Celebratio Mathematica

Tell me about your fam­ily, your early life.

I was born in Phil­adelphia, Pennsylvania on March 21, 1946. My fath­er was Ber­ney Lam­beth Mor­gan, known to all as “BL Mor­gan”. My moth­er, Ern­estine My­ers Mor­gan, was a home­maker. Mor­gan is a Welsh name. Lam­beth is a good Eng­lish name. I’m some mix­ture of Scotch, Ir­ish, Welsh, and Eng­lish.

My fath­er came from a poor fam­ily in north Texas. His fath­er worked meas­ur­ing strands of cot­ton for a cot­ton mill. In a small world story, my moth­er was in the same high school class in Hou­s­ton as George Mackey [math­em­atician at Har­vard]. Her fath­er was an ar­chi­tect. As the fam­ily le­gend has it, he had been in­volved in the design and con­struc­tion of the Cot­ton Bowl in Dal­las.

My fath­er gradu­ated from Rice Uni­versity in 1937 with a de­gree in Mech­an­ic­al En­gin­eer­ing. His first job was in New York work­ing for In­ger­soll Rand. After sev­er­al years in New York, he moved to Phil­adelphia to take a new job. My par­ents met when my fath­er was at Rice, but didn’t marry un­til after he moved to Phil­adelphia, where I was born. We left Phil­adelphia when I was about two and a half and moved back to Hou­s­ton, where I grew up. I had one sib­ling, a young­er sis­ter who, sadly, died pre­ma­turely at age 42.

Did you play sports?

Yes, I played all sorts of sports: In ele­ment­ary school, I played Pee Wee foot­ball and little league base­ball. In ju­ni­or high school, I played on the \( 5^{\prime}5^{\prime\prime} \) and un­der bas­ket­ball team.

You were only \( 5^{\prime} 5^{\prime\prime} \)?

Yes, I was short. Bas­ket­ball teams were di­vided in­to \( 5^{\prime}5^{\prime\prime} \) and un­der, and over \( 5^{\prime}5^{\prime\prime} \). Even as a seni­or — as a 9th grader — I was \( 5^{\prime}5^{\prime\prime} \) or \( 5^{\prime}6^{\prime\prime} \). In ju­ni­or high I was skinny and short. I shot up in high school.

Did you play bas­ket­ball?

Not for the high school team, I played in a church league. I star­ted play­ing a little vol­ley­ball dur­ing high school, just pick-up games. Then in col­lege, I played in­tra­mur­al sports: I played touch foot­ball, and I played vol­ley­ball and bas­ket­ball.

You men­tioned something that I was go­ing to pick up on. The church league, were you a church­go­er, your fam­ily?

My fam­ily, yes. We went to church pretty much every Sunday. I don’t think any one of us was very re­li­gious but this was just what you did.

Did you think of go­ing any­where else be­sides Rice Uni­versity?

Yes, I ap­plied to three col­leges. One of them was a safety: that was Stan­ford, back in the days when it was easi­er to get in­to. The oth­er two were Rice and Cal­tech. I was ad­mit­ted to Rice and Cal­tech and not to Stan­ford!

Be­fore de­cid­ing on Rice, I ser­i­ously thought about go­ing to Cal­Tech. I think now, “What would my life have been like if I’d gone to Cal­tech in­stead of Rice?” En­ter­ing col­lege, I thought that I was go­ing to be a chem­ic­al en­gin­eer or maybe a chem­ist. I’d won a city-wide prize in chem­istry that provided a small fel­low­ship if I were to ma­jor in chem­istry or chem­ic­al en­gin­eer­ing. In high school, the hard­est and most in­ter­est­ing sci­ence class was chem­istry, mainly be­cause we had a great teach­er. I didn’t really know any­thing about pure math­em­at­ics, and I cer­tainly didn’t know that one could make it a pro­fes­sion.

But back to the point. If I’d gone to Cal­tech, what would have happened? At Cal­tech, not to speak ill of the dead, the math de­part­ment was a some­what funny place and not what I con­sidered to be at the fore­front of the kind of math­em­at­ics that ended up in­ter­est­ing me. Phys­ics at Cal­Tech, on the oth­er hand, was full of world fam­ous people mak­ing great ad­vances to­ward what is now, 60 years later, called the Stand­ard Mod­el of Particle Phys­ics. The fac­ulty in­cluded Feyn­man and Gell-Mann to name two su­per­stars. I’m sure I would have ended up be­ing a phys­ics ma­jor. It is now clear to me, after 30 years of watch­ing and par­ti­cip­at­ing in the in­ter­ac­tions between math­em­aticians and phys­i­cists, that I’m a hell of a lot bet­ter a math­em­atician than I ever would have been a phys­i­cist.

Now, maybe if I’d gone to phys­ics gradu­ate school and learned what phys­ics gradu­ate stu­dents learn, I would have blos­somed as a phys­i­cist. But in my heart of hearts, I know that I’m more a math­em­atician — spir­itu­ally a math­em­atician — than I am a phys­i­cist.

That’s in­ter­est­ing. What year did you gradu­ate?

I was in the class of 1968, so I star­ted in the fall of 1964.

I lucked out at Rice. The uni­versity had just hired a cadre of young to­po­lo­gists: El­don Dyer, Les Glaser and Ed Con­nell were there and were do­ing ex­cit­ing al­geb­ra­ic and geo­met­ric to­po­logy. I was im­me­di­ately drawn to that.

I see, that’s in­ter­est­ing. Miles Tier­ney came for a year.

Yes, and you were there; Steve Ger­sten ar­rived.

Yes, I was there, be­cause Dyer came, and I was his stu­dent, so I was there for 1964–65 and that was your fresh­man year.

I was tak­ing what’s called “Hon­ors Fresh­man Cal­cu­lus” from Dyer. I had no idea what math­em­at­ics was. I’d seemed to have a cer­tain abil­ity for it but it was so for­eign to me, I couldn’t keep track of the terms. I con­tinu­ally con­fused “com­pact” and con­nec­ted”. The words soun­ded so sim­il­ar to me.

He was do­ing that in your cal­cu­lus class?

Yes, it was be­gin­ning hon­ors cal­cu­lus. The course was ba­sic­ally a rig­or­ous in­tro­duc­tion to mul­tivari­able cal­cu­lus with some ba­sic real ana­lys­is thrown in. We were all fresh­men, six or eight of us. Any­way, it was based on a text book that was just about to come out. Dyer had a copy of the book, but we didn’t. I re­mem­ber one prob­lem on the first semester fi­nal, where I fi­nally began to get a glimpse of what was go­ing on. I can’t even re­mem­ber how things were defined, but the prob­lem was to prove that, for a man­i­fold in Eu­c­lidean space, its tan­gent bundle and its nor­mal bundle were per­pen­dic­u­lar. As I worked through this prob­lem, I sud­denly began to get the pic­ture, and maybe that was the be­gin­ning of un­der­stand­ing real math­em­at­ics for me.

At the end of my fresh­man year, I still wasn’t sure what I was go­ing to ma­jor in. The math in Dyer’s class had been fun, but I still thought of Chem­istry and/or Phys­ics as a bet­ter ma­jor. Hav­ing done well in chem­istry, I was in­vited to work in a chem­istry lab dur­ing the sum­mer. That ex­per­i­ence con­vinced me!! I was do­ing some ex­per­i­ments hav­ing to do with va­cu­um tubes. My va­cu­um tubes nev­er held a va­cu­um, and I couldn’t blow glass to save my life. My hands were a bloody pulp. I knew I was not cut out for labor­at­ory sci­ence.

My sopho­more year was when I really blos­somed as a math­em­atician. I was tak­ing Con­nell’s course — it was a course on group the­ory out of Her­stein’s book. I re­mem­ber a prob­lem that I didn’t get right on the first ex­am: Are all Abeli­an groups of or­der 25 iso­morph­ic? I struggled and struggled and struggled, but I didn’t get it. A first in­dic­a­tion, I think, that I would be more of a geo­met­ric, to­po­lo­gic­al math­em­atician than an al­geb­ra­ic one.

Con­nell was very in­ter­ested in sur­gery the­ory and was fol­low­ing what [Wil­li­am] Browder and [Den­nis] Sul­li­van were do­ing in Prin­ceton. He was sure you could tri­an­gu­late man­i­folds us­ing sur­gery the­ory. That was his goal in life: to tri­an­gu­late man­i­folds. Oc­ca­sion­ally, he would sup­ple­ment the group the­ory course with in­tro­duct­ory ma­ter­i­al from sur­gery the­ory. I re­mem­ber he taught us what a loc­ally trivi­al fiber bundle is.

