Celebratio Mathematica

Alberto Pedro Calderón

On becoming a mathematician:
markers and decisive moments in Alberto P. Calderón’s early life

by Alexandra Bellow Calderón

Al­berto Pedro Calder­ón was born on Septem­ber 14, 1920, in Men­d­oza, Ar­gen­tina, a city at the foot of the Andes. With its strong desert cli­mate, its etern­ally snow-capped moun­tains, its pic­tur­esque vine­yards and olive orch­ards — where Al­berto roamed freely as a child — Men­d­oza left an in­delible im­print on Al­berto’s ima­gin­a­tion. He would re­turn to Men­d­oza of­ten later in life.

As a child, Al­berto was de­term­ined to un­der­stand how things “really worked” He be­came good at fix­ing all sorts of things — mech­an­ic­al, elec­tric­al, or oth­er­wise. As an adult, Al­berto was known among fam­ily and friends as “the man with the golden hands” be­cause he could fix any­thing (be it a broken down Oliv­etti type­writer, a Swiss watch, an out of tune pi­ano); in the pro­cess he fab­ric­ated the pieces that he needed and came up with the most un­ortho­dox solu­tions.

Al­berto ac­quired his early love for math­em­at­ics and mu­sic from his fath­er, a phys­i­cian, who, real­iz­ing his son’s un­usu­al gifts, tried to stim­u­late him. “At the din­ner table he would chal­lenge Al­berto, a boy of six or sev­en, to make rap­id men­tal cal­cu­la­tions; or, he would play clas­sic­al mu­sic for Al­berto and his sis­ter.” To the end of his life Al­berto could nev­er listen to the Bach Partita in D minor, the Chaconne es­pe­cially, with any­thing less than total ab­sorp­tion and rapt emo­tion. “The link between math­em­at­ics and mu­sic was there all of his life.” This makes one think of Leib­n­iz’s fam­ous say­ing: “Mu­sic is the secret arith­met­ic of the soul, un­aware of its act of count­ing”.

There were, however, hurdles on the road to be­com­ing a pro­fes­sion­al math­em­atician, “a math­em­atician’s math­em­atician”, as Al­berto Calder­ón was some­times called, be­cause oth­er math­em­aticians would come to him for help when they got stuck on a dif­fi­cult prob­lem.

“In the be­gin­ning, Al­berto seemed destined for a ca­reer in en­gin­eer­ing. His fath­er wanted him to study in the best en­gin­eer­ing school in the world, which at the time was ETH (Ei­dgenöss­is­che Tech­nis­che Hoch­schule) in Zürich. In or­der to have him over­come the lan­guage bar­ri­er and get him ready for ETH, his fath­er sent him, at the age of twelve, to a board­ing school in Switzer­land, the Montana Knaben­in­sti­tut near Zürich. It is here that Al­berto met his des­tiny in the per­son of Doc­tor Save Ber­cov­ici, the math­em­at­ics pro­fess­or. Their re­la­tion­ship began when Al­berto com­mit­ted a mis­chiev­ous act in the pres­ence of the pro­fess­or. The tra­di­tion­al pun­ish­ment was to send the cul­prit to his room for three days, dur­ing the hours when the boys used to go ski­ing. But the pro­fess­or had a dif­fer­ent idea. He saw this as a unique op­por­tun­ity to at­tract Al­berto to math­em­at­ics: he gave the boy a prob­lem in Geo­metry, prom­ising him that if he could solve it, he would be pardoned. The prob­lem was to con­struct with ruler and com­pass only, an iso­sceles tri­angle, giv­en the height and the sum of the length of the base and one of the sides. With youth­ful am­bi­tion and en­ergy, Al­berto set to work and found a con­struc­tion that solved the prob­lem. Al­berto was pardoned, Prof. Ber­cov­ici be­came Al­berto’s ment­or and math­em­at­ics moved per­man­ently to the cen­ter of Al­berto’s men­tal life. Un­for­tu­nately, fam­ily money ran out, and after two years in Switzer­land, Al­berto was called back to Ar­gen­tina.”

