Celebratio Mathematica

Paul Baum

A conversation with Paul Baum

by Peter Haskell

Haskell: Tell me a little about your grow­ing up, par­tic­u­larly about your back­ground and fam­ily life in your early years.

Baum: My par­ents had an apart­ment in a build­ing on 106th Street near Am­s­ter­dam Av­en­ue in Man­hat­tan. My earli­est memory is of sit­ting on the front steps of that apart­ment build­ing and con­tem­plat­ing a nearby candy store and an en­gin­eer­ing pro­ject. The candy store in­ter­ested me be­cause they sold a kind of chew­ing gum known as “Man­drake the Ma­gi­cian” gum. This astound­ing gum changed col­ors as you chewed it. The en­gin­eer­ing pro­ject was the tear­ing down of the Am­s­ter­dam Av­en­ue El (“El” is the “el­ev­ated train”). I was fas­cin­ated by the big ma­chines and the amaz­ing things they did.

My par­ents were in­tel­li­gent and tal­en­ted. My fath­er, Mark Baum, was born in San­ok (in what was then Aus­tria–Hun­gary and is now Po­land) in 1903. When he was four years old, the fam­ily moved to Rszezów where his ma­ter­nal grand­fath­er was a suc­cess­ful busi­ness­man. He grew up in his grand­fath­er’s im­pos­ing house sur­roun­ded by a large and pros­per­ous ex­ten­ded fam­ily. Mark was in the first gen­er­a­tion of Jew­ish boys to have a sec­u­lar edu­ca­tion. In the loc­al gym­nas­i­um, each school day began with the Lord’s Pray­er. The Cath­ol­ic boys stood and re­cited the pray­er. Each Jew­ish boy stood si­lently and de­fi­antly with his arms crossed over his chest. World War I was a huge dis­aster which dis­rup­ted everything. After the war, Mark left for Amer­ica. Trav­el­ing alone, he ar­rived in New York in 1920 at the age of sev­en­teen.

My moth­er, Celia Frank Baum, was born in Schenectady, New York, in 1907. When she was one year old, the fam­ily moved to Brook­lyn. The Rebbe of Tolna, i.e., one of the great rab­bis of nine­teenth cen­tury East­ern Europe, was her great-grand­fath­er. Her par­ents had come to the USA after the fail­ure of the 1905 at­tempt to over­throw the czar. She grew up in Brook­lyn in ex­treme poverty. A neigh­bor­hood pub­lic lib­rary, partly fun­ded by An­drew Carne­gie, be­came a refuge for her from the prob­lems and dif­fi­culties of her im­mig­rant fam­ily.

My par­ents first met on the beach in Provin­cetown, Mas­sachu­setts, in the sum­mer of 1932. The at­trac­tion between them was very strong and they im­me­di­ately began their tur­bu­lent off-again-on-again re­la­tion­ship which con­tin­ued un­til their deaths in 1997–98. My fath­er was an artist. He began by paint­ing land­scapes and city­scapes. Very gradu­ally, his paint­ing style evolved in­to com­plete ab­strac­tion. He has paint­ings in many mu­seums and private col­lec­tions, in­clud­ing two in the per­man­ent col­lec­tion of the Met­ro­pol­it­an Mu­seum of Art. (For more on my fath­er’s paint­ings go to mark­ My moth­er star­ted as a teach­er in the New York pub­lic schools. She worked her way up and even­tu­ally be­came a pro­fess­or of edu­ca­tion in Brook­lyn Col­lege. She liked to joke about her ca­reer say­ing, “Those who can, do. Those who can’t, teach. And those who can’t even teach be­come pro­fess­ors of edu­ca­tion like me.”

My par­ents mar­ried in 1935. I was born in 1936 and my broth­er Wil­li­am (Billy) Baum was born in 1939.

I went through the pub­lic schools in New York, and then was an un­der­gradu­ate at Har­vard. After a year in France, I re­turned to the USA and en­rolled in Prin­ceton as a gradu­ate stu­dent. My par­ents’ story and my per­son­al story say something about the good­ness and gen­er­os­ity of Amer­ica. I hope that Amer­ica will re­main ideal­ist­ic and good. At the present time (Ju­ly, 2019) there is some pos­sib­il­ity that this will not hap­pen.

Haskell: When did you first con­sider be­com­ing a math­em­atician?

Baum: In grade school I was dimly aware that math­em­aticians ex­is­ted, but I had no clear idea of what it was that math­em­aticians did. When I was eight years old, I began to won­der about vari­ous math­em­at­ic­al prob­lems. One prob­lem that in­ter­ested me was the is­sue of cal­cu­lat­ing the length of the di­ag­on­al of a rect­angle from the lengths of the two sides. I con­cluded that it was some sort of av­er­age of the lengths of the two sides, al­though I didn’t come up with the pre­cise for­mula.

