Celebratio Mathematica

Paul Baum

Working with Paul Baum

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.