I star­ted talk­ing to him in the even­ings about sur­gery the­ory, and he gave me an in­form­al course on the sub­ject. The sum­mer after my sopho­more year, Sul­li­van came through town on his way to Berke­ley. He gave a series of lec­tures. I re­mem­ber that, in one of them, he got stuck on some ques­tion in lin­ear al­gebra: He had a sub­space and he wanted to show that some oth­er sub­space pro­jec­ted onto the quo­tient iso­morph­ic­ally. So he said, “I can’t do this, how do you do it?” So I said, “You just do this…”. And he said, “Ah, young people. I’m too old to do stuff like that.” Den­nis, at 25!! Any­way, I was com­pletely hooked by all of this sur­gery the­ory.

Start­ing in my ju­ni­or year, I took gradu­ate math courses. It was al­most like I was in the European aca­dem­ic sys­tem. To the ex­tent I could, I didn’t take any oth­er courses ex­cept math courses from then on.

To go back, just for a mo­ment, were you va­le­dictori­an of your high school or high up [in rank­ing]?

No, I wasn’t. If I re­mem­ber the num­bers cor­rectly, my high school had roughly 550 stu­dents. I think that my class rank by grade point av­er­age was in the low fifties. I was in the top 10%, but barely.

You prob­ably got A’s in your sci­ence classes?

Yes, I got A’s in all the sci­ence classes.

My fresh­man year at Rice, I had three one pluses [A+] in math, phys­ics, chem­istry, and two threes [C] in his­tory and Eng­lish.

Yet you like his­tory and read a lot, but back then, you wer­en’t tak­ing [those courses] very ser­i­ously.

I took them ser­i­ously; I didn’t really un­der­stand what the sub­jects were about. Es­pe­cially his­tory; his­tory was very hard for me. I just didn’t see the point, mem­or­iz­ing a bunch of stuff — dates, names of wars and gen­er­als. Of course, that’s not what his­tory is about, but that’s what I thought it was about then.

Dur­ing my ju­ni­or year, I con­tin­ued my even­ing ses­sions on sur­gery the­ory with Con­nell. One even­ing I went in­to his of­fice and he said, “I’ve been giv­ing or­al ex­ams to the gradu­ate stu­dents and they’re just not do­ing well. I don’t know what’s go­ing on.” I said, “What sort of ques­tions are you ask­ing?” He star­ted ask­ing the ques­tions, I answered them. He said, “Okay, you just passed your to­po­logy qual­i­fy­ing ex­am (qual).” It was al­most like he was my thes­is ad­visor, even though I was a ju­ni­or, not a gradu­ate stu­dent!

This was the PhD or­als?

Yes, the PhD quals.

But you wer­en’t in the PhD pro­gram yet?

No.

It was up to him? If he said you’ve passed, that was a fact?

Yes. It was very in­form­al. Later, I took a more form­al or­al ex­am in al­gebra. Ger­sten was ask­ing me ques­tions about al­geb­ra­ic in­tegers, poly­no­mi­al equa­tions, etc. I was do­ing okay. Then at the end he asked, “What’s the spec­trum of a ring.” I said, “No idea.” He said, “You bet­ter learn that, but you passed.”

There was a third one, must have been ana­lys­is. But I don’t even re­mem­ber it.

At some point, you be­came a grad stu­dent?

Even­tu­ally, but at this point, I had just fin­ished my ju­ni­or year. Dur­ing the sum­mer after my ju­ni­or year, Con­nell left for Berke­ley to be a seni­or Miller In­sti­tute Fel­low, so I was left alone math­em­at­ic­ally.

In my seni­or year, Browder’s pa­per on man­i­folds of Ker­vaire in­vari­ant one [e1] came out. I spent a large por­tion of the year read­ing that pa­per. But to do that, I had to go back to all the Cartan sem­inars on co­homo­logy of \( K(\mathbb{Z}/2,n) \)’s. I was read­ing those pa­pers and try­ing to un­der­stand what Browder had done. It in­volved a lot more al­geb­ra­ic to­po­logy than I knew when I star­ted the pro­ject. Dur­ing that year, I also star­ted think­ing about open ques­tions in sur­gery the­ory.

At the end of my seni­or year, I got a BA and I was all set to go to Prin­ceton as a gradu­ate stu­dent. But, if I had done that, I would have been im­me­di­ately draf­ted for the Vi­et­nam war. I talked to the De­part­ment Chair about my situ­ation. He offered me an In­struct­or­ship in Math­em­at­ics at Rice for the fol­low­ing year (1968–1969). I ac­cep­ted the po­s­i­tion and taught there for the year, which pro­tec­ted me from the draft. Dur­ing that year, I wrote my thes­is about ho­mo­topy equi­val­ences that pull back the stable nor­mal bundle of the range to the stable nor­mal bundle of the do­main. I called these maps “tan­gen­tial ho­mo­topy equi­val­ences.”

So you got your PhD from Rice, with [Mor­ton] Curtis.

Ex­actly.

After re­ceiv­ing my PhD at Rice, in the Fall of 1969, I went to Prin­ceton as an in­struct­or.

Did you ap­ply to a bunch of places?

I ap­plied to Prin­ceton and MIT, and I in­ter­viewed at those two places. I don’t think I ap­plied any­where else. Curtis nom­in­ated me for a Har­vard Ju­ni­or Fel­low­ship, but I did not re­ceive that.

You were there for three years?

Yes. Teach­ing gave me a con­tinu­ing ex­emp­tion from the draft un­til the lot­tery came out in the Decem­ber, 1969. For­tu­nately, I re­ceived a high num­ber, so I was spared any fur­ther wor­ries about be­ing draf­ted, and I had three years as an in­struct­or at Prin­ceton.

Cap­pell was around?

He was a gradu­ate stu­dent at Prin­ceton and re­ceived his PhD in the sum­mer of 1970. It was clear that he was equi­val­ent to any young fac­ulty mem­ber. [Ju­li­us] Shaneson was there as an as­sist­ant pro­fess­or. At some point, Cap­pell and Shaneson star­ted work­ing to­geth­er — as far as I know, a col­lab­or­a­tion that con­tin­ues today.

Let me add one thing about this whole peri­od. I feel like I really missed out. I would have much pre­ferred to go to gradu­ate school at Prin­ceton, but that was not to be be­cause of the draft loom­ing over me. Of course, be­ing an in­struct­or had ad­vant­ages over be­ing a gradu­ate stu­dent. It was nice to be paid; and it was nice not to have to sub­mit to or­al ex­ams. But what I missed is the re­laxed learn­ing that you can do as a gradu­ate stu­dent that you can’t really do later. As a gradu­ate stu­dent, you have a lot of time to learn from your fel­low gradu­ate stu­dents. In the end, it wasn’t a ter­rible det­ri­ment to me but I have al­ways re­gret­ted that I had to take the path I did. I feel like there were, and prob­ably still are, big holes in my math­em­at­ic­al know­ledge that would have been remedied, at least par­tially, if I’d been a gradu­ate stu­dent, hanging around the com­mon room, talk­ing to oth­er gradu­ate stu­dents, in­stead of talk­ing to Browder, Shaneson, and Sul­li­van.

I re­mem­ber a time in the Prin­ceton com­mon room, when [Phil­lip] Grif­fiths, who was a fac­ulty mem­ber in the de­part­ment, told me that a com­pact, com­plex sub­man­i­fold of a Kähler man­i­fold is ho­mo­lo­gic­ally non-trivi­al. I couldn’t be­lieve it. Speak­ing of a hole in my back­ground! The ma­gic of the Kähler form! One of the best things that came out of those three years at Prin­ceton was — thanks to Sul­li­van — Grif­fiths and I star­ted a math­em­at­ic­al con­ver­sa­tion.

There’s that book with the four au­thors…

It’s not a book, it’s a pa­per [2]. And it’s a great story. This all star­ted when Sul­li­van came down to Prin­ceton in the spring of 1971 from MIT and gave lec­tures on ra­tion­al ho­mo­topy the­ory. His point was that you use dif­fer­en­tial forms to do ho­mo­topy the­ory over the ra­tion­als. He had vari­ous nice ex­amples, and he said he was sure that it must have an in­ter­est­ing ap­plic­a­tion to Kähler man­i­folds.

This made Grif­fiths perk up, think­ing that maybe this would help with the Hodge Con­jec­ture, be­cause dif­fer­en­tial forms are giv­ing you ho­mo­topy the­or­et­ic in­form­a­tion that goes bey­ond ho­mo­logy. Grif­fiths used to come to my of­fice to get his to­po­logy les­son al­most every day. In re­turn, he would give me a Kähler man­i­folds, al­geb­ra­ic geo­metry les­son.

