Celebratio Mathematica

When I ar­rived in Berke­ley in 1977, I had a gen­er­al no­tion of study­ing to­po­logy, but only the vaguest of ideas of what that might ac­tu­ally en­tail. At some point, I began to won­der about an ad­visor, and no­ticed the group of stu­dents stand­ing at the black­board in the com­mon room, of­ten draw­ing com­plex dia­grams and talk­ing an­im­atedly. These were not ex­clus­ively Kirby stu­dents, but who­ever they were, they seemed to be hav­ing a lot of fun do­ing math. In the spring, I star­ted go­ing to the to­po­logy sem­in­ar that Rob ran; of course as a be­gin­ner I was usu­ally lost after a few minutes. The last talk of the spring was by Rob him­self, and he presen­ted his proof with Sel­man Ak­bu­lut that the double cov­er of a ho­mo­topy \( \mathbb{R}P^4 \) con­struc­ted by Cap­pell and Shaneson was in fact \( S^4 \).

It was a beau­ti­ful and eye-open­ing lec­ture; Rob star­ted with a de­scrip­tion of handle cal­cu­lus (still only a few years old) and showed how to build a handle­body for the double cov­er. (Nobody no­ticed that the fram­ing of a last 2-handle was in­cor­rect — this was poin­ted out later by Aitchis­on and Ru­bin­stein [e1], and re­paired only after strenu­ous work of Ak­bu­lut–Kirby and Gom­pf.) It all went by very fast, but by the end he had ar­rived at Fig­ure 13 on the last page of [2]. The in­struc­tions in the pa­per read, “Fig­ure 13 is the un­link! Get 3 col­ors of chalk and a large black­board; have fun.” And that is just what we did, call­ing out sim­pli­fic­a­tions of the pic­ture un­til it fell in­to three com­pon­ents. Walk­ing out of the room, I knew that I would be a Kirby stu­dent. I think the same was true for my class­mates Tim Co­chran, Dave Schorow, and John Hughes, al­though per­haps they felt this earli­er.

Rob’s sem­in­ar was a fo­cal point for our learn­ing, and most of us gave talks on a reg­u­lar basis on our read­ings and later on our re­search. Much more took place in in­form­al read­ing groups and sem­inars. Tim, Dave, and I were already read­ing the ba­sic texts (Mil­nor’s h-cobor­d­ism the­or­em lec­tures; Rourke–Sander­son on PL to­po­logy; Rolf­sen; Cas­son–Gor­don) that were con­sidered the pre­requis­ites for fur­ther study. Rob’s handle cal­cu­lus was still fairly new, and the gen­er­a­tion of stu­dents just be­fore mine (Ak­bu­lut, Melvin, Ka­plan, Harer) had played an im­port­ant role in its de­vel­op­ment and re­fine­ment. Those more seni­or gradu­ate stu­dents took it as their re­spons­ib­il­ity to train us in the largely un­writ­ten folk­lore of the sub­ject. I re­mem­ber es­pe­cially that John Harer, who was in his last year, would but­ton­hole one of us in a hall­way, and say something like “Wait, you don’t know about spin struc­tures…,” and give an im­promptu lec­ture at a black­board with ex­er­cises to be per­formed on the spot. We could go to Chuck Liv­ing­ston or Pat Gilmer for fur­ther help and in­struc­tion and in­deed my thes­is top­ic grew out of con­ver­sa­tions I had with them. Oc­ca­sion­ally one of us would go ask Rob a ques­tion; he was gen­er­ous with his time and ex­pert­ise when asked, but some­times you could get the mes­sage that it was best for you to just dope out some of these things on your own. Once I sat down next to Rob and pulled out a re­print while we waited for a lec­ture to start. Rob looked over my shoulder at what I was read­ing and told me that I shouldn’t read pa­pers like that, but should just fig­ure out the proof my­self. In time I could re­cog­nize close par­al­lels between the way that Rob dealt with his stu­dents and his style of rais­ing his kids to be in­de­pend­ent and self-suf­fi­cient.

