Celebratio Mathematica

Paul Baum

Good old days at Brown and Penn State

by Hitoshi Moriyoshi

I met Pro­fess­or Baum for the first time in Tokyo in March 1986. Paul gave a lec­ture at the Uni­versity of Tokyo and talked about an early ver­sion of the Baum–Connes con­jec­ture and its ap­plic­a­tions. It was re­lated to to­po­lo­gic­al in­vari­ance of Pon­trja­gin classes, the Novikov con­jec­ture and the Gro­mov–Lawson con­jec­ture. I was in the audi­ence, as a gradu­ate stu­dent of the Uni­versity of Tokyo, su­per­vised by Akio Hat­tori, known for the Hat­tori–Stong the­or­em in Al­geb­ra­ic To­po­logy. At that time I had just fin­ished a Mas­ter’s thes­is on the ex­ist­ence prob­lem for pos­it­ive scal­ar curvature and I was in­ter­ested in a new de­vel­op­ment, Non­com­mut­at­ive Geo­metry ini­ti­ated by Alain Connes. I was much fas­cin­ated by Paul’s talk and his pro­found con­jec­ture re­lated to both To­po­logy and Ana­lys­is. Thus I began to think about trans­fer­ring from the Uni­versity of Tokyo to Brown, where Paul was on the fac­ulty.

In Septem­ber 1987 I moved to Brown Uni­versity. The De­part­ment of Math­em­at­ics was ac­com­mod­ated in a cozy Amer­ic­an house fa­cing Thay­er Street on the East Side of Provid­ence. In 1987–88 Paul was on leave from Brown and at Penn State. But when Paul got back to Provid­ence, he kindly spent a lot of time with a young gradu­ate stu­dent from the Far East. I re­mem­ber that there was a re­pro­duc­tion of “A View of Delft” by Ver­meer in his of­fice, which made me real­ize Paul’s soph­ist­ic­ated taste in paint­ings. It was no won­der since his fath­er, Marc Baum, is a fam­ous paint­er whose works are ex­hib­ited in the Met­ro­pol­it­an Mu­seum. I en­joyed my stu­dent life at Brown, but the De­part­ment of Math­em­at­ics had a hard time in those years. Many cel­eb­rated pro­fess­ors were about to move from Brown. It was un­for­tu­nate that W. Fulton, R. MacPh­er­son, J. Har­ris, Jean-Luc Bryl­in­ski, Ra­nee Bryl­in­ski and Paul even­tu­ally left Brown.

In sum­mer 1988, I had a chance to at­tend AMS meet­ings, the AMS Sum­mer In­sti­tute “Op­er­at­or The­ory and Op­er­at­or Al­geb­ras” in Durham, New Hamp­shire, and the AMS Sum­mer Re­search Con­fer­ence, “To­po­lo­gic­al In­vari­ants of El­lipt­ic Op­er­at­ors” at Bowdoin Col­lege in Maine. I still re­mem­ber the warm en­cour­age­ment from Paul when I gave talks.

I already men­tioned Paul’s soph­ist­ic­ated taste in paint­ings, but he was also tal­en­ted at mu­sic. Sarah, his daugh­ter, said “Daddy is a sing­er at home” when I joined his fam­ily. Later, I heard Paul’s beau­ti­ful voice in real­ity in Cor­tona, Italy, in the middle of a party at “Ana­lys­is and To­po­logy in In­ter­ac­tion 2008” or­gan­ized by Paolo Piazza. Peter Haskell told me that, when Peter was Paul’s cal­cu­lus teach­ing as­sist­ant, one Monday morn­ing he saw Paul climb on a desk in front of a large cal­cu­lus class and lead the class in singing “Morn­ing has Broken.”

