# Alberto Calderón

Alberto Calderón | |
---|---|

File:Alberto Calderon.jpeg | |

Born | Mendoza, Argentina |
September 14, 1920

Died | April 16, 1998 Chicago, Illinois, United States |
(aged 77)

Nationality | Argentinian |

Fields | Mathematics |

Alma mater | University of Buenos Aires |

Doctoral advisor | Antoni Zygmund |

Doctoral students | [[{{{1}}}]]^{[]}[[{{{1}}}]] ^{[]}Cora Sadosky |

Known for | Partial differential equations Singular integral operators Interpolation spaces |

Notable awards | Bôcher Memorial Prize (1979) Leroy P. Steele Prize (1989) Wolf Prize (1989) Steele Prize (1989) National Medal of Science (1991) |

Spouse | Mabel Molinelli Wells (m. 1950; div. 1985) Alexandra Bellow (m. 1989; wid. 1998) |

Children | María Josefina Calderón Pablo Alberto Calderón |

**Alberto Pedro Calderón** (1920–1998) was one of the leading mathematicians of the 20th century. 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.

## Contents

## Education

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. Later, upon remarrying, he coaxed the same skill set out of his younger son, Calixto Pedro Calderón, also a well-respected mathematician. Therefore, 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 Brazilian paper,^{[1]} see also On an inverse boundary value problem and the Commentary by Gunther Uhlmann,^{[2]} 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, María 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 ^{[3]} 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* ^{[4]} 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*,^{[5]} 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*,^{[6]} see also the Commentary by Paul Malliavin.^{[2]} 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*,^{[7]} and Calderón’s proof of the *boundedness of the “first commutator”*.^{[8]} These papers triggered a frenzy of activity by other mathematicians in the next decades; see also the later paper by the Calderón brothers^{[2]}^{[9]} and the Commentary by Y. Meyer.^{[2]} Calderón’s pioneering work in *interpolation theory* opened up a whole new area of research,^{[10]} see also the Commentary by Charless Feffermann and Elias M. Stein,^{[2]} and in *ergodic theory*, his elementary but basic paper ^{[11]} (see also the Commentary by Donald L. Burkholder,^{[2]} and ^{[12]}) 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.^{[2]}

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.

## Professional Positions

### Research, teaching and visiting positions

- 1949 - 1950 Rockefeller Foundation Fellow, University of Chicago
- 1950 - 1953 Visiting Associate Professor, Ohio State University, Columbus, Ohio
- 1953 - 1955 Member, Institute for Advanced Study, Princeton, New Jersey
- 1955 - 1959 Associate Professor, Massachusetts Institute of Technology
- 1959 - 1968 Professor, University of Chicago
- 1972 - 1975 Professor, Massachusetts Institute of Technology
- Visiting Professor at various times at different universities, including: University of Buenos Aires, Cornell University, Stanford University, National University of Bogotá, Colombia, Collège de France, Paris, University of Paris (Sorbonne), Autónoma and Complutense Universities, Madrid, University of Rome, Göttingen University

### Named Professorships

- 1968 - 1972,
*Louis Block Professor of Mathematics*, University of Chicago

- 1975 - 1985,
*University Professor of Mathematics*, University of Chicago - 1975 -,
*Honorary Professor*, University of Buenos Aires

## Honors, academies

- 1958 Member, American Academy of Arts and Sciences, Boston, Massachusetts
- 1959 Correspondent Member, National Academy of Exact, Physical and Natural Sciences, Buenos Aires, Argentina
- 1968 Member, National Academy of Sciences of the U.S.A.
- 1970 Correspondent Member, Royal Academy of Sciences, Madrid, Spain
- 1983 Member, Latin American Academy of Sciences, Caracas, Venezuela
- 1984 Member, National Academy of Exact, Physical and Natural Sciences, Buenos Aires, Argentina
- 1984 Foreign Associate, Institut de France, Paris, France
- 1984 Member, Third World Academy of Sciences, Trieste, Italy

## Honors, prizes

- 1969
*Latin American Prize in Mathematics*, awarded by IPCLAR (Instituto para la Promoción de las Ciencias, Letras y Realizaciones), Santa Fe, Argentina - 1979
*Bôcher Memorial Prize*, awarded by the American Mathematical Society - 1983
*Konex Prize (Science and Technology)*, Buenos Aires, Argentina - 1989
*Premio de Consagración Nacional*, Buenos Aires, Argentina - 1989 Wolf Prize, awarded by the Wolf Foundation, Jerusalem, Israel
- 1989 Steele Prize, awarded by the American Mathematical Society
- 1991 National Medal of Science, Washington D.C., U.S.A.