I don’t know that Rob de­lib­er­ately or­gan­ized his gradu­ate stu­dents to pass along know­ledge in this fash­ion, or wheth­er it de­veloped or­gan­ic­ally. It seems to have been widely un­der­stood that one was sup­posed to do the same thing for the next gen­er­a­tion. I think I was a little dis­ap­poin­ted that the stu­dents who ar­rived after me, Ian Aitchis­on and Bob Gom­pf, seemed to already know a lot and didn’t need much tu­tor­ing. Bob Pen­ner turned up for a few months as a sort of refugee from the rather nongeo­met­ric MIT grad pro­gram, and I re­mem­ber us col­lect­ively teach­ing him about handle cal­cu­lus. This en­tailed ex­plain­ing to him that the 3-man­i­fold in­vari­ant he’d dis­covered by us­ing Rob’s the­or­em was just the or­der of the first ho­mo­logy, a sad les­son that we all had to learn at some point.

As far as I know, few of Rob’s many stu­dents in that era had a sched­uled weekly meet­ing with him; I think this may have changed in sub­sequent years. The un­der­stand­ing was that when you had something to dis­cuss, you dropped by or made an ap­point­ment. Sim­il­arly, in that era, we were not giv­en (or even sug­ges­ted) prob­lems to work on; part of your edu­ca­tion was learn­ing how to find your own prob­lems. For­tu­nately, there were lots of sources: Rob’s desk — a dis­tant fore­run­ner of the ArX­iv — al­ways had a pile of re­cently ar­rived pre­prints, and we were en­cour­aged to grab something to read. If you got in­ter­ested in a pa­per, then you could give a talk on it in the sem­in­ar. My fa­vor­ite memory of this was find­ing Camer­on Gor­don’s pa­per n the \( G \)-sig­na­ture the­or­em in low di­men­sions in that pile. A few weeks later, I ran up to Rob in the hall and ex­claimed about how great it was. He smiled at my en­thu­si­asm, and asked, ”would you like to meet the au­thor”, who was stand­ing right there with him. There were of­ten vis­it­ors; Camer­on was around quite a bit, as was Larry Taylor. Larry was very help­ful to me at a late stage in writ­ing my thes­is, and also provided a key sug­ges­tion to Tim Co­chran that made his thes­is much stronger.

Once I found my way to a thes­is top­ic, I met with Rob more fre­quently. I usu­ally wanted to launch straight in­to my new­est res­ults, but after a few minutes, Rob would usu­ally ask, “Re­mind me what is a doubly slice knot again…?” or some oth­er ques­tion about the ba­sics of what I was do­ing. I might have found this a bit ex­as­per­at­ing but by the end of my gradu­ate years I de­veloped more skill at giv­ing a quick gloss over the back­ground ma­ter­i­al, a skill that trans­lated well in­to sem­in­ar talks out in the world. In gen­er­al, Rob didn’t like “ma­chinery”, al­though (or per­haps be­cause) turn­ing his geo­met­ric ideas in­to his fam­ous work on tri­an­gu­la­tions re­lied on soph­ist­ic­ated res­ults of sur­gery the­ory. I re­mem­ber lec­tur­ing in some sem­in­ar (about a fake \( \mathbb{R}P^4 \)) where I needed to ex­plain about nor­mal maps, a part of sur­gery the­ory. Rob kept press­ing me for a more geo­met­ric ex­plan­a­tion, and in the end gave up, say­ing with a laugh that I was so earn­est about it all that he would just ac­cept my ver­sion.

In ad­di­tion to the talks and sem­inars, Rob had just fin­ished com­pil­ing the first ver­sion of his prob­lem list, star­ted in 1976 at a con­fer­ence in Stan­ford, and ex­pan­ded over the years by him call­ing or writ­ing to his many friends and ask­ing for in­put. This was a fant­ast­ic re­source for us as stu­dents, es­pe­cially as there were no texts or even sur­vey art­icles to help get a grasp on a rap­idly de­vel­op­ing field. Most of the prob­lems were too hard and fun­da­ment­al for gradu­ate stu­dents, but I thumbed through the prob­lem list fre­quently to get ideas for ques­tions that I might be able to ap­proach. Even see­ing how to phrase a ques­tion, as in the short write-ups for most of the prob­lems, was a big help. Al­though most of the prob­lems were sug­ges­ted by oth­ers, Rob put it to­geth­er in a way that re­flec­ted his point of view that there was a uni­fied field of low-di­men­sion­al to­po­logy.

