Celebratio Mathematica

We all knew it at Eck­hart Hall. It was very easy, at the math­em­at­ics lib­rary, to come across Zyg­mund or any of the many first-rank math­em­aticians who aboun­ded at the Uni­versity of Chica­go in the six­ties, brows­ing through the more re­cent pub­lic­a­tions or con­sult­ing the more or less clas­sic­al works. But you al­most nev­er saw Calder­ón there. Nor did he have to be there. His work­ing habit con­sisted in read­ing only the titles and then in­vent­ing his own his­tory on them. It had been his meth­od since his youth. And it gave him very good res­ults. Ac­tu­ally, thanks to this habit he was him­self among the many math­em­at­ic­al geni­uses.

At the be­gin­ning of the fifties Ant­oni Zyg­mund, a prom­in­ent math­em­atician work­ing on Four­i­er ana­lys­is, was giv­ing a course at the Uni­versity of Buenos Aires. Calder­ón, who already had read the state­ments of the the­or­ems in the fam­ous treat­ise by Zyg­mund on tri­go­no­met­ric series and, as usu­al, had made up his own his­tory on many of them, was at­tend­ing the course with in­terest. Ob­serving the dif­fi­cult ac­ro­bat­ics of Zyg­mund in prov­ing one of the del­ic­ate res­ults of his own book, Calder­ón was as­ton­ished: “Pro­fess­or, the proof you have presen­ted us today is dif­fer­ent, and much more com­plic­ated, than the one in your book.” It was now Zyg­mund’s turn to be as­ton­ished: “What do you say? The proof I have presen­ted today is ex­actly the one in my book. Do you see any easi­er way?” So Calder­ón presen­ted Zyg­mund his own ver­sion of the the­or­em, the one he thought was in the book, a short­cut no one had thought be­fore and that opened new ven­ues on the sub­ject. Zyg­mund, who had a mag­ni­fi­cent smell for de­tect­ing the good math­em­atician, keenly sought from that mo­ment on to bring Calder­ón to Chica­go. After that, the pair Calder­ón–Zyg­mund turned in­to something as fam­ous and known in the con­tem­por­ary math­em­at­ic­al world as the pairs Astaire–Ro­gers, Tracy–Hep­burn or Laurel–Hardy can be on the screen.

Al­berto Calder­ón, who died in Chica­go on April 16, 1998, has been, without any doubt, one of the most ori­gin­al and im­port­ant math­em­aticians of the twen­ti­eth cen­tury. It would take very long to enu­mer­ate the marks of re­cog­ni­tion he re­ceived from all over the world, from the Na­tion­al Medal of Sci­ence, the highest dis­tinc­tion in the United States, to his mem­ber­ship in the Academy of Sci­ences of many coun­tries, in­clud­ing Spain. All of us in the Ibero-Amer­ic­an math­em­at­ic­al com­munity are genu­inely proud of him. Most of all, for a large part of the Ar­gen­tinean and Span­ish math­em­at­ic­al com­munit­ies do­ing re­search in math­em­at­ic­al ana­lys­is, he be­came a bridge to strongly in­teg­rate us to the cre­at­ive math­em­at­ic­al streams of our cen­tury.

Al­berto Calder­ón was born on Septem­ber 14, 1920, in Men­d­oza, Ar­gen­tina, son of a med­ic­al doc­tor of Span­ish an­ces­try. His fath­er had a spe­cial in­terest in hav­ing Al­berto de­vel­op the qual­it­ies he could per­ceive in him and his idea was that, in due time, his son should study at the Ei­dgenöss­is­che Tech­nis­che Hoch­schule (ETH) in Zurich. He thus sent him to Switzer­land to at­tend sec­ond­ary school, so that he should feel from very early on com­pletely at ease in French and Ger­man speak­ing en­vir­on­ments. Calder­ón’s par­tic­u­lar in­clin­a­tion for math­em­at­ics woke up in school when he was twelve years old. As he en­joyed telling the story, one of his teach­ers once de­cided to ab­solve him from a pun­ish­ment if he could solve a geo­metry prob­lem: “The prob­lem se­duced me, and awoke in me an eager­ness to solve more and more sim­il­ar prob­lems. This little in­cid­ent clearly showed me what my vo­ca­tion was, and had a de­cis­ive in­flu­ence in my life.”

