I met Professor Baum for the first time in Tokyo in March 1986. Paul gave a lecture at the University of Tokyo and talked about an early version of the Baum–Connes conjecture and its applications. It was related to topological invariance of Pontrjagin classes, the Novikov conjecture and the Gromov–Lawson conjecture. I was in the audience, as a graduate student of the University of Tokyo, supervised by, known for the Hattori–Stong theorem in Algebraic Topology. At that time I had just finished a Master’s thesis on the existence problem for positive scalar curvature and I was interested in a new development, Noncommutative Geometry initiated by . I was much fascinated by Paul’s talk and his profound conjecture related to both Topology and Analysis. Thus I began to think about transferring from the University of Tokyo to Brown, where Paul was on the faculty.
In September 1987 I moved to Brown University. The Department of Mathematics was accommodated in a cozy American house facing Thayer Street on the East Side of Providence. In 1987–88 Paul was on leave from Brown and at Penn State. But when Paul got back to Providence, he kindly spent a lot of time with a young graduate student from the Far East. I remember that there was a reproduction of “A View of Delft” by Vermeer in his office, which made me realize Paul’s sophisticated taste in paintings. It was no wonder since his father, Marc Baum, is a famous painter whose works are exhibited in the Metropolitan Museum. I enjoyed my student life at Brown, but the Department of Mathematics had a hard time in those years. Many celebrated professors were about to move from Brown. It was unfortunate that, , , , and Paul eventually left Brown.
In summer 1988, I had a chance to attend AMS meetings, the AMS Summer Institute “Operator Theory and Operator Algebras” in Durham, New Hampshire, and the AMS Summer Research Conference, “Topological Invariants of Elliptic Operators” at Bowdoin College in Maine. I still remember the warm encouragement from Paul when I gave talks.
I already mentioned Paul’s sophisticated taste in paintings, but he was also talented at music. Sarah, his daughter, said “Daddy is a singer at home” when I joined his family. Later, I heard Paul’s beautiful voice in reality in Cortona, Italy, in the middle of a party at “Analysis and Topology in Interaction 2008” organized by. told me that, when Peter was Paul’s calculus teaching assistant, one Monday morning he saw Paul climb on a desk in front of a large calculus class and lead the class in singing “Morning has Broken.”
In the AMS meeting at Bowdoin College, Paul and I had a walk on the campus. Peter Haskell andwere also there, I remember. We were talking about \( K \)-theory and \( K \)-homology, which Paul had been working on at that time with and . Paul explained to me the subtlety of those subjects for a while and then concluded that “\( K \)-theory is divine, \( K \)-homology human-created.” I remember the phrase very well. We know the Baum–Connes conjecture claims that a geometric \( K \)-homology group is isomorphic to the \( K \)-theory of a \( C^* \)-algebra. It goes without saying that the conjecture is based on a deep and beautiful theorem, the Atiyah–Singer index theorem. Thus, if I may say so, the Baum–Connes conjecture amounts to the following: A creation by a human can be as perfect as one by God in such beautiful mathematics.
I should add some words on the development of \( K \)-homology afterwards. Nigel and, who are now at Penn State, gave another formulation of \( K \)-homology via Paschke duality, which is a counterpart of Spanier–Whitehead duality in homotopy theory. Therefore, \( K \)-homology is realized now as the \( K \)-theory of certain \( C^* \)-algebras due to their theory even though it is not exactly the same as the geometric \( K \)-homology that appears in the conjecture.
In 1988 Paul moved to Penn State and so did Ranee and Jean-Luc Brylinski. Several Brown students also moved to the new place, which my American friend called “the middle of nowhere.” In fact, at the airport of State College, no taxis were waiting, so passengers had to make calls to their friends to get to town. But it turned out to be a wonderful circumstance for graduate students since there were no distracting activities other than Mathematics. I attended seminars on Topology, Operator Algebras and Noncommutative Geometry, which were organized by strong mathematicians at Penn State, such as, Jean-Luc Brylinski, , N. Higson, , , , and Paul. (J. Roe and joined Penn state later.)
It was very fortunate for me to spend time with Paul at Penn State. I was able to talk with him about Mathematics throughout the afternoon. Paul shared his great insight with me and also was very helpful. One day I thought that I had proved a theorem. Take a covering space whose deck transformation group is amenable and assume that the base space is a closed manifold. Then the spectrum of Laplacian on the covering space is not bounded from below, that is it contains zero. When I described the result to Paul in his office, he made a phone call to Jeff Cheeger at once and kindly asked him about the result. It turned out that it was well known to specialists in Differential Geometry. So I could not get a theorem, but felt grateful to Paul for such kind help.
After finishing my thesis at Penn State, I joined SUNY Buffalo in 1990., , and others formed an active group in Operator Algebras there. I spent one year at Buffalo and left the United States for Japan in 1991. Since then Paul has been to Japan many times, and he continues to influence me.
I am much indebted to Professor Baum for what I learned by observing the clarity of his mathematics, his earnest attitude towards mathematics and even his noble spirit of a great American. I am grateful to him beyond words.