Rob gave some mem­or­able top­ics courses, most not­ably a class on to­po­lo­gic­al man­i­folds that he gave to­wards the end of my years in Berke­ley and a course on to­po­lo­gic­al 4-man­i­folds in the im­me­di­ate af­ter­math of Freed­man’s an­nounce­ment of his clas­si­fic­a­tion res­ults in the sum­mer of 1981. The lat­ter was Rob’s way of learn­ing the very com­plic­ated proof that Cas­son handles are to­po­lo­gic­ally handles. (This was a great trick that I learned from Rob and put in­to prac­tice in my ca­reer: if you want to learn something new, give a course on it. You’d learn more than the stu­dents!) Of course we got bogged down in the de­tails by the end, and it was very much Rob’s style to get us to work through those de­tails as a group.

The course on high-di­men­sion­al to­po­lo­gic­al man­i­folds felt like a trip down memory lane; Rob de­clared that it was the last time he would ever give a course on the sub­ject, since he couldn’t see what good it would do any­one. The ma­jor prob­lems had been solved, but the ideas were still fresh and for me it was a great ex­per­i­ence. We were all eager to learn Rob’s fam­ous tor­us trick, but Rob’s present­a­tion of some of the more found­a­tion­al ideas, such as the proof of the Schoen­flies the­or­em, made them feel fresh and new. I had the sense that he had great af­fec­tion for some of that ma­ter­i­al (“mesh­ing” stands out in this way) and maybe was a bit sad that it was viewed as be­ing some­what dated. I think that Rob’s present­a­tion fol­lowed a book that he was writ­ing with Jim Kister that was in­ten­ded as an al­tern­at­ive to the more hi-tech treat­ment in the fam­ous Kirby–Sieben­mann book [1].

We did learn many pieces of cur­rent re­search from courses giv­en by postdocs and vis­it­ors. Steve Ker­ck­hoff gave a couple of courses re­lated to Thur­ston’s work on hy­per­bol­ic 3-man­i­folds and on sur­face dif­feo­morph­isms in con­nec­tion with his res­ol­u­tion of the Nielsen real­iz­a­tion prob­lem. Al­len Hatch­er gave a semester course on his proof of the Smale con­jec­ture, teach­ing us a lot of more tra­di­tion­al 3-man­i­fold the­ory along the way. Dur­ing one quarter, Rob traded courses with Jim Mil­gram, go­ing down to Stan­ford to teach 4-man­i­folds while Mil­gram taught us sur­gery the­ory. Rob told us rather poin­tedly that we were to at­tend all of the lec­tures no mat­ter how con­fused we were. I thought at the time that the Stan­ford stu­dents got the bet­ter of the deal, but Mil­gram’s course came in handy later in my ca­reer.

Low-di­men­sion­al to­po­logy felt like a fairly co­her­ent sub­ject in this peri­od, and Rob felt strongly that we should know about 3-man­i­folds and 4-man­i­folds in equal meas­ure. The ground-break­ing ideas of Freed­man and Don­ald­son in 4-man­i­fold the­ory were just over the ho­ri­zon, but Thur­ston had already star­ted a re­volu­tion in 3-di­men­sion­al to­po­logy by in­tro­du­cing vast new ho­ri­zons in hy­per­bol­ic geo­metry and oth­er geo­met­ric meth­ods. We did our best to learn this, with Rob or­gan­iz­ing reg­u­lar study groups to try to read Thur­ston’s notes that were ar­riv­ing in in­stall­ments (mi­meo­graphed, no less!) from Prin­ceton. We put a lot of ef­fort in­to this, but some­how didn’t learn as much as we might have. In ret­ro­spect, I be­lieve that we didn’t re­cog­nize how many de­tails in those notes needed to be worked out by the read­er, and that Thur­ston was try­ing to con­vey his vis­ion­ary geo­met­ric view­point rather than lay­ing out a guided route to the spec­tac­u­lar res­ults he’d an­nounced. Still, some stu­dents, like Joel Hass and Bill Menasco, wrote theses on geo­met­ric top­ics, and many of us even­tu­ally learned some por­tions of Thur­ston’s work.