In high school (Bronx High School of Sci­ence), I had a won­der­ful cha­ris­mat­ic teach­er, Ju­li­us Hlav­aty, for tenth-grade geo­metry. I loved solv­ing the geo­metry prob­lems and work­ing out the proofs of the­or­ems. The proof of the Py­thagorean the­or­em was spe­cially sat­is­fy­ing to me as it defin­it­ively solved the rect­angle prob­lem I had thought about years earli­er. Dr. Hlav­aty was the coach of the math team and I was on this team. We com­peted against math teams from the oth­er New York high schools. In tenth grade I de­cided that math was for me.

In twelfth grade I entered the West­ing­house Sci­ence Tal­ent Search. I was one of forty fi­nal­ists awar­ded a trip to Wash­ing­ton, DC, and a schol­ar­ship to be used against first-year col­lege ex­penses.

Haskell: When did you know that you would be a math­em­atician?

Baum: In my Prin­ceton Ph.D. thes­is I solved a prob­lem in al­geb­ra­ic to­po­logy that had baffled the ex­perts. Ar­mand Borel, who was one of the ex­perts on the prob­lem, ad­vised me not to work on it. I was, however, able to come up with a new ap­proach and thus solve the prob­lem. So at the time it seemed clear to me that I would be a math­em­atician.

Haskell: Did you have doubts after that?

Baum: Alas, a few years after com­plet­ing my Ph.D. thes­is it be­came clear that there was a gap in my proof. I had not made any false as­ser­tions, but one some­what tech­nic­al point which had seemed ob­vi­ous to me re­quired a proof. On the day that I real­ized there was a gap, I said to my wife, “Pack your bags. They are go­ing to kick me out of Prin­ceton.” That is ex­actly what happened. We moved to Provid­ence and I was on the fac­ulty at Brown Uni­versity.

Be­fore leav­ing Prin­ceton, I had at­ten­ded a lec­ture by Raoul Bott. Dur­ing his lec­ture I de­cided to aban­don the prob­lem of my thes­is and work on the top­ics in­tro­duced in Bott’s very in­ter­est­ing talk. Once at Brown, I went reg­u­larly to Har­vard. Bott and I worked on de­vel­op­ing his ideas.

Mean­while, vari­ous math­em­aticians worked on the pro­ject of com­plet­ing the proof in my thes­is. Sev­er­al pa­pers were pub­lished giv­ing com­plete proofs. One of these pa­pers was by my stu­dent Joel Wolf, based on his Brown Uni­versity Ph.D. thes­is. Joel went on to a long and dis­tin­guished ca­reer as an ap­plied math­em­atician.

As any­one can ima­gine, there were mo­ments of dis­cour­age­ment dur­ing all this. With the be­ne­fit of sev­er­al dec­ades of hind­sight, it is now clear that my thes­is was a break­through on a worth­while prob­lem. The work I did with Bott (fo­li­ation sin­gu­lar­it­ies) was cer­tainly of equal in­terest to the work in my thes­is.

Haskell: Were role mod­els an im­port­ant part of en­vi­sion­ing your­self as a math­em­atician? If so, how?

Baum: I had a num­ber of role mod­els who were im­port­ant to me. First, there was my in­spir­ing high school math teach­er Ju­li­us Hlav­aty. Then, while an un­der­gradu­ate at Har­vard, there were George Mackey and Oscar Za­r­iski. I loved Za­r­iski’s Yid­dish ac­cent and his un­fail­ing cheer­ful­ness. With Mackey, I ad­mired his in­tense de­vo­tion to math­em­at­ics. My role mod­el when I was a gradu­ate stu­dent at Prin­ceton was Nor­man Steen­rod. Even now, I am still amazed by the at­ten­tion and guid­ance Steen­rod gave to his stu­dents.

As a postdoc at Ox­ford, my role mod­el was Mi­chael Atiyah. The re­mark­able en­ergy and ima­gin­a­tion he brought to math­em­at­ics made a deep im­pres­sion on me.

Dur­ing my years at Brown, my role mod­el was Raoul Bott. In ad­di­tion to be­ing a mar­velous math­em­atician, Bott was a most en­ter­tain­ing and amus­ing per­son­al­ity. I really en­joyed in­ter­act­ing with him.

Sev­er­al times I vis­ited the Max Planck In­sti­tute in Bonn whose dir­ect­or was Friedrich Hirzebruch. I came to greatly ad­mire and like Hirzebruch. For me, his Riemann–Roch the­or­em was an ex­ample of truly great math­em­at­ics. He was astound­ingly none­got­ist­ic­al and de­voted great amounts of time and en­ergy to help­ing young math­em­aticians.

With Alain Connes, I ad­mired the dar­ing and ori­gin­al­ity he brought to math­em­at­ics. The mes­sage from Alain al­ways seemed to be “Go ahead. Don’t be in­hib­ited. If it in­terests you, start think­ing about it and see what will hap­pen.” The ar­ti­fi­cial bound­ar­ies between the dif­fer­ent fields in math­em­at­ics meant very little to Alain.