We star­ted work­ing to­geth­er that spring and con­tin­ued in the fall of 1972 when Grif­fiths moved to Har­vard, and I moved to MIT. It wasn’t clear what we were try­ing to prove. We had vari­ous spe­cial cases of what is now called “Form­al­ity”. The ba­sic idea was, if you think — as De­ligne did — in terms of ei­gen­val­ues of a Frobeni­us, you make an ana­logy between com­plex vari­et­ies and vari­et­ies in char­ac­ter­ist­ic \( p \), where there’s a Frobeni­us. The norm of its ei­gen­val­ues on co­homo­logy goes mul­ti­plic­at­ively with the di­men­sion. You could have cup products be­cause that op­er­a­tion is ho­mo­gen­eous with re­spect to di­men­sion. But a Mas­sey product is not ho­mo­gen­eous. It takes classes of di­men­sions \( a \), \( b \), and \( c \) and pro­duces a class of di­men­sion \( a + b + c -1 \). You can’t have a non-trivi­al Mas­sey product, be­cause the product has to com­mute with the ac­tion of the Frobeni­us and the ei­gen­val­ues don’t match.

Of course, we were work­ing with com­plex man­i­folds, so there is no Frobeni­us op­er­at­or. But the Hodge de­com­pos­i­tion can be used in this con­text in­stead of the ei­gen­value ar­gu­ment. Grif­fiths and I had vari­ous spe­cial cases of a much more gen­er­al ar­gu­ment. (What I just de­scribed above is hind­sight.)

De­ligne came to Cam­bridge to give a lec­ture in late 1972. Af­ter­wards, we’re sit­ting in Grif­fiths’ of­fice ex­plain­ing to him what we’ve done. He says, “I’m not sure about all that, but I do know one thing: the Mas­sey products van­ish be­cause of the Frobeni­us ana­logy.” Grif­fiths and I looked at each oth­er and two minutes later, we had the en­tire ar­gu­ment.

It was now ob­vi­ous to us that this prin­ciple — we called it the prin­ciple of two types — that we had worked out in vari­ous spe­cial cases, was true in com­plete gen­er­al­ity. This be­came the form­al­ity res­ult. When Sul­li­van heard about this ar­gu­ment, he found quickly a much sim­pler, slick­er ar­gu­ment. The four of us (De­ligne, Grif­fiths, Sul­li­van, and my­self) wrote a pa­per present­ing both proofs. Later, I gen­er­al­ized the prin­ciple of two types to non-com­pact, com­plex vari­et­ies [3].

I see. De­ligne’s par­ti­cip­a­tion was: he came and said the secret words.

He said the secret words which made clear the breadth of ap­plic­ab­il­ity of the ideas we had. Af­ter­wards, Grif­fiths said to me, “That’s not the first time De­ligne has done that to me!”

Where was Grif­fiths at that time?

He was at Har­vard. In 1972. I went from Prin­ceton to MIT; Grif­fiths went from Prin­ceton to Har­vard; Sul­li­van and [Daniel] Quil­len were both at MIT. I’d hes­it­ated between go­ing to MIT and go­ing to Berke­ley to be a Miller Fel­low.

In those days you were also col­lab­or­at­ing with Sul­li­van?

Sul­li­van and I did work while I was at Prin­ceton. I went up to Bo­ston to see him one time. He posed a prob­lem that he had been think­ing about. In my naïveté, I thought I had the solu­tion, when in fact, I had noth­ing be­sides a tri­vi­al­ity. Nev­er­the­less, this sparked a con­ver­sa­tion that turned in­to a pa­per. This was a pa­per on \( Z/n \)-man­i­folds [1]. These are some of my fa­vor­ite things.

Then I was at MIT for two years, 1972–74. Dur­ing the second of those two years, I was try­ing to un­der­stand De­ligne’s pa­per, “Hodge the­ory, II” [e2], so that I could gen­er­al­ize what the four of us had done for com­pact Kähler man­i­folds to open al­geb­ra­ic vari­et­ies, which I even­tu­ally did. I re­mem­ber sit­ting in my apart­ment in the South End of Bo­ston, just por­ing over De­ligne’s pa­per. It was slow go­ing but I fi­nally got there. That’s what I did for most of the aca­dem­ic year 1973–1974.

By this time, Sul­li­van had de­camped to IHES out­side of Par­is (where De­ligne was). I re­ceived a Sloan Found­a­tion Fel­low­ship and de­cided to use it to go to IHES and join them. I spent the first year at the IHES writ­ing up what I had done the pre­vi­ous year. In my second year, I taught a gradu­ate course at the Uni­versity of Par­is, Or­say. In French! That was a lot of fun.

I didn’t know you knew French that well.

Well, I didn’t really. But it was a great learn­ing ex­per­i­ence. Math­em­at­ic­ally, IHES wasn’t a good place for me. The first year I was busy writ­ing, so the math­em­at­ic­al en­vir­on­ment was not so im­port­ant. But the second year, there was too much go­ing on math­em­at­ic­ally. I felt like a particle in Browni­an mo­tion. I wasn’t think­ing ser­i­ously about any one thing, and I didn’t know what dir­ec­tion to pur­sue next. The up­side was get­ting to meet and in­ter­act with an in­cred­ible panoply of the world’s best math­em­aticians, in­clud­ing the per­man­ent mem­bers Sul­li­van, [Misha] Gro­mov, and [Alain] Connes, and the vis­it­ors like [Nick] Katz and [Spen­cer] Bloch. But since I did not have a well-defined pro­gram that I was work­ing on, I couldn’t find my feet in all this bril­liance.

Then in the Spring of 1976, [Robert] MacPh­er­son vis­ited IHES. One day he ex­plained the [Marc] Goresky–MacPh­er­son ideas on in­ter­sec­tion ho­mo­logy. It im­me­di­ately oc­curred to me how to gen­er­al­ize slightly their res­ults to do something Sul­li­van had tried to do in the late 1960s. I sat down and worked it out and ex­plained it to Sul­li­van. I nev­er wrote it up. A couple of years ago, [Graeme] Segal of Ox­ford was at the Si­mons Cen­ter. For some reas­on, we got to talk­ing about the good old sur­gery days, and I ex­plained to him what I had done but nev­er pub­lished. He was really in­ter­ested, he said, “Just write it up.” So I wrote it up, but I haven’t even put it on the web.

Is it now more ex­pos­it­ory, or is it new?

It is a dif­fer­ent, more geo­met­ric proof of a sem­in­al res­ult that Sul­li­van proved by ho­mo­topy the­or­et­ic means in the 1960s. He had tried to find a dir­ect geo­met­ric proof but he could not make it work. For any nor­mal map, you should find a col­lec­tion of, let’s say, sub­man­i­folds of the range, and put the map trans­verse to the sub­man­i­folds. For each sub­man­i­fold you get nu­mer­ic­al sur­gery in­vari­ant — either a sig­na­ture in­vari­ant or a Ker­vaire in­vari­ant. The col­lec­tions of these in­vari­ants should com­pletely de­term­ine the ori­gin­al nor­mal map up to nor­mal cobor­d­ism. Such a res­ult would be called the char­ac­ter­ist­ic vari­ety the­or­em. But it doesn’t work, at least as I have for­mu­lated it here.

In the end, Sul­li­van proved a ver­sion of this state­ment by us­ing ho­mo­topy the­ory, Con­ner–Floyd the­ory, and peri­od­icity in real \( K \)-the­ory. What I real­ized is that one could use in­ter­sec­tion ho­mo­logy to build sin­gu­lar spaces that sat­is­fied Poin­caré du­al­ity, and then one could do the same nor­mal map con­struc­tion us­ing these ob­jects. They formed a bor­d­ism the­ory dual to the the­ory of ho­mo­topy classes of maps from a space to \( G/\textit{TOP} \).

That’s what you wrote but nev­er even pos­ted?

I have been re­vis­it­ing it re­cently and maybe someday soon I will post it — it’s only 50 years out of date!

Hav­ing the bor­d­ism the­ory dual to \( G/\textit{TOP} \) — to me that’s a nice state­ment. And then, I’ve al­ways felt that you can prove that, at odd primes, this is the bor­d­ism the­ory dual to K -the­ory, us­ing [Jeff] Chee­ger’s work giv­ing a ver­sion of in­ter­sec­tion ho­mo­logy us­ing dif­fer­en­tial forms with de­cay con­di­tions along the bound­ary. But that is still a fu­ture pro­ject for me.

You went to Columbia and star­ted to get in­ter­ested in — ?

3-man­i­folds.

Right, and that really turned in­to \( \mathbb{R} \)-trees.