“Al­berto and Dr. Ber­cov­ici did not see each oth­er again un­til thirty years later. Al­berto and his first wife, Ma­bel, were trav­el­ing in Europe, they passed through Zürich, and Al­berto looked up Dr. Ber­cov­ici in the tele­phone book, find­ing his name, fol­lowed — as was the cus­tom in Switzer­land at that time — by a brief de­scrip­tion of his pro­fes­sion­al activ­it­ies. Dr. Ber­cov­ici was a col­or­ful char­ac­ter and in his case the de­scrip­tion read: ‘Math­em­at­ics, Phys­ics, Philo­sophy, Psy­cho­ther­apy.’ Dr. Ber­cov­ici re­membered Al­berto, his star pu­pil, per­fectly well. It was a very emo­tion­al, some­what chaot­ic en­counter: the ladies con­versed in Eng­lish, Dr. Ber­cov­ici spoke in Ger­man and Al­berto in French, since Dr. Ber­cov­ici un­der­stood French but could not speak it, while Al­berto un­der­stood Ger­man but could no longer speak it.” Al­berto did not tell Save Ber­cov­ici about his own aca­dem­ic ac­com­plish­ments; he was too shy, or too mod­est, or per­haps he re­membered too well how the old Math­em­at­ics Pro­fess­or used to ad­mon­ish the class: “Boys, please, do not brag! (bitte nicht protzen!)”. This was the last time that Save Ber­cov­ici and Al­berto saw each oth­er.

It is fair to say, however, that the two years that Al­berto spent in Switzer­land as a school­boy, were a mind-open­ing, life-trans­form­ing ex­per­i­ence that con­trib­uted in no small meas­ure to the breadth of in­tel­lec­tu­al in­terests and quiet self-con­fid­ence that he ex­hib­ited all his life. A testi­mony to this is the fol­low­ing. When Al­berto Calder­ón re­turned to Chica­go from Buenos Aires in 1989, to ac­cept a post-re­tire­ment ap­point­ment at the Uni­versity of Chica­go and to get mar­ried a second time, he came with only three suit­cases: one of them con­tained his fa­vor­ite pa­pers and books (most of these books showed signs of wear and tear and had been care­fully and lov­ingly re­stored by Al­berto him­self). Here is a par­tial list:

  1. “A Thou­sand and One Nights”, with il­lus­tra­tions by E. Du­lac (upon re­ceiv­ing first prize at the school in Switzer­land, 1934)

  2. Will Dur­ant, “The Great Philo­soph­ers” (upon re­ceiv­ing first prize at the school in Switzer­land, 1934)

  3. J. Rey Pas­tor, “An­ális­is Al­geb­raico”, Mad­rid, 1934

  4. Rouché et [sic] Comber­ousse, “Géo­met­rie”, Par­is, 1900

  5. A. Zyg­mund, “Tri­go­no­met­ric­al Series”, First Edi­tion, Warsaw, 1935

  6. de la Vallée Poussin, “Cours d’Ana­lyse In­fin­itési­male”, Par­is, 1937

  7. J. Mar­cinkiewicz, “Col­lec­ted pa­pers”, Warsaw. 1964

  8. S. Saks and A. Zyg­mund, “Ana­lyt­ic Func­tions”, Warsaw, 1965

  9. G. H. Hardy, “Col­lec­ted pa­pers”, Ox­ford, 1967

  10. A book of tan­gos (the lyr­ics of the best known Ar­gen­tine tan­gos). Al­berto knew most of these tan­gos by heart. At home, when listen­ing to tango mu­sic, after a glass of wine — prefer­ably a good Mal­bec from Men­d­oza — he would sing along or get in­to the step, for he was a real “tanguero”, the tango was in his blood.