In the AMS meet­ing at Bowdoin Col­lege, Paul and I had a walk on the cam­pus. Peter Haskell and N. Hig­son were also there, I re­mem­ber. We were talk­ing about \( K \)-the­ory and \( K \)-ho­mo­logy, which Paul had been work­ing on at that time with Ron Douglas and Mike Taylor. Paul ex­plained to me the sub­tlety of those sub­jects for a while and then con­cluded that “\( K \)-the­ory is di­vine, \( K \)-ho­mo­logy hu­man-cre­ated.” I re­mem­ber the phrase very well. We know the Baum–Connes con­jec­ture claims that a geo­met­ric \( K \)-ho­mo­logy group is iso­morph­ic to the \( K \)-the­ory of a \( C^* \)-al­gebra. It goes without say­ing that the con­jec­ture is based on a deep and beau­ti­ful the­or­em, the Atiyah–Sing­er in­dex the­or­em. Thus, if I may say so, the Baum–Connes con­jec­ture amounts to the fol­low­ing: A cre­ation by a hu­man can be as per­fect as one by God in such beau­ti­ful math­em­at­ics.

I should add some words on the de­vel­op­ment of \( K \)-ho­mo­logy af­ter­wards. Nigel and John Roe, who are now at Penn State, gave an­oth­er for­mu­la­tion of \( K \)-ho­mo­logy via Pasch­ke du­al­ity, which is a coun­ter­part of Span­i­er–White­head du­al­ity in ho­mo­topy the­ory. There­fore, \( K \)-ho­mo­logy is real­ized now as the \( K \)-the­ory of cer­tain \( C^* \)-al­geb­ras due to their the­ory even though it is not ex­actly the same as the geo­met­ric \( K \)-ho­mo­logy that ap­pears in the con­jec­ture.

In 1988 Paul moved to Penn State and so did Ra­nee and Jean-Luc Bryl­in­ski. Sev­er­al Brown stu­dents also moved to the new place, which my Amer­ic­an friend called “the middle of nowhere.” In fact, at the air­port of State Col­lege, no tax­is were wait­ing, so pas­sen­gers had to make calls to their friends to get to town. But it turned out to be a won­der­ful cir­cum­stance for gradu­ate stu­dents since there were no dis­tract­ing activ­it­ies oth­er than Math­em­at­ics. I at­ten­ded sem­inars on To­po­logy, Op­er­at­or Al­geb­ras and Non­com­mut­at­ive Geo­metry, which were or­gan­ized by strong math­em­aticians at Penn State, such as J. An­der­son, Jean-Luc Bryl­in­ski, R. Her­man, N. Hig­son, A. Ocneanu, C. Ogle, C. Skau, B. Tsy­gan and Paul. (J. Roe and Ping Xu joined Penn state later.)

It was very for­tu­nate for me to spend time with Paul at Penn State. I was able to talk with him about Math­em­at­ics throughout the af­ter­noon. Paul shared his great in­sight with me and also was very help­ful. One day I thought that I had proved a the­or­em. Take a cov­er­ing space whose deck trans­form­a­tion group is amen­able and as­sume that the base space is a closed man­i­fold. Then the spec­trum of Lapla­cian on the cov­er­ing space is not bounded from be­low, that is it con­tains zero. When I de­scribed the res­ult to Paul in his of­fice, he made a phone call to Jeff Chee­ger at once and kindly asked him about the res­ult. It turned out that it was well known to spe­cial­ists in Dif­fer­en­tial Geo­metry. So I could not get a the­or­em, but felt grate­ful to Paul for such kind help.

After fin­ish­ing my thes­is at Penn State, I joined SUNY Buf­falo in 1990. Toshi Nat­sume, Cathy Olsen, Lew Coburn and oth­ers formed an act­ive group in Op­er­at­or Al­geb­ras there. I spent one year at Buf­falo and left the United States for Ja­pan in 1991. Since then Paul has been to Ja­pan many times, and he con­tin­ues to in­flu­ence me.

I am much in­debted to Pro­fess­or Baum for what I learned by ob­serving the clar­ity of his math­em­at­ics, his earn­est at­ti­tude to­wards math­em­at­ics and even his noble spir­it of a great Amer­ic­an. I am grate­ful to him bey­ond words.