That’s right. Thanks to [Peter] Shalen. He and [Marc] Cull­er had stud­ied the case of a curve of rep­res­ent­a­tions in­to \( \operatorname{SL}(2,R) \). They showed that you get a com­pac­ti­fic­a­tion of this curve and the points at in­fin­ity come from ac­tions of the group on an or­din­ary tree. Shalen was vis­it­ing Columbia for the year (1981–1982). He ap­proached me and asked, “Are you will­ing to think about the case when the space of rep­res­ent­a­tions is high­er di­men­sion­al? It should be re­lated to one of the two big res­ults in Thur­ston’s the­or­em about the ex­ist­ence of hy­per­bol­ic struc­tures, namely the al­geb­ra­ic con­ver­gence of a se­quence of rep­res­ent­a­tions of the fun­da­ment­al group of a 3-man­i­fold with tor­us bound­ary.” In his view that’s what these trees would be about: What’s the se­quence of hy­per­bol­ic rep­res­ent­a­tions go­ing to con­verge to if it doesn’t con­verge to a rep­res­ent­a­tion? The an­swer should be that it con­verges to an ac­tion on an \( \mathbb{R} \)-tree.

Shalen and I de­veloped a gen­er­al the­ory of \( \mathbb{R} \)-trees and ac­tions of groups on them. Both of us also did work on this sub­ject with Cull­er and oth­ers. Then came the chal­lenge of ap­ply­ing this to prove Thur­ston’s al­geb­ra­ic con­ver­gence. That is the hard­est math­em­at­ic­al work I have ever done. I wouldn’t claim that this is the most im­port­ant or deep­est res­ult I have ever proved, but it cer­tainly was the most dif­fi­cult.

I think I may have asked you once what your fa­vor­ite pa­pers were. You brought up \( \mathbb{R} \)-trees, which in a way sur­prised me. I knew of all of the oth­er stuff you’d done. I wouldn’t have guessed \( \mathbb{R} \)-trees my­self.

If I re­mem­ber cor­rectly, this is what is men­tioned in my brief cita­tion when I was in­duc­ted in­to the Na­tion­al Academy of Sci­ences.

I’m most proud of the Hodge the­ory work and later work with [Bob] Fried­man on el­lipt­ic vari­et­ies. That’s more where I feel my heart is. A mix­ture of geo­metry and to­po­logy. It was hard to un­der­stand De­ligne’s work, but once I un­der­stood his work, it wasn’t that hard to sort out my ap­plic­a­tion. Where­as, the ap­plic­a­tion of \( \mathbb{R} \)-trees to Thur­ston’s al­geb­ra­ic con­ver­gence was, from scratch, hard. Shalen and I made real pro­gress the year he was at Columbia. But to fin­ish it off took an­oth­er sev­er­al years. I would vis­it him in Chica­go and he would vis­it me in New York or Bo­ston, and we would work es­sen­tially non-stop dur­ing our times to­geth­er. It was in­cred­ibly dif­fi­cult put­ting all the pieces of the ar­gu­ment to­geth­er.

In the end, there are oth­er ways to think about all this. Gro­mov had a way of get­ting an ul­tra fil­ter as­so­ci­ated with hy­per­bol­ic space it­self. Maybe it isn’t quite as com­plic­ated as we had made it out to be.

Well, that prob­ably took you up un­til 1982 when Don­ald­son ap­peared.

That’s right. When Don­ald­son ap­peared in 1982, I had the re­ac­tion that Kar­en Uh­len­beck once ex­pressed, which, was, “Oh, this is a beau­ti­ful the­or­em, but maybe it’s just one point where these two sub­jects (low di­men­sion­al to­po­logy and Yang–Mills the­ory) hap­pen to touch by some mir­acle that we don’t un­der­stand…”. But in ‘84 I was at MSRI and talk­ing with my col­league from Columbia, Bob Fried­man. Danny Ruber­man came in one day and said, “I just got this pre­print from Don­ald­son [e5], and I can’t un­der­stand what it’s about, but I know you can ex­plain it to me.” It was the Dol­gachev sur­faces. Dol­gachev proved that the Dol­gachev \( E(1)_{\{2,3\}} \) was not dif­feo­morph­ic to the ra­tion­al sur­face \( E(1) \), even though they are ho­mo­top­ic­ally equi­val­ent.

This is his the­or­em about \( h \)-cobor­d­isms.

That’s right. It is the fail­ure of \( h \)-cobor­d­ism; start with the el­lipt­ic sur­face \( E(1) \) over \( CP^1= S^2 \). When you do two log trans­forms of or­ders 2 and 3, you get an­oth­er com­plex sur­face \( h \)-cobord­ant to, but not dif­feo­morph­ic to, \( E(1) \).

I’m cer­tainly grate­ful to Danny for point­ing this Don­ald­son pa­per out to me be­cause it felt right up my al­ley. It is the kind of math­em­at­ics that al­ways ap­peals to me. I star­ted talk­ing to Fried­man. I think, if we had both been at Columbia, we nev­er would’ve star­ted col­lab­or­at­ing. We would have both been too busy. But at MSRI, we had time to sort out what was go­ing on. I’ve had no oth­er peri­od in my life like this. We’d go down the hill from MSRI for lunch every day and eat some­where on Tele­graph Av­en­ue. Dur­ing lunch, we would prove an­oth­er the­or­em. Every day, an­oth­er the­or­em. Just like apples fall­ing off the tree.

After our year at MSRI was over, we both re­turned to Columbia and con­tin­ued our col­lab­or­a­tion. Once we got star­ted, we kept it go­ing at Columbia for a long time. In the late 1980s, we wrote a huge tome on al­geb­ra­ic sur­faces and Don­ald­son in­vari­ants.1

At some point, you taught a course at Prin­ceton.

Yes. For sev­er­al years I was lec­tur­ing at Prin­ceton a couple of days a week. That’s how I got Peter [Oz­sváth] as a gradu­ate stu­dent. Then he con­vinced his Hun­gari­an friends to talk to me. [Zoltán] Szabó and [An­drás] Stip­sicz had come from Hun­gary to­geth­er and were at Rut­gers for gradu­ate school.

I used to say that I had two Hun­gari­an gradu­ate stu­dents, one with mul­ti­pli­city two. When Stip­sicz, Szabó, and I first star­ted meet­ing, Stip­sicz, the more talk­at­ive of the two, would start the con­ver­sa­tion and hold the floor while Szabó re­mained si­lent. Only at cru­cial mo­ments would Szabó pipe up. It took me a while to real­ize the pat­tern; to see Szabó as hav­ing his own voice, and to un­der­stand how spe­cial that voice was. Of course, all three have gone on to have stel­lar ca­reers in math­em­at­ics. Hav­ing had three stu­dents of this caliber was a great gift. It is one of the high points of my math­em­at­ic­al ca­reer.

I was go­ing to say that one of the things I re­gret — one of the missed op­por­tun­it­ies — was that Fried­man and I had com­puted, among oth­er things, the Don­ald­son poly­no­mi­als for the \( K^3 \) sur­face, which wasn’t very hard be­cause they all had to be powers of the quad­rat­ic form. We saw that the \( n \)-th one was \( Q^n/2n! \), but we nev­er wrote down the form­al series. If we’d writ­ten the series down, that would’ve led us to­ward the Kron­heimer–Mrowka struc­ture the­or­em, or at least ques­tions in that dir­ec­tion. We nev­er con­sidered am­al­gam­at­ing all the terms to­geth­er in a power series!

That’s a curi­ous miss for some­body like you.

We wer­en’t think­ing in that way. I nev­er con­sidered am­al­gam­at­ing all the Don­ald­son in­vari­ants to­geth­er in a power series. That is something the phys­i­cists do as second nature, and something that Kron­heimer–Mrowka thought to do. I’ll al­ways re­gret that over­sight.

In a sim­il­ar vein, when Fin­tushel and Stern did the blow-up for­mu­las by blow­ing up twice, I thought, “Oh my god! What a great ar­gu­ment! Why didn’t I think of that?” But I was nowhere close to think­ing along those lines.

Oz­sváth’s thes­is prob­lem was the blow-up for­mula. Can you com­pute more terms? One day Oz­sváth told me, “You know, I’ve been talk­ing to all my to­po­logy gradu­ate stu­dent friends and it seems like there are only two prob­lems to­po­logy gradu­ate stu­dents are work­ing on: it’s either the blow-up for­mu­las or the glu­ing for­mu­las.”

In fact, Tom [Mrowka] and I, we were try­ing to do glu­ing for­mu­las along the 3-tor­us for Don­ald­son the­ory. One day, I thought to my­self, “How would Fin­tushel and Stern do this? They would ap­proach it in a com­pletely dif­fer­ent way than we were.” So, I asked Stip­sicz and Szabó to study an al­tern­at­ive ap­proach in­spired by what Fin­tushel and Stern had done. Mrowka and I, on the one hand, and Stip­sicz and Szabó, on the oth­er hand, were both work­ing on glu­ing the­or­ems along the 3-tor­us from dif­fer­ent per­spect­ives. Both teams got their an­swers at ex­actly the same time. We com­pared an­swers. Sure enough, they were the same.