  11. Al­berto’s trans­la­tion in­to Eng­lish of sev­er­al be­loved poems from world lit­er­at­ure (Al­berto was flu­ent in sev­er­al lan­guages and loved po­etry). Here is the trans­la­tion in­to Eng­lish, done by Al­berto in the spring of 1989, of one of the best known poems of the great Span­ish poet Gust­avo Ad­olfo Béc­quer (Ri­mas LIII, 938):

    Again will dark swal­lows
    hang their nest from your bal­cony,
    and with their black wings
    play­fully call by your win­dowpane.
    But those which slowed their flight
    to con­tem­plate your beauty and my bliss,
    the ones that learned our names,
    those won’t come again.

    Again will vines and ivy
    climb the walls of your garden
    and per­haps bloom in spring
    more beau­ti­fully than be­fore.
    But the dark leaves
    covered with crys­tal drops of dew
    we watched tremble, roll and fall
    like tears of the day,
    those leaves won’t grow again.

    Again will your ears
    hear pas­sion­ate words of love,
    and per­haps your heart
    will wake from its slum­ber.
    But, si­lent and ec­stat­ic
    as one prays to God in church,
    the way I loved you, have no il­lu­sions
    you won’t be loved that way again.

“Al­berto fin­ished high school in Men­d­oza. Per­suaded by his fath­er that he could not make a liv­ing as a math­em­atician, Al­berto entered the Uni­versity of Buenos Aires, stud­ied en­gin­eer­ing and gradu­ated as a civil en­gin­eer. But be­ing con­stant and per­sist­ent in his af­fec­tions, he nev­er aban­doned math­em­at­ics — his great love. Hav­ing dis­covered the ‘Boletín Matemático Ar­gen­tino’ in the lib­rary of his high school in Men­d­oza, Al­berto, once in Buenos Aires, made the ac­quaint­ance of its ed­it­or, Dr. Bern­ardo Baidaff.” (Later Al­berto liked to joke that the Ro­mani­ans had played a cru­cial role in his life: the first one was Dr. Save Ber­cov­ici, the second one was Dr. Bern­ardo Baidaff, and the third one was his second wife, my­self…). “Dr. Baidaff gen­er­ously offered Al­berto the use of his lib­rary and every as­sist­ance pos­sible and this led to a last­ing friend­ship. While study­ing en­gin­eer­ing, Al­berto es­tab­lished close con­tacts with the math­em­aticians at the Uni­versity of Buenos Aires: he at­ten­ded the ad­vanced cal­cu­lus classes of Ju­lio Rey Pas­tor (the only Pro­fess­or in the In­sti­tute of Math­em­at­ics), he par­ti­cip­ated in a sem­in­ar where he be­came ac­quain­ted with the bril­liant young Span­ish refugees Lu­is San­taló and Manuel Bal­an­zat. He also met Rey Pas­tor’s as­sist­ant, the Ar­gen­tine math­em­atician Al­berto González Domínguez, a man of vast hu­man­ist­ic cul­ture, who had left be­hind Greek, Lat­in, and philo­logy for the sake of math­em­at­ics, and who be­came Al­berto’s ment­or, pro­tect­or and de­voted friend.”

“Al­berto’s wish to be­come in­de­pend­ent, to stop be­ing a fin­an­cial bur­den to his fath­er, led him, after gradu­at­ing as a civil en­gin­eer, to the state-owned oil com­pany, the YPF (Yaci­mi­en­tos Pet­rolí­fer­os Fisc­ales), where he got a job in the re­search labor­at­ory of the geo­phys­ic­al di­vi­sion. The job suited him well: he had to work on chal­len­ging, dif­fi­cult prob­lems in ap­plied math­em­at­ics, hav­ing to do with the design and use of in­stru­ments for geo­phys­ic­al pro­spect­ing. This was in line with Al­berto’s mode of think­ing, for he firmly be­lieved that math­em­at­ics is the Queen of Sci­ences, and that any good queen has to serve her sub­jects well.” It was in this Lab, in fact, that Al­berto con­ceived of the pos­sib­il­ity of de­term­in­ing the con­duct­iv­ity of a body by mak­ing elec­tric­al meas­ure­ments at the bound­ary; he did not pub­lish his res­ults un­til sev­er­al dec­ades later, in 1980, in his short Bra­sili­an [sic] pa­per, which pi­on­eered a whole new area of math­em­at­ic­al re­search on “in­verse prob­lems.”1 Al­berto’s in­de­pend­ence of spir­it and ex­cep­tion­al per­form­ance, however, an­noyed the lab dir­ect­or, who be­came en­raged when he dis­covered that, in his spare time, Al­berto was pas­sion­ately read­ing Kur­atowski’s “To­po­lo­gie.” “Al­berto fi­nally, and very re­luct­antly, resigned. It was a bless­ing in dis­guise, Al­berto used to say later, that his lab dir­ect­or made life dif­fi­cult for him, for oth­er­wise Al­berto, who en­joyed his work, would have very likely re­mained there for the rest of his act­ive life.”

