Alberto Pedro Calderón (1920 -1998) was one of 20th century leading mathematicians. He was born in Mendoza, Argentina. His name is associated with the University of Buenos Aires, but first and foremost with the University of Chicago, where Calderón and his mentor, the distinguished analyst Antoni Zygmund, started one of the longest (more than 30 years) and most spectacular collaborations in mathematical history. Together they developed the ground-breaking theory of singular integral operators, thus creating the "Chicago School of (hard) Analysis" (sometimes simply known as the "Calderón-Zygmund School"); this has been one of the most influential movements in pure mathematics, but with remarkable applications to science and engineering as well. Calderón's work, characterized by great originality, elegance and power reshaped the landscape of mathematical analysis and ranged over a wide variety of topics: from singular integral operators to partial differential equations, from interpolation theory to Cauchy integrals on Lipschitz curves, from Ergodic theory to inverse problems in electrical prospection. Calderón's work has also had a powerful impact on practical applications such as signal processing, geophysics, tomography and other areas as well.
Civil Engineering Degree, University of Buenos Aires, 1947
Doctor of Philosophy in Mathematics, University of Chicago, 1950
Biography, professional life and work:
Alberto Pedro Calderón was born on September 14, 1920, in Mendoza, Argentina, a city at the foot of the Andes. With its strong desert climate, its eternally snow- capped mountains, its picturesque vineyards and olive orchards - where Alberto roamed freely as a child - Mendoza left an indelible imprint on Alberto's imagination; he would return to it often later in life.
Alberto's father, Don Pedro, was a physician (a urologist), his mother, Haydée, a free-spirited woman who allegedly was the first woman in Mendoza to drive a car. Realizing how gifted Alberto was, his father, at the dinner table, challenged the boy to make rapid mental calculations, or alternatively played classical music for him and his older sister, Nenacha. Don Pedro must undoubtedly have subscribed to Leibniz's famous saying: "Music is the secret arithmetic of the soul, unaware of its act of counting."
Alberto's mother died unexpectedly when he was twelve years old and the father decided to send him to a boys' boarding school in Switzerland, the Montana Knabeninstitut near Zürich. The idea was to prepare him for what was regarded at that time as the finest engineering school in the world, the ETH (Eidgenössische Technische Hochschule) in Zürich. Here the boy met his destiny in the person of Prof. Save Bercovici, who awakened in Alberto his true vocation: his passion for Mathematics. Having committed an act of mischief in the presence of the professor, Alberto expected to be punished. But instead the math professor promised to pardon the boy if he could solve a problem in Geometry, namely: to construct with ruler and compass only, an isosceles triangle, given the height and the sum of the length of the base and one of its sides. Alberto solved the problem, Prof. Bercovici became his mentor and mathematics moved permanently to the center of his mental life. At age fourteen, Alberto had to return to Argentina. He finished his high school studies in Mendoza. But the two years that Alberto spent in Switzerland as a schoolboy, were a mind-opening, life-transforming experience that contributed in no small measure to the breath of intellectual interests and quiet self-confidence that he exhibited all his life.
Persuaded by his father that he could not make a living as a mathematician, he entered the University of Buenos Aires and studied engineering. After graduating as a civil engineer he got a job in the research laboratory of the geophysical division of the state-owned oil company, the YPF (Yacimientos Petrolíferos Fiscales), but he never abandoned mathematics, his great love. He became acquainted with the mathematicians at the University of Buenos Aires: Julio Rey Pastor, the first Professor in the Institute of Mathematics, his Assistant Alberto González Domínguez (who became his mentor, great fan and devoted friend), the brilliant young Spanish refugees Luis Santaló and Manuel Balanzat. The work in the Lab was interesting and stimulating. It was in fact in this Lab that Alberto conceived the possibility of determining the conductivity of a body by making electrical measurements at the boundary; he did not publish his results until several decades later, in 1980, in his short Brasilian paper (1980) "On an inverse boundary value problem", Seminar on Numerical Analysis and its Applications to Continuum Physics, Atas 12, Sociedade Brasileira de Matematica, Río de Janeiro, pp. 67-73. (see also an inverse boundary value problem and see also the Commentary by Gunther Uhlmann in SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY), which pioneered a whole new area of mathematical research on "inverse problems".
In 2007, the Inverse Problems International Association (IPIA) instituted the Calderón Prize, named in honor of Alberto P. Calderón, and awarded to a "researcher who has made distinguished contributions to the field of inverse problems broadly defined".
Two events were decisive in determining Alberto's mathematical future: His Supervisor at the YPF Lab made his life very difficult (especially after noticing that, in his spare time, Alberto was passionately reading Kuratowski's "Topologie" !), Alberto resigned and was immediately offered a job at the University of Buenos Aires. Secondly, Antoni Zygmund, one of the world's leading mathematical analysts and a professor at the University of Chicago, arrived at the University of Buenos Aires in 1948 at the invitation of Dr. Alberto González Domínguez and Alberto Calderón was assigned to him as his assistant. A great mathematical talent was in short order discovered, and Zygmund invited Calderón to come to Chicago to work with him. In 1949 Calderón arrived in Chicago, indeed, with a Rockefeller Fellowship, in order to work with Zygmund, not in pursuit of a degree. But the intervention of Marshall Stone, (a most visionary chairman), encouraged him to obtain a doctorate, without which Calderón's academic career would have been hindered. Brilliantly cutting through red tape, Stone suggested that Calderón "staple together" three recently published papers into a dissertation and thus Calderón was able to obtain his Ph.D. in Mathematics under Zygmund's supervision in 1950, only a year after arriving in Chicago. The dissertation proved momentous: each of the three papers solved a long-standing open problem in ergodic theory or harmonic analysis.