When is this? I al­ways liked the the­or­em of Fin­tushel and Stern where they got the Al­ex­an­der poly­no­mi­al in­to it. They cut out the fiber of a el­lipt­ic sur­face. The bound­ary then is a 3-tor­us and they glue in a knot com­ple­ment cross the circle. Then the Al­ex­an­der poly­no­mi­al ap­pears in the Don­ald­son in­vari­ants. Where was that in this story?

When I was think­ing, “How would they [Fin­tushel and Stern] do glu­ing for­mu­las along the 3-tor­us,” that was ex­actly the ar­gu­ment that I had in mind. Us­ing the cores, i.e., the nuc­le­us.

The nuc­le­us was [Bob] Gom­pf’s no­tion. He at­trib­utes that back to when he was talk­ing to Mrowka.

OK, in any case, Fin­tushel and Stern did great work here.

In the early to mid-1990s, I was work­ing with Szabó. Our idea was to use glu­ing for­mu­las along a sur­face times \( S^1 \) to prove a vari­ant of the Thom Con­jec­ture. Szabó and I had sketched out a plan of at­tack us­ing Don­ald­son the­ory. It was go­ing to be com­plic­ated and it wasn’t com­pletely clear we were go­ing to get all the way there, but at least we had an out­line of how we were go­ing to go about it. Then, in the fall of 1994, I vis­ited Mrowka at Cal­Tech, and he told me about the newly cre­ated Seiberg–Wit­ten (SW) the­ory. Im­me­di­ately after that, I was vis­it­ing Har­vard, and I talked to [Cliff] Taubes about the plan that Szabó and I had laid out for the Thom Con­jec­ture. Taubes and I agreed that it would be much more feas­ible to carry out this plan us­ing SW the­ory in­stead of Don­ald­son the­ory. Ma­gic­ally, all the obstacles that Szabó and I had fore­seen in the Don­ald­son-the­ory ap­proach dis­ap­peared when we re­placed Don­ald­son the­ory by SW the­ory. Taubes ex­plained to me the ba­sic ins and outs of SW the­ory. I said, “Well, let’s write it out.” That must have been dur­ing the week. When I got back to New York, I had a long con­ver­sa­tion with Szabó in which we cla­ri­fied sev­er­al parts of the ar­gu­ment us­ing SW the­ory. Then on Sunday, I got a call from Taubes; he asked me sev­er­al tech­nic­al ques­tions, which I answered. Then he said, “I think it’s all right but, now I have to tell you, that Kron­heimer and Mrowka have an­nounced a proof of the Thom Con­jec­ture.”

The two proofs were dif­fer­ent. In­deed, the two ar­gu­ments proved dif­fer­ent, but closely re­lated, things. We proved the res­ult for al­geb­ra­ic sur­faces with \( b^+ > 1 \) and classes of pos­it­ive self-in­ter­sec­tion, where­as Kron­heimer–Mrowka proved the ori­gin­al Thom Con­jec­ture for \( CP^2 \).

Which ap­par­ently Thom nev­er made.

What?

When I was talk­ing to [Arnold] Kas back in 1973, we tried to find out why it was called the Thom Con­jec­ture. Did Thom ever say this? Wasn’t in print. Nobody knew why, but the Thom Con­jec­ture is so catchy that you knew the name wasn’t go­ing to go away.

Then at some point, you star­ted talk­ing to Wit­ten?

Yes. Fried­man and I had stud­ied \( \operatorname{SL}(2) \)-bundles on el­lipt­ic sur­faces. Wit­ten came to Columbia to re­ceive an hon­or­ary de­gree, and I was des­ig­nated to es­cort him. He was sup­posed to be there by 10:00 AM. No Wit­ten. 10:15, no Wit­ten. The Hon­or­ary de­gree re­cip­i­ents were go­ing to march out to the com­mence­ment ce­re­mon­ies at 10:45. At 10:25, Wit­ten shows up. I said, “I’m really glad you came. I was a little wor­ried.” He said, “Have you ever con­sidered \( \operatorname{SL}(n) \)-bundles over the tor­us?” He wasn’t one to let any op­por­tun­ity go to waste.

That star­ted a three-way col­lab­or­a­tion of Fried­man, Wit­ten and me on what is called \( F \)-the­ory — namely, the study of vari­et­ies that are el­lipt­ic­ally fibered over some base. Fried­man and I used to go down to Prin­ceton once every couple of weeks to talk to Wit­ten. I saw him one day at the In­sti­tute. He was look­ing be­draggled and he said, “I’ve been writ­ing up our joint pa­per for the last two days and I’ve nev­er writ­ten any­thing so hard!” But of course, it took him just three days to write it up. That was the phys­ics ver­sion [4]. Fried­man and I spent six months writ­ing the math ver­sion [5]. The whole col­lab­or­a­tion was a lot of fun. Watch­ing Wit­ten do math up-close was eye-open­ing.

Then later…this is a great story…

Wit­ten asked Fried­man and me about com­mut­ing triples of ele­ments in a com­pact Lie group, which was re­lated to the things we’d done on \( G \)-bundles over the 2-tor­us. We thought about it and pretty much sor­ted it out and then wrote back to Wit­ten and said we’d worked it out. He said, “You should talk to Borel be­cause I asked him the ques­tion, too, and he is giv­ing me very sim­il­ar an­swers.” So, we went down to the In­sti­tute to talk to Borel. I’ll nev­er for­get it.

I was stand­ing at the board do­ing something about a tech­nic­al is­sue in root sys­tems, or Weyl cham­bers. I said, “You’re go­ing to do this and then you take the Levi (LEE-vi) factor.” It’s called the Levy (LEV-y) factor even though I mis­takenly said, Levi (LEE-vi). Borel looked at me dis­gustedly, and he said, “I hope you don’t say, “Lie [lye] group.” I think that if I had said, “Oh, you mean it’s not Lie [lye] group?” he would have thrown me out of his of­fice. Any­way, we con­tin­ued.

A little bit later in the con­ver­sa­tion Borel, the world’s lead­ing ex­pert in root sys­tems, says something in a tech­nic­al root-sys­tem ar­gu­ment. I said, “I don’t think that’s right. That’s not con­sist­ent with the pic­ture over here.” We talk about it for a minute and, sure enough, Borel has made a mis­take in an ar­gu­ment about root sys­tems. Borel looks at me and says, “Time to go to lunch!”

What year was that?

A year be­fore he died. Au­gust 2003.

Did you write pa­pers with Borel?

An AMS Mem­oir [6]. Fried­man and I would send him a manuscript to look at and make com­ments on. We’d get back, “I have no com­ments ex­cept for the fol­low­ing 18 dangling pre­pos­i­tion­al phrases.”

I nev­er ex­changed any word with Borel, I’m pretty sure. What kind of a guy was he?

He was very in­tim­id­at­ing, es­pe­cially to the young people around the In­sti­tute. A vign­ette that Mike Dav­is told me was that one year, when he, Mike, was fairly young, he was at the In­sti­tute and one of his friends went up to Lang­lands and Borel and said, “Do either of you know any­thing about rep­res­ent­a­tion the­ory?” Borel said, “You could say that.” He [the friend] said, “Do you mind if I ask you a stu­pid ques­tion?” Borel said, “You’ve already asked two.”

That’s Borel in a nut­shell. I told this story at a little in­form­al me­mori­al ser­vice for Borel in Switzer­land. His wife and daugh­ters were there. One of his daugh­ters said to me “Don’t you think that just covered up his in­sec­ur­ity?” I said, “Well, yes, I think it prob­ably did.” Be­cause he was, in fact, an in­cred­ibly gen­er­ous and gentle soul un­der this crusty ex­ter­i­or. And un­be­liev­ably loy­al. When El­lis Kol­chin died, we had a me­mori­al ser­vice for him on the Columbia cam­pus. Only one math­em­atician from out­side New York City came — Ar­mand Borel.

Let’s talk about Si­mons and his In­sti­tute [at Stony Brook]. How did you get in­volved?

Well, I heard that Si­mons was cre­at­ing a math/phys­ics in­sti­tute at Stony Brook. I didn’t know much about it. Den­nis talked to me and he said, “Would you be in­ter­ested in be­ing Dir­ect­or?” I said, “No, no. I’m happy where I am.” I didn’t want to leave Columbia. Ac­cord­ing to Den­nis, just as I was about to walk away, I said, “Well, is there any fun­drais­ing in­volved?” And Den­nis is in his own mind said, “Ah, look he really is in­ter­ested, though I didn’t really think I was.”

Was fun­drais­ing an at­trac­tion or dis­trac­tion?