My par­ents were friendly with Wolfgang and Dorothy Fuchs. Wolfgang was a pro­fess­or in the math­em­at­ics de­part­ment of Cor­nell Uni­versity. He and Dorothy had fled Ger­many when the Nazis came to power. Wolfgang en­cour­aged me. He gave me some math­em­at­ics books. He and Dorothy gave me a beau­ti­ful tuxedo which had been cus­tom made for Wolfgang when he was a stu­dent in Eng­land. I wore it at the form­al din­ners of Har­vard’s Eli­ot House. When my neph­ew Aaron Baum was a fresh­man at Har­vard, I gave the tuxedo to him.

Haskell: After a couple of years as an as­sist­ant pro­fess­or at Prin­ceton, you moved to Brown. How did you feel about that move at the time? How did it turn out?

Baum: When I ar­rived at Brown, I sud­denly felt lib­er­ated. I was work­ing with Bott and it seemed that a whole won­der­ful world of pure math­em­at­ics was open­ing up for me. The Brown math de­part­ment made one su­perb ap­point­ment after an­oth­er. With­in a few years of my ar­rival, the Brown math de­part­ment had be­come one of the best in the USA. We had a golden age in which to­po­logy and al­geb­ra­ic geo­metry flour­ished at Brown. Bob MacPh­er­son’s sem­in­ar played a key role dur­ing those won­der­ful years.

At Brown, Bill Fulton and Bob MacPh­er­son and I ex­ten­ded the Grothen­dieck–Riemann–Roch the­or­em to pro­ject­ive vari­et­ies which may be sin­gu­lar. Many math­em­aticians had tried to do this, but had not suc­ceeded due to an in­cor­rect over­all view of what was in­volved. The cor­rect point of view oc­curred to me one Rosh Hasha­nah when I was at­tend­ing ser­vices at my loc­al syn­agogue in Provid­ence. Since then I have al­ways said, “You see, there is no con­flict between re­li­gion and math­em­at­ics.” This may or may not be true, but what without doubt is true is that Fulton and MacPh­er­son are mar­velous math­em­aticians and we made a sub­stan­tial con­tri­bu­tion to the de­vel­op­ment of Riemann–Roch.

Dur­ing my time at Brown, after fif­teen years of try­ing, I fi­nally came up with a simple ex­pli­cit geo­met­ric defin­i­tion of \( K \)-ho­mo­logy — i.e., the dual the­ory to \( K \)-the­ory. Ron Douglas and I worked to­geth­er to find the iso­morph­ism between the BDF (Brown–Douglas–Fill­more) func­tion­al ana­lys­is mod­el of \( K \)-ho­mo­logy and my geo­met­ric mod­el.

The ba­sic idea of the func­tion­al ana­lys­is mod­el is due to Atiyah and was stated in his pa­per “Glob­al the­ory of el­lipt­ic op­er­at­ors” [e1]. Fur­ther de­vel­op­ment of the func­tion­al ana­lys­is mod­el was done by BDF and G. Kas­parov. This mod­el has the ma­jor ad­vant­age that it ap­plies to \( C^* \)-al­geb­ras which might not be com­mut­at­ive. The dis­ad­vant­age of this mod­el is that it does not have an ex­pli­cit clearly defined Chern char­ac­ter. The the­ory, due to Alain Connes, of cyc­lic co­homo­logy and spec­tral triples, ad­dresses this is­sue. Roughly speak­ing, any suf­fi­ciently smooth ob­ject for Kas­parov \( K \)-ho­mo­logy does have a Chern char­ac­ter in cyc­lic co­homo­logy. The geo­met­ric mod­el of \( K \)-ho­mo­logy comes equipped with an ex­pli­cit Chern char­ac­ter defined in terms of “clas­sic­al” char­ac­ter­ist­ic classes. Part of the Connes the­ory is that this is con­sist­ent with the cyc­lic co­homo­logy meth­od.

At Brown, I began work­ing with Alain Connes on the con­jec­ture now known as the Baum–Connes con­jec­ture. This con­jec­ture is un­usu­al for its breadth and for the many im­plic­a­tions of its valid­ity. There may be counter­examples lurk­ing out there. So far no counter­examples — for dis­crete groups and loc­ally com­pact groups — have been found. Over one hun­dred pa­pers have been pub­lished which prove the con­jec­ture for vari­ous classes of dis­crete groups and loc­ally com­pact groups. In an ex­pos­it­ory pa­per, Vaughan Jones and Henri Mo­scov­ici de­scribe the con­jec­ture as “a power­ful uni­fy­ing prin­ciple.”

I had sev­er­al ex­cel­lent Ph.D. stu­dents at Brown. Among them were Peter Haskell, Hitoshi Mor­iy­oshi, James Stormes and Joel Wolf. Hitoshi star­ted his Ph.D. thes­is at Brown and then con­tin­ued at Penn State.

Un­for­tu­nately, golden ages do not last forever. Math­em­aticians at Brown began to re­ceive out­side of­fers. The ad­min­is­tra­tion of Brown did not take strong de­cis­ive ac­tion to pre­serve the math de­part­ment. So it was time to move on. A num­ber of uni­versit­ies were in­ter­ested in me. I de­cided to take chance on Penn State.