## Honorary Degrees

- 1969
*Doctor Honoris Causa*, University of Buenos Aires, Argentina - 1989
*Doctor of Science, Honoris Causa*, Technion, Haifa, Israel - 1995
*Doctor of Science, Honoris Causa*, Ohio State University, Columbus, Ohio - 1997
*Doctor Honoris Causa*, Universidad Autónoma de Madrid, Spain

## Ph.D. students of Alberto Calderón

1958 Robert T. Seeley

1959 Irwin S. Bernstein

1959 I. Norman Katz

1959 Jerome H. Neuwirth

1960 Earl Robert Berkson

1964 Evelio Oklander

1965 Cora S. Sadosky

1965 Stephen Vagi

1966 Umberto Neri

1966 John C. Polking

1966 Nestor Marcelo Rivière

1967 Carlos Segovia Fernández

1968 Miguel de Guzmán

1968 Daniel Fife

1971 Alberto Torchinsky

1972 Keith W. Powls

1976 Josefina Dolores Alvarez Alonso

1976 Telma Caputti

1976 Robert Richard Reitano

1978 Carlos E. Kenig

1979 Angel Bartolomé Gatto

1979 Cristián Enrique Gutierrez

1980 Kent G. Merryfield

1982 Michael Christ

1982 Gerald M. Cohen

1984 María Amelia Muschietti

1985 Marta Susana Urciolo

## Selected papers

[1] **Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).**. This is one of the key papers on singular integral operators.

[2] Calderón, A. P. (1958): “Uniqueness in the Cauchy problem for partial differential equations”, *Amer. J. Math. 80*, pp. 16–36.

[3] Calderón, A. P. (1963): “Boundary value problems for elliptic equations”, *Outlines for the Joint Soviet - American Symposium on Partial Differential Equations, Novosibirsk*, pp. 303–304.

[4] Calderón, A. P. (1977): “Cauchy integrals on Lipschitz curves and related operators”, *Proc. Nat. Acad. Sci. U.S.A. 74*, pp. 1324–1327. PMC 430741

[5] Calderón, A. P. (1980): “Commutators, Singular Integrals on Lipschitz curves and Applications”, *Proc. Internat. Congress of Math. 1978, Helsinki*, pp. 85–96.

[6] Calderón, A. P. (1964): “Intermediate spaces and interpolation, the complex Method”, *Studia Math. 24*, pp. 113–190.

[7] Calderón, A. P. (1968): “Ergodic theory and translation-invariant operators”, *Proc. Nat. Acad. Sci. U.S.A. 59*, pp. 349–353. PMC 224676

[8] Calderón, A. P. (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.

## References

- ↑ Calderón, A. P. (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.* - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}^{2.5}^{2.6}(2008) SELECTED PAPERS OF ALBERTO P. CALDERON WITH COMMENTARY, Alexandra Bellow, Carlos E. Kenig and Paul Malliavin, Editors,*Amer. Math. Soc., Providence, Rhode Island*, CWORKS/21. - ↑ Calderón, A. P. and Zygmund, A. (1952), "On the existence of certain singular integrals",
*Acta Math. 88*, pp. 85-139 - ↑ Calderón, A. P. (1958), "Uniqueness in the Cauchy problem for partial differential equations",
*Amer. J. Math. 80*, pp. 16-36 - ↑ Calderón, A. P. (1963), "Boundary value problems for elliptic equations", 'Outlines for the Joint Soviet - American Symposium on Partial Differential Equations, Novosibirsk
*, pp. 303-304* - ↑ Atiyah, M. and Singer, I. (1963), The Index of elliptic operators on compact manifolds,
*Bull. Amer. Math. Soc. 69*pp. 422–433 - ↑ Calderón, A. P. (1977), Cauchy integrals on Lipschitz curves and related operators,
*Proc. Nat. Acad. Sci. U.S.A.*74, pp. 1324–1327 - ↑ Calderón, A. P. (1980), Commutators, Singular Integrals on Lipschitz curves and Applications,
*Proc. Internat. Congress of Math. Helsinki 1978*, pp. 85–96 - ↑ Calderón A. P. and Calderón, C. P. (2000), A Representation Formula and its Applications to Singular Integrals,
*Indiana University Mathematics Journal*©, Vol. 49, No. 1, pp. 1-5 - ↑ Calderón, A. P. (1964), Intermediate spaces and interpolation, the complex method,
*Studia Math. 24*pp. 113–190 - ↑ Calderón, A. P. (1968), Ergodic theory and translation invariant operators,.
*Proc. Nat. Acad. Sci. U.S.A. 59*, pp. 349–353 - ↑ (1999) HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS,
*Essays in Honor of Alberto P. Calderón*, Michael Christ, Carlos E. Kenig and Cora Sadosky, Editors,*The University of Chicago Press*, “Transference Principles in Ergodic Theory” by Alexandra Bellow, pp. 27–39

## External links

- O'Connor, John J.; Robertson, Edmund F., "Alberto Calderón",
*MacTutor History of Mathematics archive*, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. - Alberto Calderón at the Mathematics Genealogy Project
- Obituary: Alberto Calderon, University of Chicago Chronicle
- Alberto Pedro Calderón (1920–1998), Notices of the AMS
- Alberto Calderon, 77, Pioneer Of Mathematical Analysis, New York Times

- Pages with broken file links
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- Argentine mathematicians
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- National Medal of Science laureates
- Wolf Prize in Mathematics laureates
- PDE theorists