Not hav­ing to fun­draise was an at­trac­tion. I hate fun­drais­ing and I’m not good at it. Six months later, either Jim or Den­nis ap­proached me and said they were look­ing for a dir­ect­or and they were also talk­ing to people about what one might look for in a dir­ect­or and also how the in­sti­tute should func­tion, what worked at oth­er places and would I come and talk to them. Just tell them my ex­per­i­ences. So I did.

You had been chair­man of the board at MSRI for a while.

Yes. I had been quite in­volved with the Board at MSRI, in­clud­ing as chair­man, over a peri­od of sev­er­al years.

I didn’t really know Si­mons well, but I went and talk to him and his com­mit­tee. [Mike] Douglas was there, Sul­li­van was there, Mar­tin Ro&ccaronek was there, Jim was there. It was in Jim’s of­fice in New York. Af­ter­wards, I went home and I say to El­len [my wife], “That was the weird­est ex­per­i­ence I’ve ever had. It was half pick your brain, which is what they said it was, and half job in­ter­view.”

I think they talked to a lot of people about the Cen­ter. Any­way, a couple of weeks go by and I said to El­len, “Well, if it was a job in­ter­view, I flunked be­cause I haven’t heard any­thing.” Then I got a call the next day from Si­mons, who’d been out of the coun­try for a while, and he said, “Well, we want you to be the dir­ect­or. I said, “I’m not in­ter­ested, Jim.” He said, “Let’s meet and have lunch and at least talk about it.” We met and had lunch. We talked about it, and at the end, I said, “I’m not really in­ter­ested.” He said, “Well, think about it.” I went home and told El­len the saga and she said, “You are SO tak­ing this job.” I do what the boss says!

There’s two mean­ings to that. She was say­ing that your be­ha­vi­or in­dic­ated that you had de­cided to take the job or you’re tak­ing this job be­cause…

Yes, it was surely a com­bin­a­tion of the two. She was say­ing, “I can just see by the way you de­scribe the job, that you are in­trigue,” and then her feel­ing that this was a good thing for me to do. Even though I’ve heard her tell the story and I’ve told it many times, I nev­er asked her ex­actly what she had in mind. Any­way, I took it. It was fas­cin­at­ing and un­like any­thing I ever have done in my ca­reer.

You had been de­part­ment chair be­fore.

Yes. Sev­er­al times at Columbia, but this was very dif­fer­ent. In the be­gin­ning, I can’t say that I loved the job be­cause, some­times, I’d wake up in the middle of the night in a cold sweat, think­ing of things I had to do. If we were go­ing to have a pro­gram the fol­low­ing year, I had to get mov­ing and find some­body to do it. But it suited my tal­ents be­cause it was build­ing something from the ground up. What are the rules go­ing to be? Who are we go­ing to in­vite? What are the pro­grams go­ing to be? Who should we try to at­tract as per­man­ent mem­bers? How are we go­ing to de­cide who to in­vite to head-up pro­grams? How are we go­ing to struc­ture it? There was a build­ing to get built.

The money was there be­cause of Jim?

Yes. First thing Jim said is, “Come over on Sunday morn­ing and we’re go­ing to work out a budget.”

We sat there in his New York City apart­ment and said to ourselves, okay, how much are we go­ing to have for work­shops? How many work­shops are we go­ing to have? How much sub­sidy are we go­ing to have for the café and af­ter­noon teas? How much should be in a Dir­ect­or’s dis­cre­tion­ary fund? We’re just throw­ing num­bers out and neither of us had any real clue. But we made a budget. The total yearly ex­penses came out to be more or less the num­ber Jim was ima­gin­ing. So that was the start.

At this point, you don’t even have a build­ing.

The build­ing has been de­signed: a place where math­em­aticians and phys­i­cists could meet com­fort­ably and ex­change ideas. Douglas had been hired as the first per­man­ent mem­ber and there was a per­man­ent mem­ber of­fer out to [Nikita] Nekra­sov, who was already in res­id­ence at the Si­mons Cen­ter, but had not de­cided wheth­er to stay or re­turn to IHES.

I had a little bit of say over the in­teri­or of the build­ing, and I ex­pressed strong feel­ings re­gard­ing three things. I really pushed for, and got, few­er large of­fices and more smal­ler of­fices. My feel­ing is that people do not like to share of­fices and, when they do, that acts as an ex­cuse for them not to show up. So, more small of­fices. I worked hard on get­ting good black­boards. And the lun­ch­room (or as it be­came known, the café). Jim and I were walk­ing through the space one day and he said, “Well, this isn’t enough space for the café. We need to ac­tu­ally ex­pand the build­ing a little bit so that the lun­ch­room can be big­ger.”

Den­nis hooked me up with the best chef on that end of Long Is­land, who ac­ted as an un­of­fi­cial café con­sult­ant. I took him down to the In­sti­tute in Prin­ceton to show him their café. He’s French, the chef at the In­sti­tute is French, and so they were bab­bling in French in the back­ground. We took the idea for a salad ar­ray from the In­sti­tute, which was very pop­u­lar at the Si­mons Cen­ter.

But the real work was (i) hir­ing per­man­ent mem­bers and post-docs; (ii) set­ting up pro­grams; and (iii) find­ing people to run spe­cial semesters or week-long work­shops.

When is this?

In the fall of 2009. By the sum­mer of 2008, I said yes, but I’d already sched­uled a sab­bat­ic­al at Stan­ford for the aca­dem­ic year 2008–2009. I didn’t show up in Stony Brook un­til sum­mer 2009. I di­vided my day when I was at Stan­ford: Between 7:00 AM and 10:00 AM, I was call­ing people up — “[Paul] Seidel, won’t you come and vis­it for a year?” “Oh yes. I’d be in­ter­ested in do­ing that.” “[Si­mon] Don­ald­son, would you be in­ter­ested? If you were ever in­ter­ested, we’d love to have you. But I un­der­stand if you’re not. In either case, who else might we think about?” Those sorts of con­ver­sa­tions were go­ing on every morn­ing. Then the rest of the day I was a vis­it­ing mem­ber of the Stan­ford Math De­part­ment, giv­ing a sem­in­ar course.

I re­mem­ber you also had to hire some phys­i­cist per­man­ent mem­bers.

Of course, I had much less dir­ect know­ledge of phys­ics. Douglas was already there and there were of­fers out to the phys­i­cist Nikita Nekra­sov and the math­em­atician [An­drei] Okounkov.

There was some kind of com­mit­tee of Stony Brook math­em­aticians and phys­i­cists run­ning things and mak­ing these de­cisions? This wasn’t just Jim Si­mons? There was a steer­ing com­mit­tee?

Yes. A steer­ing com­mit­tee had been set up a year or so earli­er, Den­nis was the head of it. He had spear­headed all these of­fers, in­clud­ing to me as Dir­ect­or. Okounkov, by that time, had said no, and Nekra­sov had not yet de­cided.2

I think the typ­ic­al ad­min­is­trat­ive job is much more re­act­ive than pro­act­ive. There’s a crisis or the ad­min­is­tra­tion’s un­happy about this or that, and you’re deal­ing with these chal­lenges rather than im­ple­ment­ing a pos­it­ive vis­ion of what you want the in­sti­tu­tion to be. In the be­gin­ning of the Si­mons Cen­ter, it was all the lat­ter and none of the former. There were no crises be­cause there was nobody there! The ad­min­is­tra­tion at SUNY Stony Brook didn’t yet know what to make of us. Be­cause of Si­mons’ fin­an­cial back­ing for the cen­ter and the fact that he had been one of the most loy­al, big donors to the Uni­versity over the years, the Uni­versity wanted things to go well and was ready and will­ing to help. It was a lot of fun.

I re­mem­ber you talk­ing about how hard it was to re­cruit people. Well, whom did you hire?

I man­aged to con­vince [Kenji] Fukaya and Don­ald­son to come as per­man­ent mem­bers. Sev­er­al oth­ers turned us down. After a long time and much ne­go­ti­ation, Nekra­sov agreed to stay. A year later, Douglas left after a year to go to Renais­sance. I hired [Ant­on] Kapustin from Cal­Tech but he left after a year.

After sev­en years as Dir­ect­or, I turned the reins over to [Lu­is] Álvarez-Gaumé, a phys­i­cist from Cern. Sim­ul­tan­eously, we hired a young phys­i­cist [Zo­har] Ko­mar­god­ski. We were com­pet­ing with Prin­ceton, MIT, and KITP, and pos­sibly oth­er places, for him. So it felt like a coup to land him. At this point, after all the hir­ing and resig­na­tions, we had four per­man­ent mem­bers: two math­em­aticians — Don­ald­son and Fukaya — and two phys­i­cists — Nekra­sov and Ko­mar­god­ski.

You had pro­grams; what was the high­light? What are you most proud of there?

