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.