Our group was very co­hes­ive; we went to tea most days and either talked math or played speed chess, at which Rob ex­celled. Rob would some­times chal­lenge us to a con­test to see who could hold their breath the longest; he could go more than 2 minutes. We thought this was aided by his kayak­ing prac­tice but he said it was a mat­ter of will­power. We of­ten went out for a cheap lunch as a group; the Mex­ic­an res­taur­ants that gave you free salsa and chips were a fa­vor­ite for ob­vi­ous reas­ons. We ar­gued about the polit­ic­al is­sues of the day (rent con­trol, nuc­le­ar arms, fem­in­ism…) more than we talked math, and I re­call rarely be­ing on the same side as Rob. We also did in­tra­mur­al sports to­geth­er, soc­cer (coed in the fall and in the spring men only) and bas­ket­ball. Rob was a full par­ti­cipant in all of these, and brought en­dur­ance and dog­ged­ness to the soc­cer field. I told him once (truth­fully) that I’d heard one of the op­pos­ing team ask at half-time to be passed the ball be­cause he could “beat the old guy”. Rob was mad enough that not much got past him in the second half. We fielded some strong teams and won our league cham­pi­on­ship one term when José Car­los Gomez-Larrañaga, who was an in­stinct­ive goal-scorer, was vis­it­ing.

Rob took some of us on raft­ing trips on the Amer­ic­an River in the sum­mer, us­ing an old Army sur­plus raft that was ill-suited to the rap­ids we en­countered. (A friend who was tak­ing a raft­ing class told me later that her in­struct­or had made the class pause, say­ing, “Watch that raft; they’re go­ing to do everything wrong” — which we pro­ceeded to do.) We hit our first rap­id without hav­ing really learned how to con­trol the boat. Rob’s daugh­ter Kate, who was quite young, was perched in the bow and was sent fly­ing as the un­der-in­flated raft es­sen­tially fol­ded in half when we hit a big wave. My im­age is that Rob plucked her out of the air as she was head­ing over­board, and she spent the rest of the af­ter­noon hold­ing onto his leg. I didn’t ap­pre­ci­ate at the time just how ac­com­plished Rob was as a kayaker, so I was really im­pressed when he kayaked up­stream to re­trieve some life­jack­ets that had been set out to dry on the is­land where a few of the crew washed up after our ini­tial cap­size. Even­tu­ally Rob got us or­gan­ized in­to some semb­lance of co­her­ent pad­dling so we made it through the more fear­some S-shaped rap­id un­scathed.

In the sum­mer of 1978, many of Rob’s stu­dents went to Cam­bridge (UK) for a small gath­er­ing. I didn’t know enough to go along, but when a sim­il­ar event took place in 1981, I jumped at the chance. It was a large group, a mix­ture of gradu­ate stu­dents from Berke­ley and else­where plus nu­mer­ous postdocs and more seni­or fig­ures. (A group pho­to­graph from late in the sum­mer can be found on the MSP web­site.) Ray Lick­or­ish or­gan­ized this, find­ing us rooms in Pem­broke Col­lege; I think he and oth­ers were a bit over­whelmed by the some­what in­dec­or­ous and en­er­get­ic Amer­ic­an stu­dents. We played soc­cer as of­ten as we could (even, shock­ingly, dur­ing Charles and Di­ana’s wed­ding) and did math the rest of the time. (There may have been some punt­ing and trips to the pub mixed in.) The soc­cer games at­trac­ted quite a crowd; a fa­vor­ite im­age was of Kate (or pos­sibly Linda’s daugh­ter Erica) who might have been 7 at the time, fear­lessly tack­ling John Hempel, who towered over her. That sum­mer was prob­ably the ori­gin of the (not en­tirely com­pli­ment­ary) repu­ta­tion of Kirby stu­dents as fe­ro­cious soc­cer play­ers. This might have been re­lated to Tim, who brought his soc­cer ball to con­fer­ences for many years af­ter­wards.

The time in Cam­bridge had an enorm­ous in­flu­ence on us gradu­ate stu­dents. I made lifelong friends from all over, and was able to spend time with some of Rob’s former stu­dents, such as Sel­man Ak­bu­lut, Paul Melvin, and Marty Schar­le­mann, and to talk in­form­ally with any num­ber of lu­minar­ies, in­clud­ing An­drew Cas­son, Camer­on Gor­don, and John Con­way. Many oth­ers passed through for short­er stays. I met Jerry Lev­ine there, which had a great im­pact on my sub­sequent ca­reer. Linda, who was to marry Rob the fol­low­ing spring, came for the sum­mer. It was a nice chance to get to know her and her daugh­ters as they learned to nav­ig­ate the strange new world of math­em­aticians that they’d entered.