I am most proud of the fact that it is a place people like to vis­it. From the be­gin­ning, I felt that this was a cru­cial in­gredi­ent. People will only vis­it the cen­ter if it is a pleas­ant place and a good, hassle-free place to work. I asked my­self if I were go­ing to vis­it a place, how would I want to be treated? What would make it easy for me?

I star­ted off to hire a per­man­ent fac­ulty of six and I only got four, but if you told me at the be­gin­ning, “Well, you’re only go­ing to man­age to hire four per­man­ent people, but here they are.” I would have said that’s a good out­come.

Hous­ing is the biggest re­gret I have. There wasn’t any­thing I could do about it. In gen­er­al, vis­it­or hous­ing is not great in the Stony Brook area. I wanted some sort of hous­ing like IAS and KITP have. My hope was that Stony Brook would build some vis­it­or hous­ing on or near the cam­pus and we could buy, or per­man­ently rent, a part of it. Nev­er happened.

After sev­en years of build­ing the cen­ter from the ground up, I felt like it was time for me to step down. The po­s­i­tion re­quired an ex­per­i­enced ad­min­is­trat­or. I’m a math­em­atician, not an ad­min­is­trat­or. I was happy dur­ing the time I spent there and quite happy to no longer be do­ing it. I feel a real sense of ac­com­plish­ment and sat­is­fac­tion as I look back on my time as Dir­ect­or.

Again come back to what you did with MSRI. You were there in 1984–85. I reck­on I was deputy dir­ect­or then for two years [1985–1987]. Dur­ing that time, I pushed for you for either the Sci­entif­ic Ad­vis­ory Com­mit­tee or for the Board of Trust­ees.

I’m not sure what you pushed for, but I ended up as a trust­ee, prob­ably in 1987. At my first trust­ee meet­ing, I had two re­ac­tions. One was that Ka­plansky, ba­sic­ally, came in and talked, so there was no time for any board over­sight or in­put. The second was that the board was not set up to fun­draise. Both were ex­acer­bated by the fact that every spon­sor­ing in­sti­tu­tion had a board seat.

To be­come a spon­sor­ing in­sti­tu­tion, a Uni­versity had to con­trib­ute \$3,000/year to MSRI. In re­turn, they got vari­ous perks, in­clud­ing a seat on the Board of Trust­ees. I felt that MSRI had to cut this link between be­ing a spon­sor­ing in­sti­tu­tion and a seat on the board for two reas­ons. First, this cre­ated an un­work­ably large Board since there were already between 15 and 20 spon­sor­ing in­sti­tu­tions and more join­ing all the time. There was no way for such a board to per­form mean­ing­ful over­sight. Secondly, the board needed people who could either con­trib­ute to MSRI them­selves or were well-placed to help MSRI fun­draise. Even though, at that point, MSRI had good NSF fund­ing, I felt that one day in the not too-dis­tant-fu­ture, the Board was go­ing to have to fun­draise. The ex­ist­ing con­fig­ur­a­tion was an im­ped­i­ment to that. There was no way to achieve these two goals (over­sight and fun­drais­ing) un­til you cut the link between spon­sor­ing in­sti­tu­tions and a seat on the board.

This was all dur­ing your first trust­ee meet­ing?

Yes and no. The dis­cus­sions star­ted dur­ing the first board meet­ing and the ex­ec­ut­ive com­mit­tee was agreed to and im­ple­men­ted in the second year.

Who was on that com­mit­tee?

[Hugo] Rossi was Chair of the board; I was Vice Chair of the board. [Hy­man] Bass was sec­ret­ary and [An­thony] Tromba of UC Santa Cruz was the Treas­urer. Maybe oth­ers I have for­got­ten. Then Hugo resigned and sud­denly I was Chair. We had a set of meet­ings with Ka­plansky, ba­sic­ally, re­cal­ib­rat­ing the re­la­tion­ship between the Dir­ect­or and the Board. I don’t know how long I was chair­man of the Board, but it was my bril­liant idea to hire Thur­ston as the next Dir­ect­or…

In some ways, it was.

It seemed like a good idea at the time.

Well, in a sense, the NSF wanted the place opened up. With Ka­plansky, it was math. Noth­ing else. I think the NSF really wanted it opened up. Well, Thur­ston did a lot that, but he had his faults, too.

I agree. I think the NSF wanted MSRI to serve a broad­er and more di­verse group of math­em­aticians and also wanted it to mount pro­grams about math­em­at­ics for the pub­lic. I agree that Thur­ston had his faults.

It was your idea to hire Thur­ston?

I had a list of people. I’d talked to eight or ten seni­or fig­ures in the field, and the only one who showed an in­terest was [Bill] Thur­ston. I was com­pletely shocked but de­lighted when he showed an in­terest.

I have one vign­ette from when we were try­ing to hire Thur­ston. Our search com­mit­tee was [Barry] Mazur from Har­vard, [Nancy] Ko­pell from Bo­ston Uni­versity and me. It was clear Bill was in­ter­ested. The three of us went down to Prin­ceton and we talked to him. We had a long walk on the track along the canal for an hour. He talked about him­self, his vis­ion for MSRI, what was needed at MSRI. At the end of this long con­ver­sa­tion, Mazur says to him, “Bill, I can’t leave be­fore I say this. I don’t know what you think about now in the morn­ing when you brush your teeth, but if you take this job, what you think about when you’re brush­ing your teeth will not be math­em­at­ics.”

Bill took the job.

Well, he wanted a new ad­ven­ture in a way.

That’s right. I think he wasn’t able to prove the Poin­caré Con­jec­ture or the vast gen­er­al­iz­a­tion of it that he for­mu­lated. He did a hell of a lot, but he couldn’t settle the Poin­caré Con­jec­ture. That was my take on why he was will­ing to do this job.

He had ideas about how to teach. He had all sorts of ideas.

Yes, and ideas about how to make math­em­at­ics more ac­cess­ible to the gen­er­al pub­lic and how to get more money in­to math­em­at­ics, star­tup pack­ages for math­em­aticians. Yes, he had lots of ideas. I be­lieve our dis­cip­line is bet­ter off for the ideas Bill helped pro­mote at MSRI.

He once said to me, “Since I got the Fields Medal, people listen to me much more than they should.” I thought that was a good com­ment on his part be­cause it does hap­pen. It happened.

One thing we haven’t talked about is the Poin­caré Con­jec­ture.

How did we miss that?

That was — when was that? 2003 to 2008, really. What I was do­ing dur­ing that time was def­in­itely math­em­at­ics but it wasn’t ori­gin­al math­em­at­ic­al re­search in the clas­sic sense. Nev­er­the­less, it was great fun. In fact, in 2006, at the In­ter­na­tion­al Con­gress, I gave a press brief­ing. I talked about the Poin­caré Con­jec­ture, and the whole his­tory. A fairly fam­ous sci­entif­ic or math journ­al­ist from Switzer­land said, “I just have one ques­tion. You’re telling me you spent three years try­ing to sort out this ar­gu­ment and make sure it’s right, but that you haven’t cre­ated any­thing new, you hadn’t ad­ded any­thing new to the ar­gu­ment. You just sor­ted it out. Is that right?” “Yes, that’s pretty fair.” He says, “Why would you ever do that?”

I said, “Be­cause I was in­ter­ested. It was beau­ti­ful math­em­at­ics, it re­lied upon things that I un­der­stood and things that I didn’t, but I man­aged to find a col­lab­or­at­or who was much stronger in the things that I was weak in. Lastly, I just wanted to know wheth­er or not it was true and if it was, what the ar­gu­ment was. I was in­ter­ested in sort­ing all this stuff out and un­der­stand­ing it. It was beau­ti­ful stuff. That’s why I did it.”

You told me that you would read a sen­tence of Perel­man’s pa­per and think about it. Is this true? Is this not true? How would I prove it? About a week later, you’d have an ar­gu­ment worked out for why that sen­tence was cor­rect. Then you’d ask your­self, “Let’s see. If I was Perel­man and I was only al­lowed one sen­tence to cov­er this week’s work, well, that’s about what Perel­man wrote.”

That’s pretty close to ac­cur­ate, though some­times it would take a week and then I figured it out. Some­times, it would take sev­er­al weeks. Some­times, I’d go ask Klein­er or Lott if they un­der­stood it. Some­times, of course, I’d ask Tian or talk to him about it. I couldn’t al­ways fig­ure it out my­self. In the end, if I’d wanted to sum­mar­ize the whole long path I’d been through, that’s what I would have writ­ten. That, more than any­thing else, con­vinced me that Perel­man had done it all. You can’t hit ex­actly the right note over and over and over again if you haven’t sor­ted it all out. He chose, for whatever reas­on, to write what he wrote.

Perel­man gave you a long prob­lem set; prove this sen­tence, prove this sen­tence, prove this sen­tence…!

