# Calixto Pedro Calderón

### Quick Info

Mendoza, Argentina

**Calixto Calderón**is an Argentinian mathematician who has spent much of his career in the United States. He has made major contributions to ordinary differential equations, partial differential equations, harmonic analysis, mathematical biology and the history of mathematics.

### Biography

**Calixto Calderón**is the son of Pedro Juan Calderón and Matilde Angela Garcia. Pedro Calderón, from an old colonial family, was a surgeon who had trained in Buenos Aires and later in Paris under Georges Marion, the famous French surgeon and urologist. Pedro, who of course was Spanish speaking, was also fluent in French and Italian. He practiced as a surgeon in Mendoza. He married Haydée Cores and they had a son, Alberto Pedro Calderón born 1920, who became a well known mathematician [2]:-

Dr Pedro Calderón had a natural affinity for mathematics and music. He would have undoubtedly subscribed to Leibniz's famous saying that: "Music is the secret arithmetic of the soul, unaware of its act of counting." The fact is that he tried to instil in his sons, at an early age, a keen interest in mathematics and music. "At the dinner table he would challenge Alberto, a boy of six or seven, to make rapid mental calculations; or he would play classical music for Alberto an his older sister Nenacha."Haydée Calderón died at a young age in 1932 and Pedro Calderón later married Matilde García Gallo who was much younger. Matilde was the daughter of Edibert Garcia Gallo who had been born in the Basque region of France in 1881. She was Spanish speaking but also fluent in French and knew basic German. Pedro and Matilde had three children, a daughter Matilde Jr, a son Calixto, the subject of this biography, and a younger daughter María Teresa. Pedro treated Matilde Jr and Calixto in the same way at the dinner table as he had Alberto and Margarita (known as Nenacha). As well as mathematics and music, Calixto's parents gave him a love of modern languages.

Calixto was educated at the Colegio Marista San Josee de Mendoza. This religious school was founded in 1917 and by 1947, when Calixto was a primary pupil, there were 405 primary pupils and 204 secondary pupils. His favourite subject at school was physics but his father believed that to have a good career one needed to study practical subjects so, after graduating from the school, Calixto entered the University of Cuyo to study engineering. The University of Cuyo was Calixto's local university, situated in Mendoza. Calderón entered the School of Engineering and there he studied mathematics, physics, technical drawing and chemistry. Calixto's real love, however, was for mathematics and his brother Alberto Calderón, by this time Professor of Mathematics at the University of Chicago, encouraged him to switch to mathematics. Alberto Calderón advised his brother to study at the University of Buenos Aires where, he said, Alberto González Domínguez was both an outstanding teacher and very supportive of his students. Not only did Alberto Calderón give his brother strong moral support for the move but he also offered financial support. In 1961 Calixto left the University of Cuyo and began to study mathematics at the University of Buenos Aires.

The first course Calixto Calderón studied in the Faculty of Exact and Natural Sciences was Funciones Reales I, a course on Lebesgue integration given by Evelio Oklander. At this stage in his career, Oklander was an instructor and soon after delivering this course he went to the University of Chicago where he studied for a Ph.D. advised by Alberto Calderón. Calixto Calderón recorded in [4] some more of his undergraduate experiences:-

With the years, I would take Complex Analysis, Ordinary Differential Equations, Funciones Reales II, Partial Differential Equations, Topology, Functional Analysis, Projective Geometry, Differential Geometry, and other important subjects. We had a first rate faculty teaching these fundamental courses. Among them I remember: Alberto González Domínguez, Mischa Cotlar, Luis Santaló, Mario Gutiérrez Burzaco, Rafael Panzone, Agnes Benedek, Miguel Herrera, and Prof Roque Carranza (statistician). This period was replete of personal experiences. I met mathematicians like Dr Julio Rey Pastor and Beppo Levi during the mathematical teas that were held in the OEA building of Avenida de Mayo. A full floor had been loaned to the School of Exact Sciences of the University of Buenos Aires.Calixto Calderón was awarded his Licenciado en Ciencias Matemáticas in 1965. It was his one of the last courses he took before graduating which particularly influenced the direction of his future studies [4]:-

Shortly before my graduation as "Licenciado en Ciencias Matemáticas," I took a course on "Boundary Values of Analytic Functions" that was taught by Dr González Domínguez. That course was an introduction to "mes premiéres armes" in mathematics. Dr González Domínguez would become my thesis adviser later. I learned from him topics such as the Central Limit Theorem in Probability Theory, Introduction to Functions of Hille (Hermite Expansions), Laguerre Expansions, Theory of Approximation and Theory of Distributions. ... the most discernible influence on González Domínguez was Einer Hille's work "A Class of Reciprocal Functions." That influence was passed on to me and from me on to some of my own students. During my formative years, Dr Rafael Panzone and his wife Dr Agnes Benedek had a tremendous influence in developing my mathematical taste and knowledge.After graduating from the University of Buenos Aires in 1965, Calixto Calderón spent a year as a specialist in the Department of Science Affairs at the Pan American Union in Washington, D.C. After this, he was appointed as an assistant professor at the University of Cuyo at Mendoza, then at the University of San Juan in San Juan followed by the University of San Luis in San Luis. While at San Luis he taught a course on Distribution Theory and Linear Partial Differential Equations. In 1967, while teaching at the University of Cuyo, he had begun to undertake research at the University of Buenos Aires for his doctorate advised by González Domínguez. This was a little more complicated than it might have been since González Domínguez spent a year as a visiting professor at the University of Hawaii. Despite this, in only two years he had completed his thesis

