Celebratio Mathematica

Alberto Pedro Calderón

A tribute to Alberto Pedro Calderón

by Cora Sadosky

Pro­fess­or Ant­oni Zyg­mund vis­ited Buenos Aires twice, in 1948 and in 1959. The first vis­it, when he met Calderón, shaped the de­vel­op­ment of real ana­lys­is for the fol­low­ing fifty years; the second one shaped my life. In 1948 Zyg­mund and Calderón star­ted what was to be­come one of the most in­flu­en­tial part­ner­ships in math­em­at­ic­al ana­lys­is. Ten years later Zyg­mund re­turned to Ar­gen­tina to help build a math­em­at­ic­al school in a land where he knew math­em­at­ic­al tal­ent flour­ished. Calderón also star­ted to make peri­od­ic vis­its to the Uni­versity of Buenos Aires, where I was then an un­der­gradu­ate. It was at that time that I be­came one of the first stu­dents of both Calderón and Zyg­mund. Two years later I ar­rived at the Uni­versity of Chica­go to pur­sue a doc­tor­ate, with Calderón as dis­ser­ta­tion ad­visor and un­der the close su­per­vi­sion of Pro­fess­or Zyg­mund.

What a priv­ilege it was! Al­though I had missed Calderón’s sem­in­al course on sin­gu­lar in­teg­rals and its ap­plic­a­tions to hy­per­bol­ic PDE in Buenos Aires, by the time I ar­rived in Chica­go I had already been the sole be­ne­fi­ciary of a course on his new the­ory of in­ter­pol­a­tion of op­er­at­ors, later pub­lished in Stu­dia Math­em­at­ica. The fi­nal art­icles were dif­fi­cult and dense, but his lec­tures and the notes I took from his course were crys­tal­line. The ex­traordin­ary op­por­tun­ity of dis­cuss­ing ideas in the mak­ing with such a pro­foundly ori­gin­al math­em­atician was a unique gift. At the time I did not un­der­stand, and there­fore failed to ap­pre­ci­ate fully, how un­usu­al Calderón’s open­ness was, and I mar­vel now in ret­ro­spect. I think this was one of his most re­mark­able traits of char­ac­ter: he would talk math­em­at­ics openly, shar­ing freely all of his thoughts, ideas, and in­sights.

Dur­ing my years at Chica­go we had long math­em­at­ic­al talks. Un­for­tu­nately I was too stub­born and in­ex­per­i­enced to pay as much at­ten­tion as I should have. For in­stance, when Atiyah and Sing­er proved the in­dex the­or­em, Calderón was quite ex­cited, but told me he did not grasp the proof. His usu­al way to grasp a proof was to work an­oth­er one for him­self, so he told me he was in­ter­rupt­ing re­search to study al­geb­ra­ic to­po­logy and ad­vised me to join him. I did not, giv­ing pri­or­ity to my ex­ams and los­ing a great op­por­tun­ity to study along­side him. After a few months he an­nounced hap­pily that he could re­sume work, hav­ing un­der­stood the in­dex the­or­em!

When I star­ted on my thes­is pro­ject, I met weekly with Pro­fess­or Zyg­mund to re­port on my work, but I also talked with Calderón al­most daily on our way home from Eck­hart Hall. Many a time I was in­vited to stay at his home for din­ner, and while I helped his wife, Ma­bel, to set the table, he played the pi­ano. We shared a de­light in Moz­art, and after din­ner some­times he played some more for me. Oth­er times I joined his chil­dren, Pachita and Pablo, in the base­ment to watch Calderón work very ser­i­ously with a large setup of elec­tric trains he had giv­en Pablo. He was an eager en­gin­eer and be­came totally ab­sorbed in the task of con­struct­ing and man­aging the in­tric­ate mod­el rail­way.

I was not temp­ted by the dis­ser­ta­tion prob­lem pro­posed by my great teach­ers (quite fool­ish of me, since it was in­ter­est­ing enough to be de­veloped by them­selves later) be­cause I was ob­sessed with para­bol­ic sin­gu­lar in­teg­rals, which seemed the nat­ur­al ob­ject to study after Calderón’s suc­cess with el­lipt­ic and hy­per­bol­ic PDEs. Calderón en­cour­aged me in that in­terest, and, as the prob­lem was in the air, very soon af­ter­wards a first pa­per on the sub­ject ap­peared by B. Frank Jones. This did not dis­cour­age me, since I came up with a no­tion of prin­cip­al value for the in­teg­ral through a non­iso­trop­ic dis­tance, an idea which Calderón thought was “the right one”. In 1963–64 he left Chica­go for a sab­bat­ic­al year, partly spent in Ar­gen­tina, and I joined him for a three-month peri­od at the In­sti­tuto Bal­seiro in Bar­i­loche. I com­pleted there the re­search for my thes­is, while in the even­ings Calderón, his lifelong friend Al­berto González Domínguez, and Fran­ce­dillaon­ois Trèves tried, mostly in vain, to teach me how to play bil­liards. There, through C–Z cor­res­pond­ence, we found out that Zyg­mund had as­signed one of his stu­dents, Eu­gene Fabes, a prob­lem close to mine and that we had both proved the point­wise con­ver­gence of para­bol­ic sin­gu­lar in­teg­rals (by dif­fer­ent meth­ods)! Pan­ic struck; Calderón de­fen­ded my pri­or­ity on the prob­lem, but all was solved am­ic­ably, and upon my re­turn Gene and I wrote our first res­ult as a joint pa­per. Shortly af­ter­wards I de­fen­ded my thes­is and left for Ar­gen­tina, while Gene star­ted a fruit­ful col­lab­or­a­tion with Nestor Rivière an­oth­er Ar­gen­tine stu­dent of Calderón, who had been an eager listen­er of our first res­ults and who later be­came key to the de­vel­op­ment of the sub­ject.