Haskell: Talk about your time at Penn State.

Baum: A re­mark­able per­son, Richard Her­man, was head of the math de­part­ment at Penn State. He at­trac­ted ex­cel­lent math­em­aticians to Penn State and trans­formed the de­part­ment in­to one of the lead­ing USA math de­part­ments. This was a spec­tac­u­lar achieve­ment. When R. Her­man be­came head at Penn State, the de­part­ment had a very low NRC rank­ing. In the most re­cent NRC rank­ings, the Penn State math de­part­ment ranked sev­enth. In some ways, the rise of the Penn State math de­part­ment was even more astound­ing than what had happened at Brown. An im­port­ant dif­fer­ence was the much great­er com­pet­ence of the ad­min­is­tra­tion at Penn State.

For my­self, I found Penn State to be a really pleas­ant and con­struct­ive place in which to live and work. Nigel Hig­son (one of the first-rate math­em­aticians that Richard Her­man had brought to Penn State) or­gan­ized a sem­in­ar that be­came my math­em­at­ic­al home. Dur­ing my time at Penn State, I con­tin­ued to ex­plore is­sues con­nec­ted to the Baum–Connes con­jec­ture. Also — with Anne-Mar­ie Au­bert, Ro­ger Ply­men, and Maarten Sol­leveld — I began work­ing on the rep­res­ent­a­tion the­ory of \( p \)-ad­ic groups. With Erik van Erp, I worked on in­dex the­ory for a class of dif­fer­en­tial op­er­at­ors which are not el­lipt­ic.

Penn State pro­moted me from Pro­fess­or to Dis­tin­guished Pro­fess­or to Evan Pugh Uni­versity Pro­fess­or in Math­em­at­ics. I am grate­ful to Penn State for provid­ing a pos­it­ive en­vir­on­ment in which I could de­vel­op the math­em­at­ics I was in­ter­ested in.

Haskell: You were first known as an al­geb­ra­ic to­po­lo­gist. You did some work in dif­fer­en­tial and al­geb­ra­ic geo­metry. Now you are primar­ily as­so­ci­ated with the op­er­at­or al­gebra com­munity. How did you come to fol­low such an un­usu­al pro­fes­sion­al tra­ject­ory?

Baum: Yes, I have fol­lowed an un­usu­al tra­ject­ory. What happened was that one thing led to an­oth­er in a way that is not ob­vi­ous.

One day in the au­tumn of 1963, at ap­prox­im­ately noon, I de­cided to leave my of­fice in the Ox­ford Math In­sti­tute and do something about lunch. As luck would have it, Mi­chael Atiyah was also ex­it­ing the build­ing at that mo­ment. We walked out to­geth­er and pro­ceeded to­geth­er for about 100 meters un­til we ar­rived at an in­ter­sec­tion where we then veered off in op­pos­ite dir­ec­tions. Dur­ing the 100 meters, Atiyah sketched out his func­tion­al ana­lys­is defin­i­tion of \( K \)-ho­mo­logy. I thought to my­self (but did not say), “This is a lot of non­sense. \( K \)-ho­mo­logy should be defined in terms of geo­met­ric cycles and ho­mo­lo­gies of those cycles. I am go­ing to find a geo­met­ric defin­i­tion of \( K \)-ho­mo­logy.”

Dur­ing the next fif­teen years, I would briefly vis­it Ox­ford every now and then and de­scribe to Atiyah my latest at­tempt at a geo­met­ric defin­i­tion of \( K \)-ho­mo­logy. His re­sponse was al­ways the same. He would listen for about two minutes and then say, “What you have done is use­less.” As time went by, however, I began to feel that I was gradu­ally hom­ing in on a really good geo­met­ric defin­i­tion of \( K \)-ho­mo­logy. When I out­lined one of my pro­posed defin­i­tions to Bob MacPh­er­son, he made a com­ment which led me to re­vise that defin­i­tion — and now I was cer­tain that I had it. I re­turned to Ox­ford. At the black­board in Atiyah’s of­fice, I presen­ted what I be­lieved to be a cor­rect and use­ful geo­met­ric mod­el for \( K \)-ho­mo­logy. I ex­pec­ted the usu­al “What you have done is use­less.” In­stead, an eer­ie si­lence em­an­ated from Atiyah. He broke the si­lence with “That is very at­tract­ive and I want you to give a talk about that in my sem­in­ar to­mor­row.”

