ture. The way I solved it was that I put a con­di­tion on the curvature and showed that the space of con­nec­tions hav­ing bounds on curvature was of a cer­tain size; it was both open and closed. It’s the con­tinu­ity meth­od in PDE. That is, you are try­ing to solve an equa­tion and you put some to­po­logy on it, and you prove that for all the ele­ments in the to­po­logy, the space of solu­tions is open and closed, mod­ulo the oth­er con­di­tions.

The con­tinu­ity meth­od is what you might think of as the most ele­ment­ary tech­nique in non­lin­ear ana­lys­is, and it’s prob­ably one of the most valu­able. It’s kind of amus­ing, be­cause non­lin­ear ana­lys­is has all this fancy stuff like the Atiyah–Sing­er in­dex the­or­em, all this fancy to­po­logy, all these fancy fixed-point the­or­ems. But in fact, the first thing you should learn is the con­tinu­ity meth­od, which is that the space of solu­tions is both open and closed.

Jack­son: It’s something that’s con­cep­tu­ally simple.

Uh­len­beck: It’s so simple that it was by­passed, in try­ing to solve the prob­lem.

Jack­son: That was one of the pa­pers for which you re­ceived the 2007 Steele Prize. What was the oth­er one?

Uh­len­beck: It was about re­mov­able sin­gu­lar­it­ies.18 This was an in­ter­est­ing ques­tion. It star­ted with the phys­i­cists. They wrote down solu­tions to the Yang–Mills equa­tions in \( \mathbb{R}^4 \). They were able to show that those solu­tions could be con­tin­ued over in­fin­ity. These equa­tions are what is called con­form­ally in­vari­ant. This means that if you take a solu­tion in \( \mathbb{R}^4 \) and you make a con­form­al change of co­ordin­ates — which means ba­sic­ally tak­ing \( x \) to \( 1/x \) in four di­men­sions — then the equa­tions are still sat­is­fied. The ques­tion of wheth­er every time you wrote down an equa­tion in \( \mathbb{R}^4 \) it would con­tin­ue over in­fin­ity was ac­tu­ally around for quite a num­ber of years.

Jack­son: And that was the ques­tion that you re­solved in this pa­per.

Uh­len­beck: Right. I now know com­pletely trivi­al proofs. I have some ex­tra ma­chinery that I’ve learned that makes it a lot easi­er.

Jack­son: Is “Uh­len­beck com­pact­ness” part of these two pa­pers?

Uh­len­beck: Yes. Uh­len­beck com­pact­ness is easi­est to de­scribe in di­men­sions two and three. There the state­ment is that if you have bounds on the in­teg­ral of the curvature squared, then you can con­struct a sub­sequence that con­verges weakly. When you have a bound on norms of func­tions, it’s kind of a tau­to­logy in dis­tri­bu­tion the­ory that you can con­struct a se­quence that con­verges weakly. But if you are work­ing in gauge the­ory, you’ve got to fix the gauges, so con­ver­gence is not so easy to show. It’s most im­port­ant for solu­tions of equa­tions in four di­men­sions, though it’s of­ten used in oth­er set­tings. You have solu­tions of equa­tions in four di­men­sions, and if the curvature is small enough in a neigh­bor­hood, you can con­struct a sub­sequence that con­verges. Then there is a count­ing ar­gu­ment that shows that if you have a func­tion­al like the Yang–Mills func­tion­al in four di­men­sions, and if you have a bound on solu­tions, then in fact a sub­sequence will con­verge off a fi­nite num­ber of points.

This was part of the solu­tion to the min­im­al sur­face prob­lem. That is, you have a per­turb­a­tion and you look at lim­its of solu­tions, and they con­verge off a set of a fi­nite num­ber of points. Around those points you can blow them up, which is the phe­nomen­on that hap­pens both in Yang–Mills fields and in min­im­al sur­faces. This kind of thing is ac­tu­ally used quite a bit in PDE now.

The com­pact­ness the­or­em has to do with a situ­ation where you look at a solu­tion of a par­tial dif­fer­en­tial equa­tion, and you need to know something about when a solu­tion can ex­ist. And you need to know that the space of solu­tions is com­pact with an iden­ti­fi­able bound­ary. My part of Don­ald­son’s the­or­em was to prove that the space was com­pact, with a cer­tain type of bound­ary oc­cur­ring. Then Cliff Taubes proved that that kind of bound­ary really did ex­ist. The most sur­pris­ing thing to me is this kind of tech­nique of con­struct­ing geo­met­ric in­vari­ants is so ubi­quit­ous now. It was com­pletely new. This per­spect­ive of “new­ness” lim­its older math­em­aticians. Young­er math­em­aticians ac­cept it as “well known” and use the tech­niques with more con­fid­ence.

