#### by David Gauld

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.*