The an­ti­cip­ated plan could not be car­ried out. Calder­ón had to re­turn to Men­d­oza, where he fin­ished his sec­ond­ary school stud­ies and af­ter­wards stud­ied en­gin­eer­ing at the Uni­versity of Buenos Aires, as was his fath­er wish, but nev­er left his love for math­em­at­ics. His con­tacts in Buenos Aires with the Span­ish math­em­aticians Rey Pas­tor, San­taló and Bal­an­zat strongly stim­u­lated him. Later on, the spe­cial guid­ance of Al­berto González Domínguez, who suc­ceeded in in­vit­ing to Buenos Aires math­em­aticians of great prestige, such as Stone, by that time Dir­ect­or of the Math­em­at­ics De­part­ment of the Uni­versity of Chica­go, and sub­sequently Zyg­mund, provided Calder­ón with the op­por­tun­ity of show­ing his true math­em­at­ic­al power when con­fron­ted with the most im­port­ant math­em­at­ic­al prob­lems of the time.

When in­ves­ted with the de­gree of Doc­tor Hon­oris Causa by the Uni­ver­sid­ad Autónoma de Mad­rid in 1997, Calder­ón gave a talk on his math­em­at­ic­al re­min­is­cences, prais­ing what came to be known as the “Stone Age” of Eck­art Hall, a peri­od in which, thanks to Mar­shall Stone’s ef­forts as De­part­ment Dir­ect­or, took place a com­pletely un­nat­ur­al con­cen­tra­tion of prime math­em­at­ic­al per­son­al­it­ies of the time. Al­bert, Chern, Graves, Mac Lane, Stone, An­dré Weil, Zyg­mund Ka­plansky, Segal…  who in the math­em­at­ic­al world were names of the­or­ems, the­or­ies and treat­ises of great in­flu­ence and im­port­ance.

The math­em­at­ic­al tal­ent of Calder­ón had the pe­cu­li­ar­ity of gath­er­ing two com­ple­ment­ary qual­it­ies for the spe­cial­ist of math­em­at­ic­al ana­lys­is, qual­it­ies which are rarely seen to­geth­er to such a de­gree in one same per­son. On one hand, he pos­sessed an ex­traordin­ary geo­met­ric in­tu­ition that al­lowed him to look at an ana­lys­is prob­lem in spa­tial terms, thus po­s­i­tion­ing him at the heart of the prob­lem. On the oth­er hand, the most com­plic­ated equa­tions of the the­ory seemed to come to life for him, as he saw them evolving from be­gin­ning to end as in a united vis­ion.

Calder­ón’s lec­tures used to have the fla­vor of im­pro­visa­tions on the prob­lems, spe­cially around har­mon­ic ana­lys­is and its re­la­tion­ships to dif­fer­en­tial op­er­at­ors, which he knew so well. A few mo­ments of re­flec­tion suf­ficed him, prob­ably while walk­ing from his home to Eck­hart Hall, in or­der to pre­pare the vari­ations on the theme he was about to talk. His ex­pos­i­tions were gen­er­ally good, re­laxed, pro­found? [sic] but it was in­es­cap­able that from time to time, per­haps be­cause in his walk to class he had met a friend and had been de­prived of that mo­ment of re­flec­tion, Calder­ón did what we all do: stay in front of the board without find­ing the way. When this happened it was worth the trip from Min­neapol­is to at­tend the fol­low­ing lec­ture. It was surely to go from 0 to 10. Calder­ón would come with his notes, and there would be no-one who could outdo his ex­pos­i­tion.

Calder­ón very much en­joyed be­ing in Spain, the close con­nec­tion with which began in 1964. Thanks to him and his en­thu­si­ast­ic sup­port, the series of in­ter­na­tion­al con­gresses on har­mon­ic ana­lys­is that star­ted in 1979 in El Escori­al (Spain) have es­tab­lished them­selves among the most im­port­ant cen­ters of ref­er­ence in the world, and set the team work­ing on this sub­ject in our coun­try at the head of math­em­at­ic­al re­search. Calder­ón at­ten­ded al­most all these meet­ings, which have been tak­ing place every four years since then.

Calder­ón’s in­flu­ence on the math­em­at­ics in Spain will not be a passing in­cid­ent.