Jack­son: What did you think when you heard about what Si­mon Don­ald­son had done in his Ph.D. thes­is?

Uh­len­beck: It was com­pletely out of the blue. He was a gradu­ate stu­dent, and if he’d been more soph­ist­ic­ated, he prob­ably nev­er would have thought you could do any­thing like this! I was very im­pressed. Giv­en the ma­chinery, it wasn’t that hard a the­or­em for me, or for people like Taubes, who had worked in the field. But put­ting it to­geth­er and ac­tu­ally get­ting a res­ult, was amaz­ing. I can’t even wish I could have done it, be­cause it was so out­side of my think­ing.

Jack­son: You and Mike Freed­man or­gan­ized a semester-long pro­gram at MSRI to study what Don­ald­son had done.

Uh­len­beck: Yes. I met Dan Freed that year. He did a lot of the or­gan­iz­a­tion and was very in­stru­ment­al in writ­ing up the notes from the lec­tures.19 We have been friends ever since.

We had people com­ing to the sem­in­ar throughout the semester. In some sense, the top­ic got easi­er as the semester went on be­cause the prob­lem was the ana­lys­is and to­po­lo­gic­al pre­requis­ites. The ac­tu­al proof it­self, once you had all the pieces to­geth­er, is very el­eg­ant. So it wasn’t a ques­tion of the ma­ter­i­al get­ting harder and harder. The num­ber of people who knew all the tech­nic­al busi­ness was very small, be­cause you had to know ana­lys­is and you had to know to­po­logy.

I un­der­stood Don­ald­son’s proof right away. The only prob­lem I had is, I didn’t know how to prove the solu­tion space he con­struc­ted was ori­ented. Dur­ing the time at MSRI, I found out how to ori­ent the mod­uli space in a way that I really liked! It turns out there is a ca­non­ic­al way to ori­ent a space that is giv­en by a solu­tion to equa­tions and that car­ries from fi­nite di­men­sions to in­fin­ite di­men­sions.

Jack­son: This was 1982, and you still were on the fac­ulty at the Uni­versity of Illinois at Chica­go. After MSRI you spent the second semester that year at Har­vard.

Uh­len­beck: Yes. Dur­ing that time I got to know Cliff Taubes quite well. I en­joyed it. I had nev­er got­ten a chance to teach gradu­ate courses be­fore. I found it dif­fi­cult and spent many hours pre­par­ing.

Jack­son: At Uni­versity of Illinois at Chica­go, you did not teach gradu­ate courses?

Uh­len­beck: Right. I didn’t want to. There wasn’t an audi­ence for the kind of to­po­logy and ana­lys­is that I was do­ing. And I’ve nev­er been one to want to lec­ture and have nobody un­der­stand what I’m talk­ing about.

Jack­son: Was that part of the reas­on you then moved to the Uni­versity of Chica­go in 1983?

Uh­len­beck: Yes, I moved to the Uni­versity of Chica­go be­cause I wanted Ph.D. stu­dents. And it was great. I got quite a few good Ph.D. stu­dents right away. My salary went up a lot too! But that was not the primary reas­on.

Jack­son: But you didn’t stay so long at Chica­go.

Uh­len­beck: No. The com­mut­ing was a lot harder. I lived in Evan­ston. My hus­band [Robert Wil­li­ams] was a pro­fess­or at North­west­ern. I’d had more people to talk to at Uni­versity of Illinois at Chica­go. I’d nev­er been in a place like Uni­versity of Chica­go. It’s a very elit­ist place. Every­one thought very well of them­selves. I felt a lot of pres­sure, pos­sibly for the first time. The gradu­ate stu­dents, es­pe­cially the wo­men stu­dents, struggled.