Once a use­ful geo­met­ric mod­el for \( K \)-ho­mo­logy had been achieved, much then fol­lowed:

  • \( K \)-ho­mo­logy is the ana­log in to­po­logy of the Grothen­dieck group of co­her­ent al­geb­ra­ic sheaves in al­geb­ra­ic geo­metry. This ob­ser­va­tion is the basis for the res­ults on Riemann–Roch with Bill Fulton and Bob MacPh­er­son.
  • The fact that the Dir­ac op­er­at­or of a Spin\( ^c \) man­i­fold de­term­ines an iso­morph­ism between the geo­met­ric mod­el of \( K \)-ho­mo­logy and \( K \)-ho­mo­logy defined via func­tion­al ana­lys­is led to the BC (Baum–Connes) con­jec­ture.
  • V. Laf­forgue proved that BC is val­id for re­duct­ive \( p \)-ad­ic groups. This in­dic­ated that there might be an in­ter­est­ing geo­met­ric struc­ture in the smooth du­als of such groups. \begin{multline*} \text{Smooth dual of } G = \\ \{\text{Isomorphism classes of irreducible smooth representations of }G\}. \end{multline*} The ABPS (Au­bert–Baum–Ply­men–Sol­leveld) con­jec­ture as­serts that there is such a geo­met­ric struc­ture. This is rel­ev­ant to the loc­al Lang­lands con­jec­ture. Spe­cial­ists in the rep­res­ent­a­tion the­ory of \( p \)-ad­ic groups at first dis­missed ABPS as something that was “too good to be true.” ABPS has been proved for a very large class of re­duct­ive \( p \)-ad­ic groups, in­clud­ing all the clas­sic­al groups.
  • The iso­morph­ism of geo­met­ric and func­tion­al ana­lys­is \( K \)-ho­mo­logy turned out to be ex­actly what was needed (joint work with Erik van Erp) to solve the in­dex prob­lem for Fred­holm op­er­at­ors in the Heis­en­berg cal­cu­lus of con­tact man­i­folds. Many of these Fred­holm op­er­at­ors are dif­fer­en­tial op­er­at­ors which are not el­lipt­ic. A num­ber of ex­cel­lent math­em­aticians had tried to solve this in­dex prob­lem. Ap­par­ently the heat equa­tion meth­od does not pro­duce a solu­tion.

Haskell: I be­lieve that you first met Alain Connes at the 1980 King­ston, Ontario, op­er­at­or al­geb­ras con­fer­ence. What were your first im­pres­sions? De­scribe the path that led to your col­lab­or­a­tion on what is known as the Baum–Connes con­jec­ture. How have the for­mu­la­tion and status of the con­jec­ture evolved over the many years since it first ap­peared in a 1982 pre­print?

Baum: In the early sum­mer of 1980 I drove to Stony Brook in my beat-up old blue Dodge (my nick­name for the car was “the blue won­der car”), picked up Ron Douglas, and headed for King­ston. As Ron and I rolled in­to King­ston, I said to Ron, “Ex­cept for you, I don’t know any­body at this meet­ing. So I don’t have to go to any of the talks, and this is go­ing to be a great va­ca­tion for me.” However, I did go to many of the lec­tures, in­clud­ing the talk by Ron. I had as­sumed that Ron would give an ex­pos­i­tion of what he and I had done on \( K \)-ho­mo­logy. In­stead, his talk centered on is­sues in­volving Clif­ford al­geb­ras. Thus it seemed to me that it was up to me to give an ex­pos­i­tion of our joint work. My talk was sched­uled for late in the af­ter­noon of a day in the second week of the meet­ing. I did not ex­pect many people to at­tend the talk, and very few did.

Alain (whom I did not know at the time) was one of the few at­tendees of my talk. That even­ing, after din­ner, he lay in wait for me in the com­mon area of the dorm­it­ory in which we were all stay­ing. He came up to me and in­formed me that he and I (and his en­tour­age) were go­ing out to a nearby pub for drinks. The en­tour­age con­sisted of Georges Skan­dal­is, Pierre Ju­lg, and Alain Valette. Over many glasses of beer, Alain began to ex­plain his ideas to me and I struggled to un­der­stand. For the next few days, in­stead of go­ing to talks, Alain and I would meet and I would try to un­der­stand what he was telling me.

My niece, Shona Baum, was re­turn­ing from a trip to Europe and I had prom­ised my broth­er that I would meet her at the Bo­ston air­port. So I left King­ston, met Shona, and then phoned Alain to tell him that I would soon be back in King­ston. So, when I re-ar­rived in King­ston, Alain con­tin­ued to ex­plain his beau­ti­ful idea that what Ron Douglas and I had done was really just a be­gin­ning. A month later we con­tin­ued at IHES and I began to fully ap­pre­ci­ate Alain’s point of view. He was say­ing that giv­en a \( C^* \)-al­gebra arising from geo­metry-to­po­logy, there should be a for­mula for the \( K \)-the­ory of that \( C^* \)-al­gebra in terms of the ori­gin­al geo­metry-to­po­logy which had pro­duced the \( C^* \)-al­gebra.

Dur­ing the next ten years, I sporad­ic­ally vis­ited Alain at IHES and we struggled to pre­cisely for­mu­late the con­jec­ture now known as Baum–Connes. On my many vis­its to IHES, I usu­ally would not go alone. My wife Bar­bara, some of our chil­dren, and some­times my moth­er, would ac­com­pany me. A friend­ship de­veloped between Alain and my moth­er. Des­pite the ma­jor age gap, there was an un­der­ly­ing rap­port and un­der­stand­ing between them. There were many many good times dur­ing these years. Din­ner parties giv­en by Alain and his wife Dani were great fun. To be­gin, there was al­ways cham­pagne fol­lowed by a de­li­cious din­ner in the French style. For one of these oc­ca­sions, my moth­er went to a hair styl­ist who dyed her hair a bright flam­ing red-or­ange. She made quite an en­trance.

