Celebratio Mathematica

I first met Paul when I was a gradu­ate stu­dent at Penn State Uni­versity, and my first ser­i­ous in­ter­ac­tion with him was as a gradu­ate stu­dent in his in­tro­duct­ory $K$-the­ory course. This course, in vari­ous ver­sions, has giv­en gen­er­a­tions of Penn State stu­dents their first glimpses of $K$-the­ory. Paul star­ted the course with a talk he has giv­en on many oc­ca­sions to a gen­er­al audi­ence about the be­gin­nings of $K$-the­ory, how it his­tor­ic­ally split in­to to­po­lo­gic­al and al­geb­ra­ic ver­sions, and how these have re­merged via the solu­tion of Ka­roubi’s con­jec­ture on the al­geb­ra­ic $K$-the­ory of $C^{\ast }$-al­geb­ras. Al­though I did not know it at the time, Paul has played im­port­ant roles in many as­pects of this ma­ter­i­al over the last fifty years (for ex­ample, through his work on the Riemann–Roch the­or­em, on Baum–Douglas $K$-ho­mo­logy, and on the Baum–Connes con­jec­ture).

As one would ex­pect from someone who has a large per­son­al stake in the sub­ject, Paul’s course was quite dif­fer­ent from what one might find else­where, or in the stand­ard texts, on $C^{\ast }$-al­gebra $K$-the­ory. It ranged broadly, even­tu­ally con­ver­ging on a state­ment of the fam­ous Baum–Connes con­jec­ture, or “the con­jec­ture of Alain and me” as Paul gen­er­ally re­ferred to it. It also in­cluded vari­ous life les­sons: as he said “You don’t just get know­ledge in this class, but wis­dom.” Un­for­tu­nately, while most of the math­em­at­ics has stayed with me (or oc­ca­sion­ally floats up from my sub­con­scious), the life les­sons have evap­or­ated; as Paul is someone who seems to live life rather well, this is no doubt my loss.

Out­side of his form­al course, as a start­ing gradu­ate stu­dent I had al­most zero sense of how math­em­aticians in­ter­ac­ted with each oth­er, and was in­tim­id­ated by pretty much every­one. Ini­tially, Paul was cer­tainly amongst the in­tim­id­at­ors, but through no fault of his own: he has the sort of ebul­li­ent and non­judg­ment­al per­son­al­ity that breaks through such bar­ri­ers quickly. Paul was al­ways ap­proach­able (when not trav­el­ing, as he was of­ten else­where, do­ing re­search and as a highly in-de­mand lec­turer). I dis­tinctly re­mem­ber ask­ing Paul a tech­nic­al ques­tion about the $K$-the­ory of cer­tain $C^{\ast }$-al­geb­ras, and his im­me­di­ate re­sponse of “Yes, this is pre­dicted by the con­jec­ture of Alain and me!” after which he pro­ceeded to ex­plain where to look; this was very use­ful for my even­tu­al thes­is.

I should say a little more about the Baum–Connes con­jec­ture, in or­der to ex­plain what happened next in my in­ter­ac­tions with Paul. This is a deep con­jec­ture con­nect­ing al­gebra, ana­lys­is, and to­po­logy. It was ori­gin­ally for­mu­lated by Paul and Alain Connes in around 1980 us­ing the geo­met­ric pic­ture of $K$-ho­mo­logy that was worked out a little earli­er by Paul and Ron Douglas; the more mod­ern for­mu­la­tion uses Kas­parov’s $KK$-the­ory, and is due to Paul, Connes, and Nigel Hig­son. The con­jec­ture has many im­port­ant ap­plic­a­tions to al­gebra, rep­res­ent­a­tion the­ory, geo­metry, to­po­logy, and $C^{\ast }$-al­gebra the­ory. However, in around 2000, Hig­son, Vin­cent Laf­forgue and Georges Skan­dal­is proved the ex­ist­ence of counter­examples to a suit­ably gen­er­al ver­sion of the con­jec­ture “with coef­fi­cients”. These counter­examples were based on the no­tori­ous Gro­mov mon­ster groups, whose Cay­ley graphs con­tain ex­panders, lead­ing to wild ana­lyt­ic prop­er­ties.

After gradu­at­ing from Penn State in 2009, my next sub­stant­ive in­volve­ment with Paul was at the Ober­wolfach meet­ing on Non­com­mut­at­ive Geo­metry in 2011. My postdoc­tor­al ment­or Guo­li­ang Yu and I had been re­vis­it­ing the ex­amples of Hig­son, Laf­forgue, and Skan­dal­is, and we were at­tempt­ing to see “how bad” the prob­lems really were, in some sense. We were able to show that if one changed the ana­lyt­ic in­gredi­ents in­volved slightly, then the pre­vi­ous counter­examples ac­tu­ally be­come con­firm­ing ex­amples; however, the ana­lyt­ic changes we made were clearly in­com­pat­ible with known res­ults on the ori­gin­al Baum–Connes con­jec­ture, so our res­ults were in some sense just a curi­os­ity at this stage.

Non­ethe­less, Paul was very en­thu­si­ast­ic! His sup­port at this stage in my ca­reer was very much ap­pre­ci­ated, and cer­tainly a boost to my, at the time, rather fra­gile sense of math­em­at­ic­al self. Paul quickly saw how to make the next step, which was to change the ana­lyt­ic in­gredi­ents in­volved in a way in­ter­me­di­ate to the ori­gin­al Baum–Connes con­jec­ture and the res­ults of Yu and my­self. He traveled the world ask­ing vari­ous math­em­aticians wheth­er this in­ter­me­di­ate step was in­deed pos­sible (amongst oth­er things!), even­tu­ally get­ting a pos­it­ive an­swer from Eber­hard Kirch­berg. Hav­ing also joined forces with Erik Guent­ner, we were up and run­ning. Over the course of sev­er­al en­joy­able vis­its and ex­cel­lent din­ners (par­tic­u­larly at the JMM in Bal­timore in 2014, around the com­ple­tion of the pro­ject), we were able to give a nat­ur­al re­for­mu­la­tion of the Baum–Connes con­jec­ture which agrees with the ori­gin­al con­jec­ture in all pre­vi­ously known cases, to which there are no known counter­examples, and for which the pre­vi­ously known counter­examples be­come con­firm­ing ex­amples. This re­mains the cur­rent state of the re­for­mu­lated con­jec­ture.

On a more per­son­al note, throughout my in­ter­ac­tions with him, Paul has been a pleas­ure. He has been sup­port­ive and gen­er­ous with his time, know­ledge, and ideas. He was, and re­mains, most en­joy­able com­pany.