Celebratio Mathematica

Alberto Pedro Calderón

A tribute to Alberto Pedro Calderón

by Michael Christ

On a warm Chica­go af­ter­noon in the late sum­mer of 1977 a class of new gradu­ate stu­dents awaited their first lec­ture on real ana­lys­is. With na­ive curi­os­ity I awaited the ap­pear­ance of Pro­fess­or Zyg­mund, au­thor of the fat­test math­em­at­ics book I had yet en­countered. In­stead, a dis­tin­guished-look­ing gen­tle­man entered and quietly an­nounced, “I am Al­berto Calderón,” sub­sti­tut­ing for Zyg­mund. The simple greet­ing still res­on­ates in memory; its tone was not that of a teach­er ad­dress­ing a class, but of a man ad­dress­ing col­leagues.

Later, hav­ing demon­strated a dis­in­clin­a­tion to­wards al­gebra and dis­in­tu­ition to­wards geo­metry, I grav­it­ated to­wards ana­lys­is and was urged by R. Fef­fer­man to at­tend Pro­fess­or Calderón’s lec­tures. These treated primar­ily his own work: com­plex in­ter­pol­a­tion, the Cauchy in­teg­ral, the com­mut­at­ors, the real vari­able the­ory of para­bol­ic Hardy spaces, bound­ary value prob­lems for el­lipt­ic PDE, al­geb­ras of pseudodif­fer­en­tial/sin­gu­lar in­teg­ral op­er­at­ors with low reg­u­lar­ity coef­fi­cients. Re­lated work of oth­ers, such as R. Coi­f­man and Y. Mey­er, was also presen­ted. The­or­ems and full de­tails of proofs were giv­en, with only oc­ca­sion­al mo­tiv­a­tion and no ed­it­or­i­al­iz­ing. While Calderón was both ar­chi­tect and brick­lay­er, his lec­tures em­phas­ized the bricks. The pace was de­cidedly slow; the thoughts of a young stu­dent wandered.

Rarely had he vis­ible lec­ture notes. Dur­ing one mem­or­able long stretch the notes con­sisted solely of his four-page pa­per on the Cauchy in­teg­ral, car­ried in an in­side coat pock­et and sel­dom con­sul­ted. The lec­tures were clear yet un­pol­ished, with oc­ca­sion­al re­treats and emend­a­tions. Once in a great while the ar­gu­ment would founder. An irked but calm Calderón, along with the audi­ence, would seek to bridge the gap. When one such break­down led to a spir­ited dis­cus­sion among Calderón, W. Beck­ner, and P. W. Jones, I fi­nally un­der­stood: the lec­tures were planned in barest out­line. Calderón was re­think­ing the the­or­ems on the black­board be­fore us; we were ex­pec­ted to think along with him. Much later he con­firmed this, ex­plain­ing that me­tic­u­lous pre­par­a­tion early in his ca­reer had pro­duced lec­tures too rap­id for his audi­ence; he had re­solved to be un­der­stood des­pite the oc­ca­sion­al blow to his pride.

Those were glor­i­ous days for ana­lys­is in Chica­go. Calderón’s mag­net­ism had at­trac­ted a re­mark­able group of young vis­it­ors, postdocs, and ju­ni­or fac­ulty, in­clud­ing Beck­ner, R. Fef­fer­man, D. Geller, S. Jan­son, D. Jer­is­on, Jones, R. Lat­ter, P. To­mas, A. Uchiyama, D. Ull­rich, M. Wilson, and T. Wolff. There were ex­cit­ing lec­tures by Coi­f­man, C. Fef­fer­man, J. Gar­nett, Mey­er, J. J. Kohn, and oth­ers and a lec­ture course by Zyg­mund. Spec­tac­u­lar de­vel­op­ments in­cluded the de­cis­ive work of Coi­f­man, A. McIn­tosh, and Mey­er on the Cauchy in­teg­ral; the ar­rival of a pho­to­copied let­ter from A.  B. Aleksandrov on in­ner func­tions; Uchiyama’s con­struct­ive de­com­pos­i­tion of BMO; G. Dav­id’s dis­ser­ta­tion; and the work of S. Bell and E. Li­gocka on bi­ho­lo­morph­ic map­pings. Calderón rarely spoke out in sem­inars, but in private con­ver­sa­tion af­ter­wards he some­times re­vealed thoughts which went well bey­ond the lec­ture.