The Baum–Connes (BC) con­jec­ture has a cer­tain au­da­cious qual­ity to it. For loc­ally com­pact groups and dis­crete groups, the con­jec­ture had to be slightly re­for­mu­lated once nonex­act groups were proved to ex­ist. This re­for­mu­la­tion was achieved in joint work with Erik Guent­ner and Ru­fus Wil­lett. So for loc­ally com­pact groups and dis­crete groups no counter­example has been found. Gran­ted the ex­treme gen­er­al­ity of the con­jec­ture and its many im­plic­a­tions, it is cer­tainly pos­sible that even­tu­ally counter­examples will be dis­covered. If so, then the is­sue will be to de­term­ine the class of loc­ally com­pact and dis­crete groups for which the con­jec­ture is val­id.

Vari­ous points of view are pos­sible on BC. The one that I prefer is that BC is a gen­er­al­iz­a­tion of the Atiyah–Sing­er in­dex the­or­em. If the loc­ally com­pact group un­der con­sid­er­a­tion is the trivi­al one-ele­ment group, then BC be­comes Atiyah–Sing­er.

Haskell: Why do \( K \)-the­ory and \( K \)-ho­mo­logy come up again and again in the­or­ems that con­nect dif­fer­ent areas of math­em­at­ics?

Baum: The com­bin­a­tion \( K \)-ho­mo­logy-and-\( K \)-the­ory is sim­pler and more power­ful than the com­bin­a­tion ho­mo­logy-and-co­homo­logy of clas­sic­al al­geb­ra­ic to­po­logy. This is il­lus­trated by J. F. Adams’ solu­tion to the vec­tor fields on spheres prob­lem. Adams was get­ting par­tial res­ults by ap­ply­ing stand­ard ho­mo­logy-and-co­homo­logy. But his ar­gu­ments were get­ting more and more com­plic­ated as he was us­ing sec­ond­ary and ter­tiary co­homo­logy op­er­a­tions. Then Atiyah and Hirzebruch (guided by what Grothen­dieck had done in al­geb­ra­ic geo­metry) brought \( K \)-the­ory in­to to­po­logy. Adams ap­plied \( K \)-the­ory to ob­tain his much-cel­eb­rated solu­tion of the vec­tor field prob­lem. In \( K \)-the­ory primary co­homo­logy op­er­a­tions suf­ficed. To un­der­stand the sol­ar sys­tem, a mod­el with the earth at the cen­ter can be used — but everything gets more and more com­plic­ated with cycles and epi-cycles and epi-epi-cycles, etc., etc. The Co­per­nic­an he­lio­centric mod­el is sim­pler and more power­ful.

\( K \)-ho­mo­logy-and-\( K \)-the­ory seems to cap­ture in just the right way the in­dex-the­or­et­ic prop­er­ties of many in­ter­est­ing math­em­at­ic­al and phys­ic­al prob­lems. An ex­ample of this is the RR the­or­em of Baum–Fulton–MacPh­er­son. Ac­cord­ing to this the­or­em, a co­her­ent al­geb­ra­ic sheaf \( \mathcal{F} \) on a com­plex pro­ject­ive al­geb­ra­ic vari­ety \( X \) (\( X \) may have sin­gu­lar­it­ies) de­term­ines an ele­ment in the \( K \)-ho­mo­logy of the un­der­ly­ing loc­ally com­pact Haus­dorff space of \( X \). The in­dex-the­or­et­ic prop­er­ties of \( \mathcal{F} \) are to­po­lo­gic­ally ex­pressed by this \( K \)-ho­mo­logy ele­ment.

An ex­tremely pleas­ant prop­erty of \( K \)-ho­mo­logy-and-\( K \)-the­ory is that it ex­tends very eas­ily and nat­ur­ally to non­com­mut­at­ive con­texts. BC is based on this. With­in the non­com­mut­at­ive set­ting, G. Kas­parov has merged and uni­fied \( K \)-ho­mo­logy and \( K \)-the­ory with his bivari­ant \( KK \)-the­ory.

In gradu­ate school we study ho­mo­logy-and-co­homo­logy. Many math­em­aticians are un­aware that \( K \)-ho­mo­logy-and-\( K \)-the­ory is sim­pler and more power­ful. The found­a­tion of \( K \)-ho­mo­logy-and-\( K \)-the­ory is Bott peri­od­icity. Thus Bott peri­od­icity has emerged as cent­ral to much of mod­ern math­em­at­ics and the­or­et­ic­al phys­ics.

Haskell: Both in one-on-one con­ver­sa­tions and in talks to audi­ences, you have a dis­tinct­ive style of start­ing from the be­gin­ning and linger­ing on defin­i­tions when oth­er speak­ers would have hur­ried on to the­or­ems. To what ex­tent is this style a com­mu­nic­a­tion tac­tic and to what ex­tent does it il­lus­trate how you think about math­em­at­ics?

