My Research Story

Like Ludwig van Beethoven's, my creative life falls into three periods, but unlike his, my middle one was perfectly fallow : an unsightly hiatus mars my list of publications in the seventies, the hippie decade.

Rather than trying your patience with some unlikely explanation for this shameful blot on my escutcheon, I shall concentrate on outlining the contents of the first and third periods. Roughly speaking, the former is abstract and theoretical, delighting in concepts and valuing form over substance, while the latter is concrete and computational, using modest tools and preferring ugly facts to shapely froth.

This points to a second difference between Ludwig and myself : while he stubbornly followed his genius, I found nothing better to follow than the fashion of the day. In the 1950's the accepted wisdom was that mathematics was a deductive science based on axioms. Here at UBC (where I started and ended my career), there were still some dinosaurs who took pride in not knowing what a Banach algebra was, but the in-crowd cultivated general topology and ring theory. Robert Langlands, a fellow student, occasionally liked to organise informal seminars to plough through something he wanted to learn. In the summer of 57 he had chosen Jabobson's new book on the structure of rings -- just my cup of tea.

That autumn, at McGill University, I met a very different kind of mathematical animal: Hans Zassenhaus, who could talk rings around any ring theorist, and in the next few breaths, write down a complete proof of the formerly first, but now second, inequality of class field theory (using L-functions). Now I knew what I wanted to do: become like that, but clean up the act, i.e., base all that on abstractions. I was relieved to hear, in a talk of Tim O'Meara, that there were such things as local and global fields -- neatly axiomatized. That gave me the green light to look into things arithmetical. Little did I suspect that my clean-up fantasy was actually being realized by people like John Tate and André Weil, not to mention the gathered might of Bourbaki. My supervisor Zassenhaus did not tell me (of course, he knew I was not in that league). Instead he gave me something to do around envelopping algebras of Lie algebras. I can't remember exactly what it was, but it did eventually lead me to my thesis: central simple algebras constructed around purely inseparable field extensions (via an appropriate Lie-algebra of derivations). That mediocre work figures as items [2] and [3] in my list of publications.

I had followed Zassenhaus from McGill through Caltech to Notre Dame du Lac in Indiana, and didn't know whether I was coming or going. At Caltech, Olga Taussky had made me present a paper by Schur in the slick functorial mode (which was not her own); at Notre Dame, Louis Mordell (who was there temporarily) heaped scorn on André Weil for having done just that with "his" theorem, and Alex Heller showed up with tales of the miraculous Grothendieck. I wanted to go to Paris, but Zassenhaus suggested Hamburg, his former home base and my native city, which I had not seen since early childhood. There were Artin, Hasse, and Witt (and other greats farther afield), the mathematical institute was small and congenial. It was culture shock for me: my conversation partners (for instance Harder and Knebusch) were still students, but surefooted in anything from harmonic analysis, over the cohomology of idèles, to Dynkin diagrams, sheaf theory, the latest simple groups, etc... amazingly non-specialized. Now I was in the very crucible which had formed my teacher Zassenhaus, but this did not suffice to make me a really good mathematician: I was too busy running after what the others had just understood and/or invented.

(To be continued)