Also in 1950, Calderón married Mabel Molinelli Wells, a mathematics graduate whom he had met while both were students at the University of Buenos Aires. They had a daughter, Maria Josefina who now lives in Paris and a son, Pablo who lives in Connecticut. The collaboration begun by Zygmund and Calderón in 1948 reached fruition in the Calderón-Zygmund Theory of Singular Integrals and lasted more than three decades. This legendary collaboration is reminiscent of the famous Hardy-Littlewood collaboration of the earlier part of the 20th century, but with the added typically American feature that the protagonists in this case were brilliant immigrants from different parts of the world. The Calderón-Zygmund memoir (1952) "On the existence of certain singular integrals", Acta Math. 88, pp. 85-139 continues to be one of the most influential papers in the modern history of analysis; it laid the foundations of what became internationally known as the "Calderón-Zygmund School of Analysis" (or Chicago School of (hard) Analysis) which developed methods with far-reaching consequences in many different areas of mathematics. A prime example of such a general method is one of their first joint results, the famous Calderón-Zygmund decomposition lemma, invented to prove the "weak-type continuity" of singular integrals of integrable functions, which is now widely used throughout analysis and probability theory. The Calderón-Zygmund Seminar has been for several decades and continues to be an important tradition in the mathematical life of Eckhart Hall at the University of Chicago.
By the mid sixties the theory of singular integrals was firmly established thanks to Calderón's epoch-making contributions to the theory of differential equations, such as: His proof of the uniqueness in the Cauchy problem (1958) Uniqueness in the Cauchy problem for partial differential equations", Amer. J. Math. 80, pp. 16-36 using algebras of singular integral operators, his reduction of elliptic boundary value problems to singular integral equations on the boundary (the method of the Calderón projector (1963) "Boundary value problems for elliptic equations", Outlines for the Joint Soviet - American Symposium on Partial Differential Equations, Novosibirsk, pp. 303-304), and the crucial role played by algebras of singular integrals (through the work of Calderón's student R. Seeley) in the initial proof of the Atiyah-Singer Index Theorem (1963) Atiyah (M.), Singer (I.), The Index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 pp. 422-433, see also the Commentary by Paul Malliavin in SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY. The development of pseudo-differential operators by Kohn-Nirenberg and Hörmander also owed a great deal to Calderón and his collaborators R. Vaillancourt and J. Alvarez-Alonso. However Calderón insisted that the focus should be on algebras of singular integral operators with non-smooth kernels to solve actual problems arising in physics and engineering, where lack of smoothness is a natural feature. This led to what is now known as the "Calderón program" whose first important accomplishments were: Calderón's seminal study of the Cauchy integral on Lipschitz curves (1977) Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74, pp. 1324-1327, and Calderón's proof of the boundedness of the "first commutator" (1978) Commutators, Singular Integrals on Lipschitz curves and Applications, Proc. Internat. Congress of Math. Helsinki, pp. 85-96. These papers triggered a frenzy of activity by other mathematicians in the next decades; see also the Commentary by Y. Meyer in SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY. Calderón's pioneering work in interpolation theory opened up a whole new area of research (1964) Intermediate spaces and interpolation, the complex method, Studia Math. 24 pp. 113-190, see also the Commentary by Charless Feffermann and Elias M. Stein in SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY, and in ergodic theory, his elementary but basic paper (1968) Ergodic theory and translation invariant operators,. Proc. Nat. Acad. Sci. U.S.A. 59, pp. 349-353 (see also the Commentary by Donald L. Burkholder in SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY, and (1999) Harmonic Analysis and Partial Differential Equations, Transference Principles in Ergodic Theory by Alexandra Bellow, pp. 27-39) formulated a transference principle that reduced the proof of maximal inequalities for abstract dynamical systems to the case of the dynamical system of the integers.
In his academic career, Calderón taught at many different universities, but primarily at the University of Chicago and the University of Buenos Aires. Calderón together with his mentor and collaborator Zygmund, maintained close ties with Argentina and Spain, and through their doctoral students and their visits, strongly influenced the development of mathematics in these countries. Calderón retired early from the University of Chicago, in 1985, and returned to Argentina, where his wife Mabel, who had been seriously ill, died. In 1989 Calderón came back to the University of Chicago on a post-retirement appointment. He also remarried in 1989: his second wife was the mathematician Alexandra Bellow, now Professor Emeritus at Northwestern University. For more details on Calderón's biography, professional life and work, see also the two introductory articles in SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY.
Calderón was recognized all over the world for his outstanding contributions to Mathematics as attested to by his numerous prizes and membership in various academies. He gave many invited addresses to universities and learned societies. In particular he addressed the International Congress of Mathematicians: a) as invited lecturer in Moscow in 1966 and b) as plenary lecturer in Helsinki in 1978. The Instituto Argentino de Matemática (I.A.M.), based in Buenos Aires, a prime research center of the National Research Council of Argentina (CONICET), now honors Alberto Calderón by bearing his name: Instituto Argentino de Matemática Alberto Calderón.
Alberto Pedro Calderón, Professor Emeritus of Mathematics of the University of Chicago, and Honorary Professor of the University of Buenos Aires, died on April 16, 1998, at the age of 77, in Chicago, after a brief illness. He was one of the towering figures of 20th-century mathematics.
Ph.D. students of Alberto Calderón:
Selected papers and publications: