by Max Karoubi
I don’t remember when I met Paul for the first time, probably at the Institute des Hautes Études Scientifiques (IHES) which he visited many times while collaborating with Alain Connes. At that time, almost 40 years ago, I formulated a conjecture predicting an isomorphism between algebraic and topological \( K \)-theory of stable \( C^{*} \)-algebras. Paul was puzzled by this conjecture, probably in relation with his famous “Baum–Connes conjecture”. Paul loves to show his interests to people around him: Manusz Wodzicki, made aware by Paul about my problem, finally solved it (with Andrei Suslin). Therefore, although Paul did not work on my question, he was very influential in finding the right people to solve it!
As is well known, the Baum–Connes conjecture (BCC) is still open, although many interesting cases are solved. There is also a “real” version (RBCC) of BCC where the base field is the field of real numbers instead of complex numbers. Since Paul knew of my interest in real topological \( K \)-theory, he started to discuss with me about RBCC. After some time with some help from John Roe, we proved RBCC is true in the cases where BCC is valid. As Paul emphasized to me, RBCC implies the stable Gromov–Lawson–Rosenberg conjecture about Riemannian metrics of positive scalar curvature for a large class of fundamental groups.
On another occasion, Paul was interested in applications of noncommutative geometry to classical algebraic topology. As a matter of fact, Paul’s thesis was in algebraic topology and he asked me whether my version of “noncommutative cochains” could provide a better understanding of the subject of his thesis. We still do not succeed on this project but Paul will never give up if he has the intuition of a “right” point of view!
This last project gives me the opportunity to comment about Paul’s unique personality. Paul loves mathematics with a passion he shares with his students and his collaborators. He always wants to understand deeply a subject he is working on. But this passion is not at all austere: he is playing with it, going instantly from his thoughts to his immediate surroundings. I remember once we were dining in a Paris restaurant, working and laughing at the same time. Our laughs (especially Paul’s) were so loud that our next table was puzzled by these special guys: since they wanted to know our profession, we asked them to guess (with a hint that the first letter is an M). We were very proud that they thought we were musicians. On another occasion, also in a restaurant, our next table thought that Paul was a Russian prince…
As a conclusion, when looking at Paul’s bibliography, one is impressed by the variety of subjects Paul has worked on, together with outstanding mathematicians, e.g., Raoul Bott and Alain Connes, to mention a few. His most recent work with Anne-Marie Aubert and Roger Plymen on group representations is remarkably innovative.
