by Alain Connes
Paul Baum is both a scholar and a mathematician of great talent and achievements. His contributions contain several really outstanding results in the area of topology in connection with analysis and he has been a key actor in the early development of noncommutative geometry. I know him quite well and admire his geometric insight.
My encounter with Paul Baum at the Kingston conference in the summer of 1980 is one of a handful of these unexpected instances of great luck in my life. He looked like one of the pioneer aviators of the early twentieth century, with his round glasses and charming smile, always ready to discuss and learn new stuff with enthusiasm.
It was a time when the elucidation of the \( K \)-theory of foliations through the associated \( C^* \)-algebra was just beginning. The hint from geometry was coming from the construction of idempotents from geometric transversals. The leaf spaces of nontrivial foliations have the basic feature that their effective cardinality as sets is actually strictly larger than the continuum and this makes them quite different from the usual spaces which occur in differential geometry. At the same time Paul Baum was working with Ron Douglas on a version of \( K \)-homology based on geometric cycles and cobordism. When we met in Kingston for the first time we realized that these geometric cycles could be adapted to generalize the notion of transversals of foliations and could be organized in a group of purely geometric nature. The point there is that while it is hard to go from a leaf space to an ordinary space, it is easy to go from an ordinary space to a leaf space, using a suitable notion of cocycle. Moreover the geometric cycles defined by Baum and Douglas, with the map to the target suitably reinterpreted, were easy to organize into a group using cobordism and Bott periodicity. We then constructed the map from this geometric group to the analytic group of \( K \)-theory of the \( C^* \)-algebra of the foliation (there is Poincaré duality at work behind the scene) and started a very long and fruitful collaboration centering around properties of this map and extensions to many other cases going from Lie groups to crossed products by discrete groups. The ensuing Baum–Connes conjecture has been a central topic in the development of the subject since the time we made the conjecture in 1982.
Paul was coming regularly to work with me at the Institut des Hautes Études Scientifiques (IHES) and he was often accompanied by his mother, Celia. In my mind it would be unfair to both of them to omit her from the picture. While Paul always prided himself as “Monsieur le bon exemple” (concerning other matters than maths) his mother, in spite of her age, was a wild bird and a lovable person! I remember vividly when the three of us (Paul, Celia and myself) celebrated her 90th birthday concomitant with my own 50th and how around that time she was driving her electric wheelchair among the cars along the road from Gif-sur-Yvette to Bures. The pair of Paul and his mother were a great example of what our civilization can produce at its best. With this pair close-by there would always be something exciting going on!
She, as a poet who loved people and wine drinking.
He, as a mathematician of great insights, with outstanding achievements but always remaining modest, curious and open to new ideas, a scholar in the best sense of the word. More recently he pioneered another subject, that of the role of \( K \)-theory of \( C^* \)-algebras in the theory of representations of \( p \)-adic groups and again his work shows his great originality and talent.
