Celebratio Mathematica

At the be­gin­ning of Septem­ber 1965 I flew in three hops a bit over 11,000km from Auck­land, New Zea­l­and, to Los Angeles. There were no wor­ries about se­cur­ity those days: after say­ing good­bye to fam­ily who came to see me off I walked across the tar­mac and climbed up in­to the plane that flew me to Fiji where I stayed a couple of nights. Then I flew on to Hawai’i for an­oth­er couple of nights be­fore the fi­nal hop to Los Angeles. Com­mer­cial planes couldn’t fly more than about 5000km, hence the breaks in the jour­ney.

My first few days in Los Angeles were a shock! 10 mil­lion people lived in the city then, about four times the pop­u­la­tion of New Zea­l­and, and they pro­duced massive amounts of smog so most days it was hard to breathe and to see more than a couple of kilo­metres. Gradu­ally, I sup­pose, I got a bit more used to this. For­tu­nately most of the time there was a sea breeze which ten­ded to blow the worst of the smog in­land and away from UCLA but oc­ca­sion­ally the wind would run out of puff and there would be a smog alert.

After find­ing some­where to live near the cam­pus with two oth­er be­gin­ning PhD stu­dents who had just ar­rived in the USA, one from In­dia and one from Ir­an, I was ready for ac­tion. It was a short walk to the Math­em­at­ics De­part­ment where I was a half-time Teach­ing As­sist­ant and half-time stu­dent. To­po­logy had in­spired me back in Auck­land so I made sure I en­rolled in a to­po­logy course. There were courses in real and com­plex ana­lys­is and al­gebra as well, all aimed at the qual­i­fy­ing ex­am­in­a­tions. The al­gebra course seemed to cov­er stuff I had long since covered at Auck­land so I tried to move up to the next course but it clashed with my teach­ing as­sign­ment. I ended up in a sem­in­ar taught by a vis­it­ing al­geb­ra­ist, thereby earn­ing the un­deserved repu­ta­tion that I must be an al­gebra whizz. The only dam­age done by go­ing in­to the wrong al­gebra course was that as part of my teach­ing du­ties the fol­low­ing semester I was as­signed grad­ing of stu­dent home­work in the course I should have en­rolled in that first semester. That duty en­sured that my know­ledge of the sub­ject rap­idly went from zero to plenty thanks to my own at­tempts at an­swer­ing the home­work prob­lems and in­teg­rat­ing where ne­ces­sary the solu­tions of the best stu­dents in the class.

The to­po­logy classes were fun, though S.-T. Hu’s lec­tur­ing style was some­what un­usu­al. At the time he was writ­ing books for gradu­ate stu­dents pre­par­ing for the PhD qual­i­fy­ing ex­ams so ba­sic­ally he read from his book on point set to­po­logy in the first semester while in the second he handed out about 700 mi­meo­graphed pages, writ­ten in his neat and large hand­writ­ing, of his yet-to-be-pub­lished book on al­geb­ra­ic to­po­logy. For the second semester the grad­ing was in­ter­est­ing: at the pen­ul­tim­ate lec­ture he called the class roll and those who answered re­ceived A grades, with B grades prom­ised for the oth­ers. Just af­ter­wards I met a class­mate who had missed all but the first lec­ture, and when I told her how he was al­loc­at­ing grades she vis­ited Pro­fess­or Hu and con­vinced him to call the roll a second time at his last lec­ture, thereby giv­ing those who would oth­er­wise have got B grades the chance for an A.

In those days it was deemed im­port­ant for a math­em­atician to know two for­eign lan­guages and we were warned by the De­part­ment chair at the be­gin­ning of sum­mer 1966 that the tests for these lan­guages were to be­come harder at the be­gin­ning of the new aca­dem­ic year. I spent some time over sum­mer get­ting to the stage where I could at least pass the ex­am in French, leav­ing Ger­man for later. After fail­ing in my first at­tempt at Ger­man I left it to one side un­til ex­am stand­ards for the second lan­guage were re­laxed some­what, re­quir­ing only that the can­did­ate trans­late a rel­ev­ant art­icle to the sat­is­fac­tion of the su­per­visor. Rob told me to find an art­icle, trans­late it and give it to him. With no Google trans­late avail­able in those days I gave him my hon­est at­tempt which he re­turned a week later with the com­ment that he had checked the math­em­at­ics in my trans­la­tion and it made sense so I must have trans­lated cor­rectly.