As it turned out, this un­for­tu­nate epis­ode for­tu­nately led to Al­berto be­com­ing a pro­fes­sion­al math­em­atician and to his spec­tac­u­lar math­em­at­ic­al ca­reer and fame. “Upon resign­ing his job at the YPF, he got an ap­point­ment at the In­sti­tute of Math­em­at­ics of the Uni­versity. Soon after that, when Ant­oni Zyg­mund, the fam­ous ana­lyst from the Uni­versity of Chica­go, came to vis­it the Uni­versity of Buenos Aires in 1948, Al­berto was auto­mat­ic­ally as­signed to Zyg­mund as his as­sist­ant.” A great math­em­at­ic­al tal­ent was wait­ing to be “dis­covered.” The rest of the story is, if not his­tory, cer­tainly math­em­at­ic­al his­tory, and is elo­quently told in the next chapter.

As Al­berto’s wife, I shared with him the last nine years of his life (his last Chica­go peri­od) and worked with him too, so per­haps some de­tails about Al­berto’s style of do­ing math­em­at­ics would be in or­der:

It was sheer joy to dis­cuss a math­em­at­ic­al prob­lem with Al­berto, for his mind nev­er traveled the beaten path and could move with great ease from a con­crete prob­lem to the most soph­ist­ic­ated ab­stract set­ting. He sel­dom read oth­er math­em­aticians’ pa­pers or books. If he learnt of a new the­or­em that in­ter­ested him, he would try to prove it by him­self and the tech­niques that he de­veloped in the pro­cess would of­ten have un­ex­pec­ted, far-reach­ing con­sequences else­where. This was the search for know­ledge at its most au­then­t­ic and in­spir­ing.

Al­berto seemed to be ob­li­vi­ous to math­em­at­ic­al pre­ju­dices and fads. He was simple and re­served in his man­ner; he treated col­leagues and stu­dents alike with kind­ness and re­spect. His open­ness and gen­er­os­ity in shar­ing math­em­at­ic­al ideas were le­gendary.

Al­berto Calder­ón had strong emo­tion­al and math­em­at­ic­al ties with his nat­ive coun­try, Ar­gen­tina, and with Spain as well, but his en­dur­ing “math­em­at­ic­al home” was un­doubtedly the Uni­versity of Chica­go, an in­sti­tu­tion he re­spec­ted, ad­mired and un­equi­voc­ally loved all his life long.

The year we were mar­ried, 1989, turned out to be a kind of “In­ter­na­tion­al Calder­ón Fest­iv­al,” for Al­berto re­ceived a lot of re­cog­ni­tion that year: the Wolf Prize (from Is­rael), the Pre­mio de Con­sagra­ción Nacion­al (from Ar­gen­tina), the Steele Prize (from the Amer­ic­an Math­em­at­ic­al So­ci­ety). What an un­usu­al “dowry” to bring to a mar­riage — friends mused. I knew that Al­berto re­garded so­ci­ety’s in­sa­ti­able ap­pet­ite for fame and im­mor­tal­ity with be­nign amuse­ment and a touch of irony. When I marveled at how he could re­main so un­as­sum­ing des­pite all the ac­claim, he would simply an­swer “I know how little I know”. It was the an­swer of a math­em­atician’s math­em­atician.2