Celebratio Mathematica

My first (not in per­son) en­counter with Paul Baum was via the Baum–Connes con­jec­ture. This con­jec­ture came up in talks I listened to as a stu­dent and as a be­gin­ning postdoc. It pre­dicts a power­ful con­nec­tion between two quite dif­fer­ent worlds: al­geb­ra­ic to­po­logy via the K-ho­mo­logy of a clas­si­fy­ing space is pre­dicted to be iso­morph­ic to the K-the­ory of a group $C^{\ast }$-al­gebra.

In my 1996 thes­is, I had stud­ied in­dex prob­lems on non­com­pact man­i­folds. To deep­en my know­ledge of this, in 1998 I star­ted a postdoc with John Roe (sponsored by the Ger­man Aca­dem­ic Ex­change agency). John had just moved to Penn State, and so with my whole fam­ily, in­clud­ing three small chil­dren, I moved to cent­ral Pennsylvania, as well. I was very de­lighted to ob­serve that Baum of the fam­ous Baum–Connes con­jec­ture was a real per­son and, much bet­ter, was a pro­fess­or at Penn State and also a mem­ber of the Geo­met­ric Func­tion­al Ana­lys­is group. Paul got in­ter­ested in par­tic­u­lar in my counter­example to the un­stable Gro­mov–Lawson–Rosen­berg con­jec­ture: there is a 5-di­men­sion­al closed spin man­i­fold which does not ad­mit a Rieman­ni­an met­ric with pos­it­ive scal­ar curvature, al­though the Rosen­berg in­dex, which takes its value in the right-hand side of the Baum–Connes con­jec­ture, is zero. This leads to a cor­rec­tion of the pre­dic­tion that the Baum–Connes con­jec­ture would im­ply the Gro­mov–Lawson–Rosen­berg con­jec­ture. This is true only if one adds the cru­cial ad­ject­ive “stable” to “Gro­mov–Lawson–Rosen­berg con­jec­ture”.

Very soon, Paul took on a role as my ment­or, and it was Paul with whom I talked most at Penn State. We star­ted the pro­ject to un­der­stand the left-hand side of the Baum–Connes con­jec­ture, the equivari­ant K-ho­mo­logy of spaces with prop­er ac­tion. Ac­tu­ally, it was Paul in his gen­er­os­ity who in­vited me to his pro­ject: un­der­stand this in the spe­cial case that the group is a $p$-ad­ic Lie group like ${\mathrm{Sl}}_n({\mathbb{Q}}_p)$ or a more gen­er­al totally dis­con­nec­ted Lie group. Once a week, we would meet for lunch at the Al­len Street Grill and dis­cuss the ques­tions and the pro­gress — again Paul was gen­er­ous and most of the time ex­plained his in­sights to me. The idea, by now es­sen­tially thought through but still wait­ing to be writ­ten up com­pletely, is to find a new and geo­met­ric mod­el of this K-ho­mo­logy group. The cycles are a some­what un­ex­pec­ted vari­ant of the Baum–Douglas cycles for usu­al K-ho­mo­logy.

The $(M,E,\phi )$-the­ory of Paul Baum and Ron Douglas is a fam­ous geo­met­ric de­scrip­tion of K-ho­mo­logy. K-ho­mo­logy is ini­tially giv­en in ab­stract-ho­mo­topy-the­or­et­ic terms, or equi­val­ently via Kas­parov’s power­ful, but com­plic­ated and ana­lyt­ic, KK-the­ory. When talk­ing about the new cycle mod­el for the ${\mathrm{Sl}}_n({\mathbb{Q}}_p)$-equivari­ant the­ory, we also had to look at the old Baum–Douglas the­ory; and when look­ing at the proofs of the main prop­er­ties we de­cided that the ori­gin­al pa­pers were more sketchy than de­sir­able. So, in our dis­cus­sion we de­veloped a more rig­or­ous and com­plete treat­ment of that the­ory. It seemed worth­while to write this up, but again be­ing slow it was per­haps only with the ad­di­tion­al push of our third coau­thor, Nigel Hig­son, that this pa­per fi­nally ap­peared in 2007 in Pure Ap­pl. Math. Q.1 By then, I had long since left Penn State, moved back to Münster and then in 2001 to Göttin­gen.