In the second year things really began to get ex­cit­ing as I con­cen­trated on to­po­logy courses. There were in­ter­est­ing courses offered by UCLA fac­ulty mem­bers — Robert Brown and Ed­mund Staples, for ex­ample — and oth­ers taught by vis­it­ors. I was for­tu­nate to have two out­stand­ing pro­fess­ors that year. One was Nor­man Steen­rod, who was vis­it­ing from Prin­ceton and lec­tured us on the the­ory of fibre bundles and on ho­mo­logy and co­homo­logy the­ory, with a cul­min­at­ing series on his eponym­ous squares. The oth­er was Rob, only a year out of his own PhD but already a clear lec­turer whose geo­met­ric de­scrip­tions made ho­mo­topy the­ory and char­ac­ter­ist­ic classes seem so nat­ur­al. It was in these lec­tures that I got to know two oth­er PhD stu­dents in to­po­logy, Ted Turn­er and Bob Hall, with whom I could spend time talk­ing about these ex­cit­ing top­ics. Un­for­tu­nately, though, I also had to pass that drat­ted qual­i­fy­ing ex­ams which were loom­ing at the end of the first quarter, so the top­ics of these lec­tures had to take a back seat to my pre­par­a­tion for those ex­ams. Once they were out of the way I asked Rob to be my thes­is su­per­visor and he agreed.

There were oth­er great vis­it­ors in my second and third years: One day I was present­ing a sem­in­ar when someone I didn’t know entered the room and sat listen­ing. At the end he asked a ques­tion, but aimed it at Jim Kister who had been my ment­or for the talk. The mys­tery man was John Mil­nor who, like Kister and Ed­win Span­i­er, was giv­ing an ex­cit­ing series of lec­tures at UCLA that time. We also had a series of lec­tures whose aim was to com­plete a proof of the Poin­caré con­jec­ture but in the last week the lec­turer had to con­fess that he had found a fatal flaw in his proof.

In­ter­est­ing as all the lec­tures and sem­inars were, my main job was to deep­en my know­ledge by read­ing a range of pa­pers and by meet­ing weekly with Rob. In those days, well be­fore the in­ter­net, one typ­ic­ally wrote to the au­thors of art­icles one was in­ter­ested in, asked for a copy of a par­tic­u­lar pa­per and any­thing else re­lated, and anxiously waited the week or so it took to get a reply — a much hoped-for bundle of re­prints and pre­prints (ac­tu­ally prin­ted on pa­per!). I spent time di­gest­ing the pa­pers of Brown and Mazur, which laid out their (re­spect­ive) proofs of the gen­er­al­ised Schoen­flies the­or­em (in the lat­ter case with an ad­di­tion by Morse), Zee­man’s IHES lec­ture notes on PL to­po­logy, and pa­pers by Brown and Gluck on stable homeo­morph­isms and their con­nec­tion to the then an­nu­lus con­jec­ture.

By late 1968 I was well in­to a re­search top­ic in PL to­po­logy work­ing some­what on my own as Rob was in Prin­ceton for the quarter. We main­tained loose con­tact through hand­writ­ten let­ters. Just be­fore Rob was due back I re­ceived a flurry of let­ters from him full of math­em­at­ics de­scrib­ing some amaz­ing dis­cov­er­ies he had just made, mostly jointly with Larry Sieben­mann. These dis­cov­er­ies fol­lowed on from Rob’s tor­us trick which had en­abled him to solve the an­nu­lus con­jec­ture, one of sev­en prob­lems put for­ward in 1963 by Mil­nor at a con­fer­ence on dif­fer­en­tial and al­geb­ra­ic to­po­logy as his “can­did­ates for the toughest and most im­port­ant prob­lems in geo­met­ric to­po­logy.” (The list was later in­cor­por­ated in­to a 1965 pa­per ed­ited by Lashof and pub­lished in the An­nals of Math­em­at­ics.) By fur­ther use of the tor­us trick and oth­er ideas, he and Sieben­mann had man­aged to solve two more of the sev­en. Each of Rob’s let­ters in­cluded a note say­ing that he would re­turn to LA a couple of days later than pre­vi­ously planned. I had the hon­our of passing on these new res­ults to col­leagues at UCLA.

