by Thomas Schick
My first (not in person) encounter with Paul Baum was via the Baum–Connes conjecture. This conjecture came up in talks I listened to as a student and as a beginning postdoc. It predicts a powerful connection between two quite different worlds: algebraic topology via the K-homology of a classifying space is predicted to be isomorphic to the K-theory of a group \( C^* \)-algebra.
In my 1996 thesis, I had studied index problems on noncompact manifolds. To deepen my knowledge of this, in 1998 I started a postdoc with John Roe (sponsored by the German Academic Exchange agency). John had just moved to Penn State, and so with my whole family, including three small children, I moved to central Pennsylvania, as well. I was very delighted to observe that Baum of the famous Baum–Connes conjecture was a real person and, much better, was a professor at Penn State and also a member of the Geometric Functional Analysis group. Paul got interested in particular in my counterexample to the unstable Gromov–Lawson–Rosenberg conjecture: there is a 5-dimensional closed spin manifold which does not admit a Riemannian metric with positive scalar curvature, although the Rosenberg index, which takes its value in the right-hand side of the Baum–Connes conjecture, is zero. This leads to a correction of the prediction that the Baum–Connes conjecture would imply the Gromov–Lawson–Rosenberg conjecture. This is true only if one adds the crucial adjective “stable” to “Gromov–Lawson–Rosenberg conjecture”.
Very soon, Paul took on a role as my mentor, and it was Paul with whom I talked most at Penn State. We started the project to understand the left-hand side of the Baum–Connes conjecture, the equivariant K-homology of spaces with proper action. Actually, it was Paul in his generosity who invited me to his project: understand this in the special case that the group is a \( p \)-adic Lie group like \( \mathrm{Sl}_n(\mathbb{Q}_p) \) or a more general totally disconnected Lie group. Once a week, we would meet for lunch at the Allen Street Grill and discuss the questions and the progress — again Paul was generous and most of the time explained his insights to me. The idea, by now essentially thought through but still waiting to be written up completely, is to find a new and geometric model of this K-homology group. The cycles are a somewhat unexpected variant of the Baum–Douglas cycles for usual K-homology.
The \( (M,E,\phi) \)-theory of Paul Baum and Ron Douglas is a famous geometric description of K-homology. K-homology is initially given in abstract-homotopy-theoretic terms, or equivalently via Kasparov’s powerful, but complicated and analytic, KK-theory. When talking about the new cycle model for the \( \mathrm{Sl}_n(\mathbb{\mathbb{Q}}_p) \)-equivariant theory, we also had to look at the old Baum–Douglas theory; and when looking at the proofs of the main properties we decided that the original papers were more sketchy than desirable. So, in our discussion we developed a more rigorous and complete treatment of that theory. It seemed worthwhile to write this up, but again being slow it was perhaps only with the additional push of our third coauthor, Nigel Higson, that this paper finally appeared in 2007 in Pure Appl. Math. Q.1 By then, I had long since left Penn State, moved back to Münster and then in 2001 to Göttingen.
The collaboration between Paul and myself continued and still continues; we have met many times at different occasions all over the world, from Moscow to Warsaw and Paris to the United States. In 2009 Paul and I overlapped as “professeurs invitées” at Université Blaise Pascal in Clermont-Ferrand. Each time we met, we discussed and pushed our old projects further, adding and moving focus to new ones as time passed. The month in Clermont-Ferrand led to the development of a geometric model for equivariant K-homology for compact Lie group actions, joint with our host Hervé Oyono-Oyono and my then student Michael Walter.
In Göttingen, colleagues and I started the Courant Research Center “Higher-order Structures in Mathematics”. It contained an extensive guest program, and we had the pleasure of inviting Paul Baum several times. During the resulting visits Paul and I continued our fruitful discussions. During these visits, I came to appreciate another of Paul’s qualities: his humour. Baum can be very funny and we had many good laughs at dinner parties in my home. In 2012, Paul spent a full couple of months in Göttingen as visiting Courant professor. The discussion of \( \mathrm{Sl}_n(\mathbb{Q}_p) \)-equivariant K-homology led us to the study of the corresponding equivariant K-theory. Paul had constructed with Peter Schneider a very nice Chern character for equivariant K-theory for compact totally disconnected groups. We studied the extension to the locally compact case, with a number of significant difficulties. This, like several other of our discussions, led to a paper which is 80% finished — getting the final work done might keep us busy for another while.
As I write this essay, Paul has again returned to Göttingen; this time as the Gauss professor of the Göttinger Akademie der Wissenschaften. We are working on a geometric model for twisted equivariant K-theory, of interest in string theory, and suggested by Paul together with Alan Carey and Bai-Ling Wang. Ironically, for a long while Paul and I agreed that this model should not work, despite claims that it does in a paper by Wang — and we spent considerable effort to try to prove this fact. However, the Gauss professorship paid off: we made the observation that a version for twisted K-theory of a theorem of Hopkins and Hovey will actually lead to a proof of Wang’s conjecture. And, together with Michael Joachim and Mehdi Khorami, we were indeed able to establish this twisted Hopkins–Hovey theorem. Now, we are again at the critical stage of writing down the details. In a team of four this should be done in a reasonably short time, and we will be able to move to new, or back to the half-finished old, projects.
What did I learn from Paul Baum? The beauty of mathematics: I have benefited a lot from the clarity and beauty of Paul’s approach to mathematics. He also taught me that ideas should be presented clearly so that other mathematicians can understand and appreciate them. He did not teach me to be efficient and to quickly write up our results for publication. There is still a lot to develop, and it is a lot of fun to do this together with Paul. I consider myself fortunate to have benefited from Paul’s collaboration, and I look forward to many more years of doing so. (Vietoris wrote his last publication at the age of 103.)