That’s, ba­sic­ally, what it was: a Moore prob­lem set for his stu­dents. In­deed, it had a little bit of Moore-type to­po­logy, the Al­ex­an­drov the­ory in it.

What did Hamilton miss?

What Hamilton missed was deep in the para­bol­ic evol­u­tion equa­tion. Perel­man had es­sen­tial new in­sights in­to that flow equa­tion that Hamilton just hadn’t seen. Perel­man in­tro­duced a com­pletely new func­tion­al or more pre­cisely an in­fin­ites­im­al ver­sion of a func­tion­al, un­like any­thing Hamilton had ever done — that gave one much bet­ter con­trol over the re­gions where the curvature blows up, i.e., goes to in­fin­ity. That was al­ways the is­sue — con­trolling the re­gion where the curvature is blow­ing up. Hamilton had proved that, as long as the curvature stays bounded, the flow con­tin­ues smoothly, but when the curvature blows up, that’s where you’re go­ing to hit the sin­gu­lar­ity.

This new func­tion­al that Perel­man in­tro­duced al­lowed him to get much bet­ter con­trol over those re­gions. That’s where you do the sur­ger­ies. You need to know what these re­gions look like, both to­po­lo­gic­ally and geo­met­ric­ally, and then cut them out and su­ture in a stand­ard piece and ex­tend the flow past the sin­gu­lar­it­ies. Then, you have to worry about what hap­pens at in­fin­ity, and that’s where the col­lapsing hap­pens. What hap­pens is ex­actly Thur­ston’s pic­ture. There are big pieces of the man­i­fold, as you go off to in­fin­ity, that con­verge to hy­per­bol­ic man­i­folds. But then there are oth­er pieces that start col­lapsing, and they can col­lapse to two-di­men­sion­al man­i­folds or to one-di­men­sion­al man­i­folds. Then you need a little bit of Al­ex­an­drov space the­ory, to un­der­stand these re­gions.

Perel­man had proved a the­or­em about three-di­men­sion­al Al­ex­an­drov spaces, which was quite deep and hard. I think he thought, when start­ing on the Ricci Flow pro­ject, that this was the ad­vant­age that would al­low him to prove the Poin­caré Con­jec­ture where Hamilton hadn’t been able to. But, in fact, he didn’t need this deep res­ult in Al­ex­an­drov-space the­ory at all.

Klein­er and Lott, they were work­ing through his pa­per, filling in de­tails and do­ing a sim­il­ar sort of thing to what you were do­ing. You were do­ing this sim­ul­tan­eously?

Yes, at the same time. Oc­ca­sion­ally, we would get to­geth­er and com­pare notes. The four of us (Klein­er, Lott, Tian,and I) had a meet­ing at Prin­ceton (that would have been be­fore 2006) where we com­pared notes. Each of us came away con­vinced that Perel­man had done it. There was one ba­sic dif­fer­ence in our ap­proaches, though. They were go­ing line by line and ex­pand­ing it out. Where­as, to me, the is­sue that I wor­ried about from the be­gin­ning arose from the way Perel­man presen­ted the ar­gu­ment. He in­tro­duced a new func­tion­al and then proved res­ults about the sin­gu­lar­it­ies that first ap­pear in this flow. Then he wrote a second pa­per, which star­ted off by, “Okay, this is what the first sin­gu­lar­ity looks like be­cause of all this work we did. Now we cut out these sin­gu­lar­it­ies and su­ture in something to com­plete the man­i­fold and re­start the flow. Everything we did in the first pa­per works in this more gen­er­al con­text, as well.”

I said to my­self, “If there’s a mis­take, that’s where it is.” I’ve done this to my­self. I do a simple case, and then I have to ap­ply the same ar­gu­ment in a more com­plic­ated situ­ation. But there is something in the ori­gin­al situ­ation that is so ob­vi­ous that I don’t even real­ize that it was a ne­ces­sary hy­po­thes­is. But this con­di­tion doesn’t hold any­more in the gen­er­al con­text. To avoid this type of mis­take, you should do the gen­er­al case from the be­gin­ning. You shouldn’t first do this simple case and then say, well, okay, it gen­er­al­izes to this oth­er case and works the same way.

In the end, Klein­er and Lott veri­fied the ar­gu­ment, too. I was just more com­fort­able start­ing with the gen­er­al case.

John, I’m guess­ing you did most of the writ­ing.

Prob­ably the ma­jor­ity but nowhere near all of it.

What was Tian like to work with?

I have the highest re­spect for him as a math­em­atician, and I think we were a very good team. He is not al­ways ex­tremely care­ful in the de­tails but he has tre­mend­ous in­sight and power so, in the es­sen­tials, he was al­ways cor­rect. There were things about which I just didn’t have any clue how to pro­ceed, and he would know either from ex­per­i­ence or could see from his in­genu­ity, or whatever, how to go about it. After we talked, I would al­ways sort out for my­self if there were places where more ar­gu­ment was needed. My care­ful­ness and the fact that I wasn’t an ex­pert in the stuff that he was so very good at, meant that I had to go more slowly, and that brought him back down to earth at times in a way that worked well.

It was a very com­fort­able, fruit­ful, and pro­duct­ive col­lab­or­a­tion.

You were not in the same place.

I would go down to Prin­ceton. He would come up to New York. We’d sort of trade off. Usu­ally, I would come in open­ing with “I want to talk about this. I don’t un­der­stand how this works or I think I see how this works, but I can’t quite put the ar­gu­ment to­geth­er,” and then we’d talk about it.

Just as with run­ning the Si­mons Cen­ter for Geo­metry and Phys­ics, I’m very happy I did pur­sue a de­tailed un­der­stand­ing of the proof of the Poin­caré Con­jec­ture. It’s un­like any­thing else I ever did or will ever do again. It wasn’t re­search but I got tre­mend­ous pleas­ure in do­ing it, and I learned a lot of math­em­at­ics.

Yes, well, it was the biggest prob­lem around.

Yes. Right. It was worth spend­ing sev­er­al years on. Gosh, I’ve had a long ca­reer. I’ve been around a long time!

You’re only 78.

What’s that? 55 years. I got my PhD when I was 23, so I’ve been at it 55 years. That is a long time. I have been in­cred­ibly lucky. I stumbled in­to something that I love do­ing and am reas­on­ably good at. It has giv­en me a life with so many bless­ings, not the least of which is to be my own boss and work at my own pace on what I want to work on. Also, it in­tro­duced me to coun­tries and cul­tures that I nev­er would have en­countered oth­er­wise. It in­tro­duced me to the greatest math­em­aticians of our time. I am good enough to ap­pre­ci­ate how spe­cial they are. It has been a great pleas­ure to know these people and watch them make in­cred­ible leaps of ima­gin­a­tion. It has also been a great pleas­ure to play a part in the de­vel­op­ment of math­em­at­ics over this last half cen­tury.

Chil­dren. A few words about your kids be­cause that’s part of your life.

I have a stepson. He was three when I met him. Mi­chael Jac­ob Kirsch. He’s Jake to every­one. And Bri­anna, our daugh­ter. She’s 35. Jake is 42.

Let’s start with Jake be­cause he’s older. Where’d he go to col­lege?

He went to Stan­ford.

Be­fore that he went to private school in New York. When El­len and Jake moved from Bo­ston to New York, he went to Field­ston in River­dale, which is a very leafy part of the Bronx. Whenev­er he talked to his friends in Bo­ston, and they’d say, “Well, where do you go to school?” He said, “I go to school in the Bronx.” El­len would say, he goes to school out of the city. I tried to tell her that the Bronx was part of New York City.

Jake went all the way through Field­ston from second grade through high school. Then, he went to Stan­ford. He blos­somed there. He star­ted the Stan­ford Cook­ing Club and learned to be a ser­i­ous chef. He met his now-wife, Nicole; He star­ted to be­come an ath­lete for the first time in his life un­der her tu­tel­age.

What sport or sports?

Well, he’s al­ways been a ski­er. I taught him to ski, and he taught Nicole to ski. She was on the Stan­ford row­ing team, and she was a swim­mer in high school. The sports Jake and Nicole do to­geth­er now are bik­ing, run­ning, and ski­ing.

Okay. What’s he do­ing now?

He works for AB In­Bev, which is the Brazili­an–Bel­gian con­glom­er­ate that owns An­heuser-Busch.

And he still cooks, and he now has two kids — a boy and a girl — which is very ex­cit­ing.

And Bri­anna?

Bri­anna went to med­ic­al school at Stony Brook, and then did her res­id­ency at the Chil­dren’s Hos­pit­al of Col­or­ado in Den­ver. She is now a pe­di­at­ri­cian in private prac­tice in Den­ver. She loves to ski and loves the moun­tains and is thrilled to be in Den­ver.

It’s a great pleas­ure to see your kids grow up to be happy and suc­cess­ful in their lives.