What a happy time that had been! I re­turned to Buenos Aires, leav­ing be­hind an am­bi­ence I cher­ished and some very in­ter­est­ing prob­lems on para­bol­ic max­im­al func­tions Calderón had sug­ges­ted for work­ing to­geth­er. A loss to me, but not to math­em­at­ics, since those prob­lems were suc­cess­fully solved by Calderón and Al­berto Torch­in­sky, an­oth­er Ar­gen­tine stu­dent of Calderón, who came to Chica­go later. In the mean­time, Calderón had dir­ec­ted the thes­is of Car­los Segovia, one of the stu­dents se­lec­ted by Zyg­mund in Buenos Aires, who is now a pro­fess­or there. While Calderón also had, in later years, sev­er­al doc­tor­al stu­dents in Buenos Aires, the ma­jor­ity of his Ar­gen­tine stu­dents re­ceived their Ph.D.s from the Uni­versity of Chica­go. Al­though both Calderón and Zyg­mund de­voted them­selves to strength­en­ing ana­lys­is in Ar­gen­tina and later in Spain, the res­ults of their ef­forts, due to polit­ic­al and oth­er cir­cum­stances, were very dif­fer­ent in the two coun­tries. Nowadays only one of Calderón’s Ar­gen­tine stu­dents is on the fac­ulty of the Uni­versity of Buenos Aires.

After gradu­ation I did not hes­it­ate to go back home, since the op­por­tun­it­ies for re­search and teach­ing in Ar­gen­tina were good. The flour­ish­ing of in­tel­lec­tu­al life un­der demo­cracy, however, las­ted only one more year. In 1966, after a blood­less mil­it­ary coup, the School of Sci­ences of the Uni­versity of Buenos Aires was bru­tally at­tacked by the po­lice, four hun­dred fac­ulty mem­bers left, and our sci­entif­ic dreams were shattered. In the fol­low­ing years tol­er­ance de­creased as mil­it­ary re­pres­sion in­creased. Un­able to find an­oth­er aca­dem­ic job in Ar­gen­tina, I was forced out of math­em­at­ics for some years, and to re­turn to it I had to leave the coun­try. In the mean­while, Calderón’s fam­ily had settled in Buenos Aires, where he stayed for longer peri­ods after the on­set of his wife’s even­tu­ally fatal ill­ness. For sev­er­al years he was dir­ect­or of the IAM (Ar­gen­tine In­sti­tute of Math­em­at­ics). In these years we had hardly any con­tact.

The cir­cum­stances of Ar­gen­tina changed for the bet­ter in the mid-1980s, and we met again in Buenos Aires, but for a time we did not know how to re­new our friend­ship. Then Calderón found a way in the un­der­stated mode so typ­ic­al of him. One day at his IAM of­fice he handed me a cas­sette, say­ing, “This is some of the Moz­art you used to love when I played it years ago, only bet­ter played. I copied it for you.” We were friends again. That gave me the joy of shar­ing some of the won­der­ful mo­ments of his last years, when he basked in the hap­pi­ness of be­ing with Al­ex­an­dra, his second wife.

Al­berto Calderón was a very un­as­sum­ing man of nat­ur­al charm, a per­son of great el­eg­ance and re­straint, and won­der­ful com­pany. Math­em­at­ic­ally Calderón was ex­cep­tion­al not only for the strength of his tal­ent but for his pe­cu­li­ar way of grasp­ing math­em­at­ics. He re­did whole the­or­ies by him­self, got to the core of what he wanted to know by him­self, found al­ways his own way. His ideas and the meth­ods he de­veloped were al­ways ex­tremely ori­gin­al and power­ful. Al­though he was an in­di­vidu­al­ist to the core, he in­flu­enced pro­foundly the work of oth­ers, who de­veloped what is known as the “Calderón pro­gram”. He shared his know­ledge freely with his stu­dents, yet did not closely fol­low their ca­reers. Calderón was mod­est, sure of him­self, and quite in­dif­fer­ent to com­pet­i­tion. He was al­ways happy to have been an en­gin­eer and con­served a real in­terest in ap­plic­a­tions. In one of our last con­ver­sa­tions he told me how in­trigued he was that his work was per­ceived to be in the found­a­tion of wave­let the­ory. I think this pleased Calderón very much.