Alberto P. Calderón (1920-98)

Credit: UNIV.CHICAGO

Alberto Calderón, one of the twentieth century's greatest mathematicians, died on 16 April this year after a brief illness. He was 77. Calderón was a pioneer in the areas of Fourier analysis and partial differential equations, and their interconnections.

Born in 1920, in Mendoza, Argentina, Calderón received his early education in Switzerland as well as in his native country. He began his career as an engineer, and one can clearly see the resulting influence in his mathematics and in the huge impact of the applications of his work.

Calderón's transition from engineer to mathematician is associated with a wonderful story, involving Antoni Zygmund, his future mentor. In 1948, Zygmund — a giant of mathematical analysis, and author of the leading book on Fourier analysis — delivered some lectures in Buenos Aires. During one of the lectures, attended by Calderón, Zygmund reproduced the proof of one of the most fundamental results of the field, the boundedness of the Hilbert transform on L^p. The Hilbert transform is one of the most important operators of analysis. At the same time, it expresses the passage from the real to the imaginary part of a complex analytic function, and is intimately related to the partial sums of a Fourier series. L^p estimates are estimates of the size of functions in terms of the integral of the pth power of their absolute value.

Calderón approached Zygmund after the lecture, asking why the argument given in the talk was so much more complicated than that in the book. Zygmund, who was completely mystified, replied that the proof presented in the talk was the one in the book. It turned out that Calderón had studied Zygmund's book according to his usual custom, which was to cover up the proof of each theorem, trying to produce his own argument, and then checking back to compare it with the book. In this case, Calderón had forgotten to check the proof in Zygmund's book, and had come up with a far superior argument. When it became clear to Zygmund what had happened, he recognized Calderón's potential, and persuaded him to study mathematics at the University of Chicago, where Zygmund was professor.

This was the start of one of the most fruitful and celebrated collaborations in the history of mathematics. Calderón earned his PhD under Zygmund's guidance in 1950, and they soon embarked on work that was to revolutionize analysis — the theory (now known as the Calderón-Zygmund theory) of singular integrals. It turns out that many of the most important processes of nature (such as electrical and heat conduction) are expressed in terms of certain integrals which look divergent or infinite. However, when properly interpreted, these integrals possess just the right cancellation of their positive and negative parts so that they are actually not merely well defined and finite, but well behaved.

These singular integrals have as their simplest example (in one dimension) the Hilbert transform. The proof that the Hilbert transform is well behaved was initially accomplished by arguments involving the theory of analytic functions of a complex variable. However, for a great many applications, one needs to handle higher-dimensional singular integrals, where complex methods do not apply. Calderón and Zygmund developed an entire programme, based on real variable theory, to analyse singular integrals, and this spectacular accomplishment came to fruition in 1952 with the appearance of a classic paper in Acta Mathematica, “On the existence of certain singular integrals”.

Applications of tremendous significance appeared immediately. For example, the analysis (in terms of L^p estimates) of the most important partial differential operator, the Laplacian, was made possible by this theory. Later, Calderón observed that the Littlewood-Paley theory, one of the most sophisticated tools of harmonic analysis at that time, was a simple consequence of the theory of singular integrals, if only one was willing to view the integrals as being vector valued. Another remarkable application came in the form of pseudodifferential operators and their use in geometry, most notably in addressing the Atiyah-Singer index theorem. Calderón was also able to apply singular integrals to prove the uniqueness of a solution for a very general class of partial differential equations with respect to the so-called Cauchy problem.

There were numerous further applications, too many to mention here. But one general remark should be made about the proof of the Calderón-Zygmund theorem. The proof proceeds by showing how to take a function, and cut it up into its large and small parts, a method known as the Calderón-Zygmund decomposition. This sounds simple, but it turns out that the correct way to split a function is not at all obvious. One needs to come to grips with the averages of a function over cubes, rather than just its value at individual points. This understanding of the splitting of functions, obtained by Calderón and Zygmund, is even more basic than any individual application.

Calderón had a marked tendency to be the exception to any given rule. For instance, mathematics is said to be a young person's game, but some of Calderón's greatest work was done when he was nearly 60 years old. In an attempt to solve elliptic boundary value problems in general, and to understand the theory of complex analytic functions in domains with sharp corners, he courageously attacked problems involving singular integrals which were not translation invariant. His brilliant success in the late 1970s represented proof positive that he was still at the height of his powers at an age when many people are contemplating retirement.

Calderón the man was as exceptional as Calderón the mathematician. He was extremely distinguished looking, and could seem aloof and distant to students and even some colleagues. But to those who knew him at all, he was remarkable for his warmth and kindness. He had a keen sense of humour, and was strikingly modest; above all, he had the greatest inner strength and character. This wonderful man, who touched so many by his mathematical work, will be remembered and sorely missed by those who had his friendship and knew him by his generosity.