#### from the University of Chicago News Office

Alberto Calderón, a University of Chicago mathematician widely regarded as one of the most influential mathematicians of the 20th century, died Thursday in Northwestern Memorial Hospital after a brief illness. He was 77 and lived in Chicago.

Calderón, University Professor Emeritus of Mathematics, is best known for his contributions to mathematical analysis, the large branch of mathematics that includes calculus, infinite series and the analysis of functions. Together with his mentor, Antoni Zygmund, he founded the Chicago school of analysis, the most influential school in that branch of mathematics in the 20th century.

“He was one of the most original and profound mathematical analysts of the past 50 years,” said Felix Browder, the Max Mason Distinguished Service Professor Emeritus of Mathematics at the University of Chicago, and a former vice-president of Rutgers University. “Calderón was one of the central links between two major areas of mathematical analysis, namely Fourier analysis and partial differential equations.He made outstanding contributions to both fields and laid much of the groundwork for other people’s work in these areas.”

Browder added, “Calderón was also a man of really remarkable upright character. He was universally respected and admired because of his extreme probity and generosity.”

Calderón’s many honors include the 1991 National Medal of Science; the 1989 Wolf Prize; the 1989 Steele Prize from the American Mathematical Society; Argentina’s Consagracion Nacional Prize, awarded in 1989; and the 1979 Bôcher Memorial Prize from the American Mathematical Society.

Calderón was born in Argentina on Sept. 14, 1920. As a child, he was educated in Argentina and in Switzerland, where he first became interested in mathematics. He received a degree in civil engineering at the University of Buenos Aires in 1947, where in 1948 he first met world-renowned University of Chicago mathematician Antoni Zygmund, who would become his mentor.

According to Robert Fefferman, Louis Block Professor and Chairman of Mathematics at the University of Chicago, Zygmund gave a seminar in Buenos Aires in which he discussed a proof of an important classical theorm that was included in his celebrated book. Calderón approached Zygmund after the lecture and asked him why the proof he had given in the seminar was so much longer than the one he had outlined in his book. Zygmund told him it was the same proof.

“It turned out that Calderón had a habit of reading mathematics and doing proofs on his own,” said Fefferman, “and later checking with the book to make sure he’d done it correctly. Only this time he didn’t check the book and it turned out he had come up with a much simpler and more elegant proof than the standard one in Zygmund’s book.”

Zygmund recognized Calderón’s brilliance as a mathematician, and brought him back to Chicago in 1949 as his student. Calderón completed a thesis in remarkably short order, receiving his Ph.D. in 1950.

In pioneering work with his mentor, Calderón formulated a theory, now known as the Calderón–Zygmund theory, of what are called singular integrals. “These singular integrals have tremendous applications within pure mathematics, as well as very important applications in industrial problems,” said Fefferman. “It is unquestionably one of the most important developments in analysis in the 20th century.”

Singular integrals are mathematical objects that look infinite, but when interpreted properly are finite and well-behaved. Calderón later showed how these singular integrals could be used to obtain estimates of solutions to equations in geometry and to analyze functions of complex variables. He also showed how singular integrals could provide entirely new ways to study partial differential equations, which are widely used to solve problems in physics and engineering.

In order to interpret signals or waves, such as those obtained when an image is represented by a digital signal, the classic method of analysis was to convert the signals into a series of sine or cosine waves through Fourier transforms. But it was shown that the tools developed in Calderón–Zygmund theory could be used to represent these signals more effectively, breaking down the complicated signal into smaller building blocks called wavelets. Wavelets are often far more useful than Fourier transforms because they reveal a signal’s localized behavior much more effectively.

When an image is transmitted over a communication channel, for example, the rough details can be sent first, then finer textures are added at smaller scales.This is a real-life implementation of the Calderón–Zygmund decomposition.

After his graduation from Chicago, Calderón served as a visiting associate professor at Ohio State University, a visiting member of the Institute for Advanced Study and then an associate professor at M.I.T. He returned to Chicago as Professor of Mathematics in 1959, and aside from brief periods in which he returned to M.I.T. and to Argentina, has been at Chicago ever since. He has also been an honorary professor at the University of Buenos Aires since 1975.

In addition to his research, Calderón’s influence as a teacher extended worldwide. “Calderón was instrumental in promoting a renewal of mathematical spirit in Spain,” said Miguel de Guzmán, a former student of Calderón’s who is a mathematics professor at Universidad Complutense in Madrid. “He brought many Spanish students to Chicago who worked together with him and Antoni Zygmund on Fourier analysis.”

Calderón was a member of the National Academy of Sciences; the American Academy of Arts and Sciences; the National Academy of Exact, Physical and Natural Sciences in Argentina; the Academie des Sciences in France; the Royal Academy of Sciences in Spain; the Latin American Academy of Sciences in Venezuela; and the Third World Academy of Sciences in Italy.

Calderón is survived by his wife, noted mathematician Alexandra Bellow (née Bagdasar), recently retired from Northwestern University, whom he married in 1989; and two children from his first marriage, Mary Josephine, of St. Charles, Ill., and Pablo, of New York, N.Y. His first wife, Mabel (née Molinelli Wells), to whom he was married for 35 years, died in 1985.