Alberto Calderón's (born 1920) revolutionary influence turned the 1950s trend toward abstract mathematics back to the study of mathematics for practical applications in physics, geometry, calculus, and many other branches of this field. His award-winning research in the area of integral operators is an example of his impact on contemporary mathematical analysis.

# Alberto P. Calderón

Widely considered as one of the twentieth century's foremost mathematicians, Alberto Calderón's career spans more than 45 years, during which he has left behind many seminal works and ideas.

Calderón was born on September 14, 1920, in Mendoza, Argentina, a town at the foot of the Andes. His father was a descendant of notable nineteenth-century politicians and military officers and was a renowned medical doctor who helped found and organize the General Central Hospital of Mendoza.

After completing his secondary education in his hometown and in Zug, Switzerland, under Dr. Save Bercovici, who encouraged Calderón's interest in mathematics, he enrolled in the School of Engineering of the National University of Buenos Aires, from which he graduated in 1947. He soon became a student of Alberto González Domínguez and of the celebrated mathematician Antoni Zygmund, who was a visiting professor in Buenos Aires in 1948. He continued his mathematical studies at the University of Chicago with a Rockefeller Foundation fellowship, and received his Ph.D. there in 1950.

Calderón began his academic teaching career as an assistant to the Chair of electric circuit theory at the University of Buenos Aires in 1948, and after graduating in the United States, continued it as a visiting associate professor at Ohio State University from 1950 to 1953. Calderón was also a member of the Institute for Advanced Study in Princeton (1954-1955) and later served as an associate professor at the Massachusetts Institute of Technology (MIT) between 1955 and 1959. He then moved to the University of Chicago, where he served as professor of mathematics from 1959 to 1968, Louis Block professor of mathematics from 1968 to 1972, and chairman of the mathematics department from 1970 to 1972.

By that time, Calderón's prestige was well established in scientific circles, and his research in collaboration with his longtime mentor Zygmund had already been dubbed "the Chicago School of Analysis," also known today as "the Calderón-Zygmund School of Analysis." Their contribution, which profoundly affected modern mathematics, included reversing a predominant trend towards abstraction and turning back to basic questions of real and complex analysis. This work, completed in tandem with Zygmund, came to be known as "Calderón-Zygmund theory."

A landmark in Calderón's scientific career was his 1958 paper titled "Uniqueness of the Cauchy Problem for Partial Differential Equations," which the American Mathematical Society has called "a real watershed in the theory of singular integral operators, taking it beyond its traditional role in the study of elliptic equations." Two years later, he used the same method to build a complete theory of hyperbolic partial differential equations.

His theory of singular operators, which is used to estimate solutions to geometrical equations, contributed to link together several different branches of mathematics. It also had practical applications in many areas, including physics and aerodynamic engineering. This theory has dominated contemporary mathematics and has made important inroads in other scientific fields, including quantum physics. Although some authors have introduced and used the notion of pseudo-differential operator, which is a sum of compositions of powers of the Laplacian with singular integral operators with kernels which are infinitely differentiable off the diagonal, the original idea and basic applications remain credited to Calderón.

Calderón's extensive work has transformed contemporary mathematical analysis. In addition to his work with singular integral operators, he also did fundamental work in interpolation theory, and was responsible, together with R. Arens, for what is considered one of the best theorems in Banach Algebras. Calderón also put forth an approach to energy estimates that has been of fundamental importance in dozens of subsequent investigations, and has provided a general model for research in this area.

## A Brief Return Home

In 1971-1972, Calderón briefly returned to his home country to serve as professor and direct mathematical doctoral dissertation studies at his alma mater, the National University of Buenos Aires. He continued to encourage mathematics students from Latin America and the United States to pursue their doctoral degrees, in many instances directly sponsoring them. Some of his pupils, in turn, have become reputed mathematicians, as, for example, Robert T. Seeley, whose extension of the Calderón-Zygmund results to singular operators on manifolds became the foundation of the now-famous Atiyah-Singer index theorem.

After his stay in Argentina, Calderón returned to MIT as a professor of mathematics, and in 1975 he became University Professor, a special position, at the University of Chicago until his retirement in 1985. Between 1989 and 1992, he was a professor emeritus, with a post retirement appointment at that same institution. In 1979 he was awarded the Bôcher prize for a paper on the Cauchy integral on Lipschitz curves. In 1989 he shared the Mathematics Prize of the Wolf Foundation of Israel with his American colleague John W. Milnor. He received innumerable other honors around the world. The American Mathematical Society honored Calderón again with the prestigious Steele Prize (fundamental research paper category) in 1989, and former U.S. president George Bush, in granting him the 1991 National Medal of Science, cited "his ground-breaking work on singular integral operators leading to their application to important problems in partial differential equations."

## Author and Lecturer

As an author, Calderón has published more than 75 scientific papers on various topics, from real variables to partial differential equations and singular integrals. A number of those papers were written in collaboration with his teacher Antoni Zygmund. Calderón has lectured in major cities the world over.

A member of the American Mathematical Society for over 40 years, Dr. Calderón has served as member-at-large of its Council (1965-1967) and in several of its committees. He has also been associate editor of various important scientific publications, such as the *Duke Mathematical Journal,* the *Journal of Functional Analysis,* and others.

Dr. Calderón married in 1950. With Mabel Molinelli, his first wife who died in 1985, he had two children: María Josefina, who holds a doctorate in French literature from the University of Chicago, and Pablo Alberto, also a mathematician who studied in Buenos Aires and New York. In 1989 Calderón married again. His second wife, Dr. Alexandra Bellow, is also a distinguished mathematician and a professor of mathematics at Northwestern University in Evanston, near Chicago.

## Further Reading on Alberto P. Calderón

Atiyah, M. F. and Singer, I. M., "The Index of Elliptic Operators on Compact Manifolds," *Bulletin of the American Mathematical Society,* 69, 1963, pp. 442-53.

González Domínguez, Alberto, "Dr. Alberto P. Calderón—Premio Bocher 1979," *Ciencia e Investigación,* 34, November-December 1978, Buenos Aires, pp. 221-23.

Beals, R. W., Coifman, R. R., and Jones, P. W., "Alberto Calderón Receives National Medal of Science," *Notices of the American Mathematical Society,* 39, No. 4, April 1992.

*Chicago Tribune,* September 17, 1991.