S.-T. Yau was on the fac­ulty at the IAS in Prin­ceton at that time. His wife had a job in San Diego, and she didn’t want to live in New Jer­sey. So Yau planned a grand pro­ject to build a geo­metry group in San Diego. He in­volved lots of people, in­clud­ing my­self, Rick Schoen, Richard Hamilton, Le­on Si­mon. I spent a winter in San Diego. The moun­tains were close so you could ski. I took surf­ing les­sons, al­though I must have been al­most 40 at the time! I was all set to move to San Diego. And then everything fell apart. Maybe it was bound to fall apart. It wasn’t clear that every­body at San Diego was happy with this in­flux of people. This was around 1986.

Jack­son: Didn’t Mike Freed­man also go to San Diego?

Uh­len­beck: He was already there. Richard Hamilton was already there too and stayed for a few years. But the point is, I had kind of agreed to leave Chica­go. I had not signed any­thing, but I had got­ten it in my mind to leave. My move to the Uni­versity of Texas was in the wake of this. At that time Texas was try­ing out all the big shots. Den­nis Sul­li­van and S.-T. Yau spent semesters at the Uni­versity of Texas around this time. They in­vited an aw­ful lot of people to vis­it and made a big fuss about it. I guess I was at­trac­ted to the change. My hus­band grew up in Aus­tin and had fam­ily there. That was not the reas­on, but it cer­tainly was a pos­it­ive reas­on that a lot of oth­er people didn’t have to go there. It seemed like an ad­ven­ture. The pres­ence of Steve Wein­berg and Harry Swin­ney in phys­ics and Camer­on Gor­don in math­em­at­ics was at­tract­ive to both me and my hus­band.

And I really did not like com­mut­ing in Chica­go. When I was at Uni­versity of Illinois at Chica­go, I got an hour of walk­ing a day, to and from the train sta­tion, and I could work on the train. But go­ing to Uni­versity of Chica­go, it was a ques­tion of driv­ing. I would have stayed there if I’d been able to do all my com­mut­ing in the morn­ing, but that late night com­mute, es­pe­cially in the middle of winter, was bad. I’m an out­door per­son. I did not want to live in Hyde Park. We had a great big yard in Evan­ston. Also, the two Chica­go fac­ulty mem­bers I had be­friended, Peter Jones and Jerry Bona, both left. That cer­tainly in­flu­enced me. They were not in my field but close enough that we could talk.

Why does KdV arise everywhere?

[D]ue to what was now an addiction to intellectual excitement, I tried to follow the influence of physics on geometry which is associated with the name of Ed Witten. My work in integrable systems grew out of this connection with physics.

Jack­son: How did you get in­ter­ested in in­teg­rable sys­tems?

Uh­len­beck: It’s very im­me­di­ate, be­cause in­teg­rable sys­tems play an im­port­ant part in con­form­al field the­ory in phys­ics. I did learn quite a bit about in­teg­rable sys­tems and worked in the area, but my mo­tiv­a­tion was to un­der­stand things in phys­ics, and that didn’t pan out. That is not to say that I didn’t en­joy work­ing on it. My friend Chuu-Li­an Terng and I have writ­ten some nice pa­pers on in­teg­rable sys­tems. She’s mar­ried to my thes­is ad­viser, so I have known her for a long time. Al­though she is from Taiwan and grew up there, we have a lot of in­terests and ex­per­i­ences in com­mon. We’ve worked to­geth­er for quite a few years now. Our col­lab­or­a­tion is also partly con­nec­ted with the Wo­men in Math pro­gram.

Start­ing in the 1990s I got in­volved in a lot of non-math­em­at­ic­al things. I was one of the hand­ful of people who star­ted the Park City Math­em­at­ics In­sti­tute (PCMI). Dan Freed and I had be­come very friendly, and he moved to Texas. He was young and en­er­get­ic and wanted to get in­volved in all these things. So he in­stig­ated our in­volve­ment with the PCMI.

Through the PCMI I got in­volved with wo­men’s is­sues. I had nev­er been in­volved in them be­fore be­cause I wasn’t in­ter­ested in polit­ics. But at some point, I thought, Here I am, in my 40s, suc­cess­ful — Where are all the wo­men? We wo­men math­em­aticians thought, yes, it has been a little dif­fi­cult, people wer­en’t so friendly to us, but things are go­ing to change. But in the early 1990s, many of the wo­men roughly in my gen­er­a­tion were at that point the last wo­men hired in their de­part­ment. We didn’t see large num­bers of wo­men com­ing after us. I felt I did owe something for my suc­cess. So Chuu-Li­an Terng and I star­ted work­ing to­geth­er on in­teg­rable sys­tems, and we also star­ted work­ing in the Wo­men in Math pro­gram. It was ori­gin­ally as­so­ci­ated with the PCMI, and later the IAS took it on and sup­por­ted it. Chuu-Li­an and I did math­em­at­ics and also or­gan­ized the Wo­men in Math pro­gram.