Baum: It is primar­ily a com­mu­nic­a­tion tac­tic. To a lim­ited ex­tent, it is also how I think about math­em­at­ics. In math­em­at­ics we try to make pre­cise defin­i­tions. Grothen­dieck be­lieved that once really good defin­i­tions have been for­mu­lated, proofs of the­or­ems will then eas­ily and nat­ur­ally fol­low. Raoul Bott used to say that if you have got it right, math­em­at­ics is like be­ing in row­boat and not row­ing, “float­ing down the river with the cur­rent.”

Un­for­tu­nately many math talks are only com­pre­hens­ible to those spe­cial­ists in the audi­ence who already have a sol­id know­ledge of the top­ics covered in the talk. This is a ser­i­ous weak­ness in our sub­ject. Math­em­aticians (es­pe­cially young math­em­aticians) un­der­stand that they will be giv­en little or no cred­it for good ex­pos­i­tion. In my opin­ion, there is a le­git­im­ate place in math­em­at­ics for clear and well-or­gan­ized ex­pos­i­tion. It is worth­while to note that some of the most em­in­ent math­em­aticians write books and give talks in which the ex­pos­i­tion is ex­cel­lent. Over the years and dec­ades, I have de­veloped a point of view on in­dex the­ory. This point of view has been elab­or­ated in a series of pa­pers with my cowork­er Erik van Erp [3], [2], [1]. I hope that Erik and I will write a book based on these pa­pers. Of course, I also hope that in such a book the ex­pos­i­tion will be OK.

Haskell: Grothen­dieck’s in­flu­ence on in­dex the­ory seems per­vas­ive. The to­po­lo­gic­al proof of the Atiyah–Sing­er in­dex the­or­em is an ex­ample in which the in­flu­ence is ex­pli­citly ac­know­ledged. The power of Gen­nadi Kas­parov’s \( KK \)-the­ory de­rives from us­ing \( K \)-the­or­et­ic ob­jects as ho­mo­morph­isms of \( K \)-the­or­et­ic groups is an­oth­er ex­ample, which seems fur­ther re­moved. Is it too simplist­ic to say that Grothen­dieck’s con­tri­bu­tion is the idea that (co-)ho­mo­logy the­or­ies are not just homes for to­po­lo­gic­al in­vari­ants but can play a dir­ect role in geo­met­ric con­struc­tions?

Baum: Grothen­dieck’s in­flu­ence on in­dex the­ory stems from his Riemann–Roch the­or­em (GRR). In or­der to state and prove GRR, Grothen­dieck in­tro­duced and defined \( K \)-the­ory. So from day one, \( K \)-the­ory has been closely linked to in­dex the­ory. I think that Grothen­dieck’s main con­tri­bu­tions to this area of math­em­at­ics are

  • defin­i­tion (with­in al­geb­ra­ic geo­metry) of \( K \)-ho­mo­logy and \( K \)-the­ory;
  • func­tori­al point of view.

Grothen­dieck aimed to give a proof of the Hirzebruch–Riemann–Roch the­or­em which would be func­tori­al and very nat­ur­al. He suc­ceeded bril­liantly. The ana­log for the Atiyah–Sing­er in­dex the­or­em is much closer than was in­dic­ated in the “to­po­lo­gic­al proof” of Atiyah and Sing­er. I have giv­en sev­er­al talks ex­plain­ing this, and I hope to present it in a book.

The Kas­parov \( KK \)-the­ory brings trans­vers­al­ity in­to the non­com­mut­at­ive set­ting. It is re­mark­able that such a geo­met­ric concept (i.e., trans­vers­al­ity) can be de­veloped us­ing func­tion­al ana­lys­is.

Haskell: In his Cel­eb­ra­tio pa­per, Jonath­an Rosen­berg re­marks that your math­em­at­ic­al con­tri­bu­tions show the in­flu­ence of Al­ex­an­der Grothen­dieck, me­di­ated per­haps by your work, with Bill Fulton and Bob MacPh­er­son, on a Riemann–Roch the­or­em for sin­gu­lar vari­et­ies. To what ex­tent were you con­scious of Grothen­dieck’s in­flu­ence? How did your Riemann–Roch work in­flu­ence your ca­reer tra­ject­ory?

Baum: Bill Fulton and Bob MacPh­er­son and I took Grothen­dieck’s proof of GRR and re­vised it so as to make it even more nat­ur­al and func­tori­al. When you do this, GRR be­comes a the­or­em which ap­plies to pro­ject­ive vari­et­ies which may have sin­gu­lar­it­ies. Grothen­dieck was surely one of the great math­em­aticians of the twen­ti­eth cen­tury. I am just one of a vast num­ber of math­em­aticians who were in­flu­enced by his in­spir­ing ideas.