The col­lab­or­a­tion between Paul and my­self con­tin­ued and still con­tin­ues; we have met many times at dif­fer­ent oc­ca­sions all over the world, from Mo­scow to Warsaw and Par­is to the United States. In 2009 Paul and I over­lapped as “pro­fes­seurs in­vitées” at Uni­versité Blaise Pas­cal in Cler­mont-Fer­rand. Each time we met, we dis­cussed and pushed our old pro­jects fur­ther, adding and mov­ing fo­cus to new ones as time passed. The month in Cler­mont-Fer­rand led to the de­vel­op­ment of a geo­met­ric mod­el for equivari­ant K-ho­mo­logy for com­pact Lie group ac­tions, joint with our host Hervé Oy­ono-Oy­ono and my then stu­dent Mi­chael Wal­ter.

In Göttin­gen, col­leagues and I star­ted the Cour­ant Re­search Cen­ter “High­er-or­der Struc­tures in Math­em­at­ics”. It con­tained an ex­tens­ive guest pro­gram, and we had the pleas­ure of in­vit­ing Paul Baum sev­er­al times. Dur­ing the res­ult­ing vis­its Paul and I con­tin­ued our fruit­ful dis­cus­sions. Dur­ing these vis­its, I came to ap­pre­ci­ate an­oth­er of Paul’s qual­it­ies: his hu­mour. Baum can be very funny and we had many good laughs at din­ner parties in my home. In 2012, Paul spent a full couple of months in Göttin­gen as vis­it­ing Cour­ant pro­fess­or. The dis­cus­sion of ${\mathrm{Sl}}_n({\mathbb{Q}}_p)$-equivari­ant K-ho­mo­logy led us to the study of the cor­res­pond­ing equivari­ant K-the­ory. Paul had con­struc­ted with Peter Schneider a very nice Chern char­ac­ter for equivari­ant K-the­ory for com­pact totally dis­con­nec­ted groups. We stud­ied the ex­ten­sion to the loc­ally com­pact case, with a num­ber of sig­ni­fic­ant dif­fi­culties. This, like sev­er­al oth­er of our dis­cus­sions, led to a pa­per which is 80% fin­ished — get­ting the fi­nal work done might keep us busy for an­oth­er while.

As I write this es­say, Paul has again re­turned to Göttin­gen; this time as the Gauss pro­fess­or of the Göttinger Akademie der Wis­senschaften. We are work­ing on a geo­met­ric mod­el for twis­ted equivari­ant K-the­ory, of in­terest in string the­ory, and sug­ges­ted by Paul to­geth­er with Alan Carey and Bai-Ling Wang. Iron­ic­ally, for a long while Paul and I agreed that this mod­el should not work, des­pite claims that it does in a pa­per by Wang — and we spent con­sid­er­able ef­fort to try to prove this fact. However, the Gauss pro­fess­or­ship paid off: we made the ob­ser­va­tion that a ver­sion for twis­ted K-the­ory of a the­or­em of Hop­kins and Hovey will ac­tu­ally lead to a proof of Wang’s con­jec­ture. And, to­geth­er with Mi­chael Joachim and Me­hdi Khorami, we were in­deed able to es­tab­lish this twis­ted Hop­kins–Hovey the­or­em. Now, we are again at the crit­ic­al stage of writ­ing down the de­tails. In a team of four this should be done in a reas­on­ably short time, and we will be able to move to new, or back to the half-fin­ished old, pro­jects.

What did I learn from Paul Baum? The beauty of math­em­at­ics: I have be­nefited a lot from the clar­ity and beauty of Paul’s ap­proach to math­em­at­ics. He also taught me that ideas should be presen­ted clearly so that oth­er math­em­aticians can un­der­stand and ap­pre­ci­ate them. He did not teach me to be ef­fi­cient and to quickly write up our res­ults for pub­lic­a­tion. There is still a lot to de­vel­op, and it is a lot of fun to do this to­geth­er with Paul. I con­sider my­self for­tu­nate to have be­nefited from Paul’s col­lab­or­a­tion, and I look for­ward to many more years of do­ing so. (Vi­et­or­is wrote his last pub­lic­a­tion at the age of 103.)