The first half of 1969 was ex­tremely busy. Rob and Larry presen­ted lots of lec­tures on their re­cent dis­cov­er­ies, while Bob Hall and I took co­pi­ous notes which were typed up1 and made avail­able to any­one else in­ter­ested in this de­scrip­tion of their work. At the same time I had to drop my mod­est ef­forts in PL to­po­logy and pick up a new top­ic, which even­tu­ally formed the ma­jor part of my PhD thes­is. By June the Kirby–Sieben­mann notes were in good or­der and I was on my way back to a job in New Zea­l­and with a fresh PhD cer­ti­fic­ate in my bag­gage.

My time in the US had spanned an event­ful peri­od. Soon after be­com­ing Gov­ernor of Cali­for­nia, Ron­ald Re­agan, with oth­ers, had moved to dis­miss the Pres­id­ent of the Uni­versity of Cali­for­nia, Clark Kerr. The Vi­et­nam War had gained mo­mentum, as had op­pos­i­tion to it, and many male US cit­izens were get­ting mar­ried to avoid be­ing draf­ted (and, for our part, we at UCLA were re­ceiv­ing some­what trau­mat­ised vet­er­ans of the war as stu­dents). Lyn­don John­son had de­cided not to run for an­oth­er term as Pres­id­ent. Mar­tin Luth­er King and Robert Kennedy were both as­sas­sin­ated, with the lat­ter dy­ing the day I had my first PhD or­al ex­am­in­a­tion.

Be­ing in South­ern Cali­for­nia had also brought ad­vant­ages: a chance to ex­plore some of the state’s nat­ur­al beau­ties, even though these were gen­er­ally not very close to LA. As a stu­dent, I had sorely missed ac­cess to places where I could go tramp­ing (the NZ word for hik­ing). I did get to Death Val­ley, Yosemite and Se­quoia Na­tion­al Parks a few times, and, with an­oth­er New Zea­l­and PhD stu­dent in math­em­at­ics, climbed Mt. Whit­ney from Road’s End in a single day. (Mt. Whit­ney is the highest peak in the con­tigu­ous United States; to­po­lo­gists might guess the math­em­at­ic­al con­nec­tion.) On an­oth­er oc­ca­sion, Rob poin­ted my wife and me to Lake Ed­iza and Mt. Ritter on the east­ern side of the Si­er­ras, a bit south of Yosemite. It is a beau­ti­ful spot but Rob’s ob­ser­va­tion that if it rained then usu­ally the rain came in the af­ter­noon “only for an hour or so” proved mis­lead­ing: we had 24 hours of con­tinu­ous rain dur­ing one of our days there! We did get to the top of Ritter and even took a movie (on film; no di­git­al avail­able!) which Rob was keen to look at. It turned out that he planned to take his fath­er there the next month so he was in­ter­ested to get an idea of the route from his scouts.

I owe Rob a lot, both for his guid­ance when I was his PhD stu­dent and for his in­flu­ence on me as a teach­er. Re­gard­ing the lat­ter, Rob’s lec­tures have al­ways aimed to cla­ri­fy spe­cif­ic con­cepts for his audi­ence — par­tic­u­larly, as is ap­pro­pri­ate in to­po­logy, geo­met­ric ones. I have tried to fol­low his ex­ample throughout my ca­reer, even though I can’t help but no­tice that for many oth­ers the goal seems just the op­pos­ite: to im­press an audi­ence by baff­ling it. Thank you, Rob.

Born and raised in the for­es­ted hills of the cent­ral North Is­land of New Zea­l­and, Dav­id Gauld fol­lowed stud­ies at the Uni­versity of Auck­land with his PhD stud­ies at UCLA be­fore re­turn­ing to Auck­land and his alma ma­ter, where he served as Head of the Math­em­at­ics De­part­ment as well as As­sist­ant Vice-Chan­cel­lor be­fore re­tir­ing in 2017.