by Rufus Willett
I first met Paul when I was a graduate student at Penn State University, and my first serious interaction with him was as a graduate student in his introductory \( K \)-theory course. This course, in various versions, has given generations of Penn State students their first glimpses of \( K \)-theory. Paul started the course with a talk he has given on many occasions to a general audience about the beginnings of \( K \)-theory, how it historically split into topological and algebraic versions, and how these have remerged via the solution of Karoubi’s conjecture on the algebraic \( K \)-theory of \( C^* \)-algebras. Although I did not know it at the time, Paul has played important roles in many aspects of this material over the last fifty years (for example, through his work on the Riemann–Roch theorem, on Baum–Douglas \( K \)-homology, and on the Baum–Connes conjecture).
As one would expect from someone who has a large personal stake in the subject, Paul’s course was quite different from what one might find elsewhere, or in the standard texts, on \( C^* \)-algebra \( K \)-theory. It ranged broadly, eventually converging on a statement of the famous Baum–Connes conjecture, or “the conjecture of Alain and me” as Paul generally referred to it. It also included various life lessons: as he said “You don’t just get knowledge in this class, but wisdom.” Unfortunately, while most of the mathematics has stayed with me (or occasionally floats up from my subconscious), the life lessons have evaporated; as Paul is someone who seems to live life rather well, this is no doubt my loss.
Outside of his formal course, as a starting graduate student I had almost zero sense of how mathematicians interacted with each other, and was intimidated by pretty much everyone. Initially, Paul was certainly amongst the intimidators, but through no fault of his own: he has the sort of ebullient and nonjudgmental personality that breaks through such barriers quickly. Paul was always approachable (when not traveling, as he was often elsewhere, doing research and as a highly in-demand lecturer). I distinctly remember asking Paul a technical question about the \( K \)-theory of certain \( C^* \)-algebras, and his immediate response of “Yes, this is predicted by the conjecture of Alain and me!” after which he proceeded to explain where to look; this was very useful for my eventual thesis.
I should say a little more about the Baum–Connes conjecture, in order to explain what happened next in my interactions with Paul. This is a deep conjecture connecting algebra, analysis, and topology. It was originally formulated by Paul and Alain Connes in around 1980 using the geometric picture of \( K \)-homology that was worked out a little earlier by Paul and Ron Douglas; the more modern formulation uses Kasparov’s \( KK \)-theory, and is due to Paul, Connes, and Nigel Higson. The conjecture has many important applications to algebra, representation theory, geometry, topology, and \( C^* \)-algebra theory. However, in around 2000, Higson, Vincent Lafforgue and Georges Skandalis proved the existence of counterexamples to a suitably general version of the conjecture “with coefficients”. These counterexamples were based on the notorious Gromov monster groups, whose Cayley graphs contain expanders, leading to wild analytic properties.
After graduating from Penn State in 2009, my next substantive involvement with Paul was at the Oberwolfach meeting on Noncommutative Geometry in 2011. My postdoctoral mentor Guoliang Yu and I had been revisiting the examples of Higson, Lafforgue, and Skandalis, and we were attempting to see “how bad” the problems really were, in some sense. We were able to show that if one changed the analytic ingredients involved slightly, then the previous counterexamples actually become confirming examples; however, the analytic changes we made were clearly incompatible with known results on the original Baum–Connes conjecture, so our results were in some sense just a curiosity at this stage.
Nonetheless, Paul was very enthusiastic! His support at this stage in my career was very much appreciated, and certainly a boost to my, at the time, rather fragile sense of mathematical self. Paul quickly saw how to make the next step, which was to change the analytic ingredients involved in a way intermediate to the original Baum–Connes conjecture and the results of Yu and myself. He traveled the world asking various mathematicians whether this intermediate step was indeed possible (amongst other things!), eventually getting a positive answer from Eberhard Kirchberg. Having also joined forces with Erik Guentner, we were up and running. Over the course of several enjoyable visits and excellent dinners (particularly at the JMM in Baltimore in 2014, around the completion of the project), we were able to give a natural reformulation of the Baum–Connes conjecture which agrees with the original conjecture in all previously known cases, to which there are no known counterexamples, and for which the previously known counterexamples become confirming examples. This remains the current state of the reformulated conjecture.
On a more personal note, throughout my interactions with him, Paul has been a pleasure. He has been supportive and generous with his time, knowledge, and ideas. He was, and remains, most enjoyable company.