After au­di­tion­ing in an or­al ex­am­in­a­tion, I asked for a dis­ser­ta­tion prob­lem. Calderón sug­ges­ted the bounded­ness of the Cauchy in­teg­ral on Lipschitz curves of large con­stant — long the main fo­cus of his own re­search. While I half-listened in shock, he gen­er­ously shared an idea for a line of at­tack, offered en­cour­age­ment, and prom­ised an al­tern­at­ive prob­lem if I made no head­way, as in­deed I did not. The shock was un­war­ran­ted, for this merely il­lus­trated both the at­ti­tude of genu­ine re­spect with which Pro­fess­or Calderón in­vari­ably treated oth­ers and his con­cen­tra­tion on the most fun­da­ment­al prob­lems. To me he later men­tioned two oth­er po­ten­tial re­search top­ics: the re­stric­tion of the Four­i­er trans­form to curved sub­man­i­folds of \( \mathbb{R}^n \), and \( L^p \) es­tim­ates for solu­tions of subel­lipt­ic par­tial dif­fer­en­tial equa­tions. Today both re­main ma­jor, fun­da­ment­al open prob­lems, des­pite the fas­cin­at­ing res­ults ob­tained by many in­vest­ig­at­ors.

Calderón’s high stand­ards for him­self pre­ven­ted some of his work from ever see­ing the light of day. He once asked about the \( \bar{\partial} \)-Neu­mann prob­lem, ex­plain­ing that he had ob­tained res­ults dif­fer­ent from those he had found in the lit­er­at­ure. His in­flu­en­tial pa­per on an in­verse bound­ary prob­lem in elec­tro­stat­ics ap­par­ently lan­guished for dec­ades be­fore fi­nally be­ing pub­lished.

Calderón rarely offered ad­vice. Per­haps he con­sidered it a pre­sump­tion, feel­ing that oth­ers should be left free to at­tack prob­lems on their own terms, just as he him­self wished to be. Once, un­able to sup­ply even a single back­ground ref­er­ence for a prob­lem he sug­ges­ted, he apo­lo­get­ic­ally ex­plained that study­ing the lit­er­at­ure could be con­fin­ing; he felt more likely to find ori­gin­al ideas by work­ing in­de­pend­ently in com­plete free­dom.

Des­pite his rig­or­ous per­son­al stand­ards, Pro­fess­or Calderón en­cour­aged a young stu­dent strug­gling to find him­self. After sug­gest­ing a prob­lem and of­fer­ing an ini­tial sug­ges­tion, he al­lowed me to work in com­plete in­de­pend­ence, but was genu­inely pleased to hear re­ports of even minor pro­gress. A long and some­times chaot­ic present­a­tion of a dis­ser­ta­tion was en­dured with un­flag­ging cour­tesy.

I was too over­awed and too nat­ur­ally reti­cent to glean more than rare glimpses of his per­son­al life. In­tro­du­cing his son, Pablo, he glowed with simple pride. The death of his first wife was, in his words, a ter­rible blow; for a time it was dif­fi­cult to con­tin­ue to work. Years later, re­sur­rec­ted in the com­pany of Al­ex­an­dra Bel­low, Calderón was re­laxed and full of good hu­mor. In his lec­tures Pro­fess­or Calderón taught one to work with the bricks and mor­tar and to ap­pre­ci­ate their beauty. But in his quiet way, by ex­ample through his own life, he taught deep­er les­sons.