ICMI

Bulletin No. 47

December 1999

**Miguel de Guzmán**

*Translation of an article which has appeared in the Spanish newspaper
El Pais, May 27, 1998.*

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? 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.

**Miguel de Guzmán**

Facultad de Ciencias Matemáticas

Universidad Complutense, Madrid, Spain

mdeguzman@bitmailer.net