Celebratio Mathematica

I don’t re­mem­ber when I met Paul for the first time, prob­ably at the In­sti­tute des Hautes Études Sci­en­ti­fiques (IHES) which he vis­ited many times while col­lab­or­at­ing with Alain Connes. At that time, al­most 40 years ago, I for­mu­lated a con­jec­ture pre­dict­ing an iso­morph­ism between al­geb­ra­ic and to­po­lo­gic­al $K$-the­ory of stable $C^{\ast }$-al­geb­ras. Paul was puzzled by this con­jec­ture, prob­ably in re­la­tion with his fam­ous “Baum–Connes con­jec­ture”. Paul loves to show his in­terests to people around him: Manusz Wod­zicki, made aware by Paul about my prob­lem, fi­nally solved it (with An­drei Suslin). There­fore, al­though Paul did not work on my ques­tion, he was very in­flu­en­tial in find­ing the right people to solve it!

As is well known, the Baum–Connes con­jec­ture (BCC) is still open, al­though many in­ter­est­ing cases are solved. There is also a “real” ver­sion (RB­CC) of BCC where the base field is the field of real num­bers in­stead of com­plex num­bers. Since Paul knew of my in­terest in real to­po­lo­gic­al $K$-the­ory, he star­ted to dis­cuss with me about RB­CC. After some time with some help from John Roe, we proved RB­CC is true in the cases where BCC is val­id. As Paul em­phas­ized to me, RB­CC im­plies the stable Gro­mov–Lawson–Rosen­berg con­jec­ture about Rieman­ni­an met­rics of pos­it­ive scal­ar curvature for a large class of fun­da­ment­al groups.

On an­oth­er oc­ca­sion, Paul was in­ter­ested in ap­plic­a­tions of non­com­mut­at­ive geo­metry to clas­sic­al al­geb­ra­ic to­po­logy. As a mat­ter of fact, Paul’s thes­is was in al­geb­ra­ic to­po­logy and he asked me wheth­er my ver­sion of “non­com­mut­at­ive co­chains” could provide a bet­ter un­der­stand­ing of the sub­ject of his thes­is. We still do not suc­ceed on this pro­ject but Paul will nev­er give up if he has the in­tu­ition of a “right” point of view!

This last pro­ject gives me the op­por­tun­ity to com­ment about Paul’s unique per­son­al­ity. Paul loves math­em­at­ics with a pas­sion he shares with his stu­dents and his col­lab­or­at­ors. He al­ways wants to un­der­stand deeply a sub­ject he is work­ing on. But this pas­sion is not at all aus­tere: he is play­ing with it, go­ing in­stantly from his thoughts to his im­me­di­ate sur­round­ings. I re­mem­ber once we were din­ing in a Par­is res­taur­ant, work­ing and laugh­ing at the same time. Our laughs (es­pe­cially Paul’s) were so loud that our next table was puzzled by these spe­cial guys: since they wanted to know our pro­fes­sion, we asked them to guess (with a hint that the first let­ter is an M). We were very proud that they thought we were mu­si­cians. On an­oth­er oc­ca­sion, also in a res­taur­ant, our next table thought that Paul was a Rus­si­an prince…

As a con­clu­sion, when look­ing at Paul’s bib­li­o­graphy, one is im­pressed by the vari­ety of sub­jects Paul has worked on, to­geth­er with out­stand­ing math­em­aticians, e.g., Raoul Bott and Alain Connes, to men­tion a few. His most re­cent work with Anne-Mar­ie Au­bert and Ro­ger Ply­men on group rep­res­ent­a­tions is re­mark­ably in­nov­at­ive.