The months after Cam­bridge were as math­em­at­ic­ally con­sequen­tial as any peri­od I ex­per­i­enced in my pro­fes­sion­al life. Mike Freed­man an­nounced his res­ults on to­po­lo­gic­al 4-man­i­folds in Au­gust (I learned of this from Rob’s course de­scrip­tion tacked up on a bul­let­in board in Berke­ley), and in Novem­ber or so Rob told us about a re­mark­able res­ult of Si­mon Don­ald­son us­ing gauge the­ory to show that some 4-man­i­folds (which Freed­man had just con­struc­ted to­po­lo­gic­ally) could not be smoothed. The sum­mary of Don­ald­son’s proof had been passed from Atiyah to Freed­man to Kirby by tele­phone, and, just as in the kids game, had got­ten com­pletely garbled. Des­pite the in­com­plete sum­mary, Rob and Mike un­der­stood quickly that the com­bin­a­tion of Don­ald­son’s and Freed­man’s the­or­ems im­plied the ex­ist­ence of an exot­ic \( \mathbb{R}^4 \). (Rob’s re­col­lec­tions of this peri­od can be found in Freed­man’s Cel­eb­ra­tio volume, in the sec­tion on Exot­ic \( \mathbb{R}^4 \)s.) A more co­her­ent re­port emerged by the spring, and some of us scrambled to learn the ba­sics. Dur­ing this en­tire peri­od, there was an act­ive and in­flu­en­tial sem­in­ar at Berke­ley in gauge the­ory run by Is Sing­er and fea­tur­ing a bright young phys­ics postdoc named Ed Wit­ten. Sadly, there was vir­tu­ally no in­ter­ac­tion between Rob’s group and this sem­in­ar at a time when each might have done well to learn from the oth­er. Who knew!

Without know­ing that these great achieve­ments were about to oc­cur, Rob and Camer­on Gor­don had already planned a con­fer­ence on 4-man­i­folds for the fol­low­ing sum­mer of 1982 in Durham, New Hamp­shire. They re­or­gan­ized the pro­gram to fea­ture lec­ture series by the prin­cip­al fig­ures in the new de­vel­op­ments, in­clud­ing Freed­man, Don­ald­son, and Cliff Taubes. At that con­fer­ence, Frank Quinn proved the 4-di­men­sion­al ver­sion of the an­nu­lus con­jec­ture that Rob had proved in high­er di­men­sions in 1969. Be­cause of the tight sched­ule, Quinn’s lec­ture had to be giv­en in the even­ing; I re­call a mara­thon ses­sion in a crowded and sweaty base­ment classroom that was mer­ci­fully brought to an end after many hours when Rob re­cog­nized that most of the audi­ence was about to ex­pire, and sent us home.

The years that I’ve de­scribed were form­at­ive and spe­cial to me and my fel­low stu­dents. The stresses of find­ing our way through gradu­ate school and our early ca­reers were oc­ca­sion­ally over­whelm­ing, but there was a great sense of gen­er­os­ity and open­ness in the field. There were many lead­ers (and fu­ture lead­ers) who con­trib­uted to that feel­ing, but Rob’s per­son­al­ity and lead­er­ship stand out and had a great in­flu­ence even bey­ond his im­port­ant math­em­at­ic­al con­tri­bu­tions. Al­though his polit­ic­al and so­cial views were all about in­di­vidu­al­ism, he ac­ted in a way to cre­ate and nur­ture a re­mark­able sense of com­munity, where name and repu­ta­tion were sec­ond­ary to what you could bring to the study of a won­der­ful sub­ject. Rob rarely told us dir­ectly that we were do­ing well, but by treat­ing us as col­leagues, he made us feel that we were already full mem­bers of that com­munity.

Daniel Ruber­man re­ceived his Ph.D. from UC Berke­ley in 1982 un­der the dir­ec­tion of Rob Kirby. After a postdoc at the Cour­ant In­sti­tute and a year at MSRI, he joined the fac­ulty at Bran­de­is Uni­versity, where he is Pro­fess­or of Math­em­at­ics.