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.
