In my recent review of the Oeuvres of Jacques Tits, I noted that the course summaries from the Collège de France, where Tits lectured for many years, were not included because they would appear in a separate volume. Here is that volume.
The Collège de France is not a university. It was established by royal decree in 1530, at which point the idea was to appoint lecturers who would explore new ideas that were not yet part of the university curriculum. Nowadays, the Collège has 57 chairs who give lecture series on their current research. The lectures are open to the public, though of course they are often quite advanced. The idea is to give access to “knowledge in the making.”
As their website explains,
The Collège de France publishes the full text of inaugural lectures (collection Fayard), proceedings of symposia and series of lectures (collection Odile Jacob), the Letter of the Collège de France, and a “yearbook” which summarizes each professor’s teaching for the year.
Tits had the Group Theory chair at the Collège from 1973 to 2000, and of course the summaries of each of his courses appeared in the Annuaire (the “yearbook” in the quote above). These summaries, say the editors, “contain many results that cannot be found elsewhere,” and so it makes sense to extract them from the annual volumes and put them together in this collection. (Particularly since older editions of the Annuaire do not seem to be available online.)
The topics that got most attention over the years were Tits’s theory of buildings, the sporadic simple groups, discrete subgroups of real Lie groups, and quasi-reductive groups. But Tits followed his interests freely, so there are some surprises. The course from 1978–79 was on “Problems in the theory of groups arising from Einsteinian relativity and chronogeometry,” for example.
Two kinds of numbers appear in the outside margin: the page numbers from the original publication, and numbers indicating the editors’ notes. Some of the course summaries are quite short, while others include a lot of detail. In addition to notes, the editors have provided an index of names cited and a terminological index.
This book will be most interesting to those whose research is in the areas listed above. It is also a valuable historical record of mathematics in the last quarter of the twentieth century.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.