Jack­son: That’s nice to do both.

Uh­len­beck: Yes! I was mo­tiv­ated by one of the mys­ter­ies to me in math­em­at­ics, which is why KdV comes up all over the place, in all sorts of geo­met­ric and phys­ic­al prob­lems.

Jack­son: Do you have any ink­lings of why that’s the case?

Uh­len­beck: Not really. The only guess I have is that it’s a nat­ur­al con­struc­tion on an in­fin­ite-di­men­sion­al tor­us some­where. So it ap­pears in a lot of places where you have an in­fin­ite num­ber of com­mut­ing vari­ables that are some­how in­volved in the struc­ture of a tor­us some­where. An­oth­er pos­sib­il­ity is that, once it ex­ists as a math­em­at­ic­al mod­el, it serves to ex­plain pat­terns that ap­pear in cal­cu­la­tions for a time. When this be­comes in­ad­equate, new mod­els are found or de­veloped that are more use­ful. One should nev­er for­get that math­em­at­ics is dif­fer­ent from what it is used to mod­el.

Jack­son: Can you tell me about your work with Chuu-Li­an on in­teg­rable sys­tems?

Uh­len­beck: We came to a cer­tain un­der­stand­ing of what scat­ter­ing the­ory and in­verse scat­ter­ing the­ory were for dif­fer­ent kinds of geo­met­ric PDEs, like KdV, sine-Gor­don, and so forth. We dis­covered that, rather than just a lot of com­plic­ated for­mu­las, there was ac­tu­ally a Riemann–Hil­bert prob­lem un­der­ly­ing most of the con­struc­tions. We were able to get defin­i­tions, which pleased us geo­met­ric­ally, of things like the Vi­ra­s­oro al­geb­ras and func­tions. So you could mo­tiv­ate a lot of the con­struc­tions that you could find in the lit­er­at­ure by factor­ing com­plex mat­rix func­tions.

Chuu-Li­an is really a clas­sic­al geo­met­er, not a PDE spe­cial­ist, so she knows a lot about the con­struc­tions of geo­met­ric sur­faces and geo­met­ric sub­man­i­folds that these equa­tions de­scribe. I was mo­tiv­ated by the fact that they ap­pear in quantum field the­or­ies. The two per­spect­ives helped our re­search.

Jack­son: Is there an area of math­em­at­ics where you think, “I would have loved to have worked in that”?

Uh­len­beck: Al­geb­ra­ic geo­metry. When I was a stu­dent at Bran­de­is, the al­geb­ra­ic geo­metry that was taught there was ab­stract. My gen­er­a­tion said there are two kinds of al­geb­ra­ic geo­met­ers: those who learned it from Hartshorne,20 and those who learned it from Grif­fiths and Har­ris.21 When I was a stu­dent, my op­por­tun­ity was to learn al­geb­ra­ic geo­metry from Hartshorne, which didn’t make any sense to me, where­as the Grif­fiths and Har­ris ap­proach has all these won­der­ful ex­amples of spaces. If I were to do it again, I would make a big ef­fort to learn al­geb­ra­ic geo­metry.

Jack­son: What is it about al­geb­ra­ic geo­metry that ap­peals to you?

Uh­len­beck: The in­ter­est­ing geo­met­ric struc­tures that come up.

Jack­son: Do you think geo­met­ric­ally?

Uh­len­beck: Yes and no. My think­ing is very con­crete, mean­ing whenev­er I work on a prob­lem, I al­ways do baby cal­cu­lus prob­lems around it, to see what is go­ing to hap­pen. I don’t think ter­ribly ab­stractly. I think a lot by ex­amples, find­ing simple con­structs that mod­el com­plic­ated phe­nom­ena. I am will­ing to learn the com­plic­ated stuff, but I like to be mo­tiv­ated by something that I can un­der­stand that is ba­sic­ally very simple. I also work a lot with in­equal­it­ies. I don’t name each term in an in­equal­ity with a word, but I think of them some­how as in­di­vidu­als. After you work at it a while, they are very per­son­al. They are your good friends.