Haskell: You are al­most cer­tainly best known for the Baum–Connes con­jec­ture. Why has that con­jec­ture cap­tured the ima­gin­a­tion of so many math­em­aticians? Is it fair to say that the con­jec­ture’s sig­ni­fic­ance arises from the ques­tions about rep­res­ent­a­tion the­ory, ri­gid­ity state­ments in to­po­logy, geo­met­ric group the­ory, and op­er­at­or al­geb­ras that it draws at­ten­tion to, of­ten by a pro­cess of us­ing one area to cast light on an­oth­er area?

Baum: As men­tioned above, the Baum–Connes (BC) con­jec­ture is un­usu­al for its breadth and for the many im­plic­a­tions of its valid­ity. When val­id for a loc­ally com­pact or dis­crete group \( G \) it im­plies some pre­vi­ously stated con­jec­tures: Novikov high­er sig­na­ture con­jec­ture (to­po­logy of man­i­folds), Gro­mov–Lawson–Rosen­berg pos­it­ive scal­ar curvature con­jec­ture (dif­fer­en­tial geo­metry), Kadis­on–Ka­plansky idem­potent con­jec­ture (\( C^* \)-al­geb­ras). In ad­di­tion, BC sug­gests that for any Lie group \( G \) and for any re­duct­ive \( p \)-ad­ic group \( G \) there should be an in­ter­est­ing geo­met­ric struc­ture in the set of equi­val­ence classes of ir­re­du­cible rep­res­ent­a­tions of \( G \). So the con­jec­ture is (with­in pure math­em­at­ics) very in­ter­dis­cip­lin­ary.

Haskell: On the oth­er hand, your con­jec­ture has no ap­plic­a­tions out­side of math­em­at­ics (ex­cept per­haps to some very the­or­et­ic­al phys­ics), has little or no in­ter­ac­tion with ma­chine com­pu­ta­tion, and might be dis­missed by the ma­jor­ity of math­em­aticians as an ex­ample of the kind of late-twen­ti­eth-cen­tury math­em­at­ics that, like the fine and lib­er­al arts, is of little sig­ni­fic­ance in an in­creas­ingly prac­tic­al age. How do you re­spond to such as­ser­tions?

Baum: I re­spond by say­ing that we should not kill the goose who is lay­ing the golden eggs. Ap­plic­a­tions of sci­ence and math­em­at­ics are very im­port­ant. Some of these ap­plic­a­tions have giv­en us tech­no­lo­gic­al mir­acles and are truly won­drous. The ob­vi­ous para­dox is that in or­der to have really good ap­plic­a­tions it is ne­ces­sary to have a thriv­ing cul­ture of pure sci­ence and math­em­at­ics. Ein­stein for­mu­lated his the­ory of re­lativ­ity in an ef­fort to un­der­stand what is true about the phys­ic­al uni­verse. For many years there were no prac­tic­al ap­plic­a­tions of Ein­stein’s the­ory. Now vari­ous en­gin­eer­ing pro­jects (e.g., dig­ging of the tun­nel un­der the Eng­lish Chan­nel) do make re­lativ­ist­ic cor­rec­tions. Num­ber the­ory seemed to be the purest of the pure — a sub­ject that nev­er ever would have any prac­tic­al ap­plic­a­tions. Now res­ults from num­ber the­ory are used in cryp­to­graphy. The point is that over and over again math­em­at­ics and sci­ence that at first ap­pear to have no ap­plic­a­tions even­tu­ally do have very use­ful and amaz­ing ap­plic­a­tions. A so­ci­ety that fails to vig­or­ously sup­port pure math and sci­ence will, in the long run, lose out on ap­plic­a­tions.

There is, however, an even more ba­sic is­sue. Ap­plic­a­tions are very im­port­ant, but are not some all-im­port­ant everything. What are the ap­plic­a­tions of Molière’s plays? What are the ap­plic­a­tions of Moz­art’s mu­sic­al com­pos­i­tions? What are the ap­plic­a­tions of Dylan Thomas’ po­etry? Great art and mu­sic are life-en­rich­ing and should be val­ued and ap­pre­ci­ated for what they are, and not in terms of some pos­sible prac­tic­al ap­plic­a­tions. The same is true for pure math­em­at­ics and sci­ence.


[1] P. F. Baum and E. van Erp: “\( K \)-ho­mo­logy and in­dex the­ory on con­tact man­i­folds,” Acta Math. 213 : 1 (2014), pp. 1–​48. Ded­ic­ated to Sir Mi­chael Atiyah on the oc­ca­sion of his 85th birth­day with ad­mir­a­tion and af­fec­tion. MR 3261009 Zbl 1323.​58017 article

[2] P. F. Baum and E. van Erp: “\( K \)-ho­mo­logy and Fred­holm op­er­at­ors, II: El­lipt­ic op­er­at­ors,” Pure Ap­pl. Math. Q. 12 : 2 (2016), pp. 225–​241. MR 3767216 article

[3] P. F. Baum and E. van Erp: “\( K \)-ho­mo­logy and Fred­holm op­er­at­ors, I: Dir­ac op­er­at­ors,” J. Geom. Phys. 134 (2018), pp. 101–​118. MR 3886929 article