*Transformación múltiple de Weierstrass y sumabilidad de las series múltiples de Hermite y de Laguerre*Ⓣ which he submitted in April 1969. In the thesis he wrote:-

On this occasion, I want to thank Dr Alberto González Domínguez, who proposed the problem to me, for his assistance during the preparation of the work and for his opportune suggestions to improve some proofs.After the award of the degree of Ph.D., in August 1969 he went to the University of Minnesota where he had a one-year visiting appointment. He wrote [4]:-

At that time I met Eugene Fabes, who would have a great influence on my career. I expanded my professional interests as to include Singular Integrals, Differentiation Theory and, much later in time, Differential Equations. This was the time when I became acquainted with the Navier Stokes Equations.Returning to Argentina, he was appointed as an Assistant Professor of Mathematics at the University of Buenos Aires in 1970. He became the thesis advisor of Luis Caffarelli who writes [3]:-

... Calixto became my mentor. Beyond mathematics, Calixto taught us how to be an analyst, to seek the heart of an issue, to understand a problem. In fact, Calixto is a non-conventional mathematician and his work has expended in diverse areas, and presented new insights. By that time Calixto had already started to think on new issues concerning harmonic analysis and partial differential equations.In 1972 Calixto Calderón left Buenos Aires and returned to the University of Minnesota where he had been offered a permanent appointment. Caffarelli submitted his thesis

*Sobre conjugación y sumabilidad de series de Jacobi*Ⓣ to the University of Buenos Aires and he was awarded a Ph.D. in 1972. He followed Calderón to the University of Minnesota in January 1973 and they wrote two joint papers:

*Weak type estimates for the Hardy-Littlewood maximal functions*(1974); and

*On Abel summability of multiple Jacobi series*(1974). Caffarelli tells us what the University of Minnesota was like at that time in [10]:-

I came to the United States to the University of Minnesota in January of 1973. There was no email, no fax, and even the telephone was very expensive. But I found at Minnesota and in the mid-west an extraordinary group of people. My colleagues were extremely generous, dedicated, and friendly, and they taught me much of what I know. They shared their ideas and gave me guidance as I began my research programme.Calixto Calderón found his time at the University of Minnesota very positive in developing his career, particularly the collaboration he began with his two colleagues Néstor Marcelo Rivière (1940-1978) and Eugene Barry Fabes (1937-1997). They published the joint paper

*Maximal smoothing operators*in 1974. Calderón left the University of Minnesota in the autumn of 1974 to take up a tenured position at the University of Illinois Chicago Circle. Caffarelli writes [3]:-

Once more, he embedded himself in new problems and ideas, some connected to his past research, others bold and new. From these times are his articles on existence of solutions to Navier Stokes Equations in a very particular norm and through the years Calixto published articles in diverse areas such as tumour growth, multiscale species competition, probability. For Calixto, doing mathematics never felt like an obligation but always an enjoyment. He also took a strong interest at this time in the history of science and, in particular, the early contributions to science of the Hispanic culture.The historical papers are

*The sixteenth century Iberian calculatores*(1990),

*Alvaro Thomas and the Iberian calculatores*(1991), and

*Copernico el Mito y la Controversia*Ⓣ (2006). His work on mathematical biology has been much more extensive and is reviewed in detail in [1]. Evans Afenya, who was a Ph.D. student of Calderón's at the University of Illinois Chicago and is now an internationally leading professor researching mathematical modelling of cancer and cancer treatments, begins the introduction as follows:-

Aside from his primary areas of interest in harmonic analysis, it is important to mention that at some point in the career of Prof Calderón, he started getting interested in the applications of mathematical ideas and techniques to problems of biology and medicine. It had always been the view of Prof Calderón that biology and medicine could benefit from systematised mathematical support just as how the disciplines of physics and engineering benefitted immensely from such support many years ago. It is, therefore, not surprising that a number of doctoral candidates produced by Prof Calderón completed their dissertations on topics related to mathematical biology.The article [1] gives details of Calixto Calderón work in three sections: Solid Tumour Modelling; Disseminated Cancer Modelling; and Other Biomathematical Works of Prof Calderón. Among his Ph.D. students in this area are: Tor Anthony Kwembe,

*Nonlinear Diffusion and its Application to Biomathematics*(1989), Marwan Saeed Abual-rub Abualrub,

*Nonlinear Partial Differential Equations Applied to Diffusion Problems arising in Mathematical Biology*(1992), and Evans Afenya,

*Modelling Granulocytopoiesis, Normal and Leukemic States*(1992).

Calixto Calderón continued to hold the professorship at the University of Illinois Chicago until he retired in 2003. He did, however, hold several other positions during these years. He was a visiting professor at Rice University in Houston, Texas in 1979, a visiting professor at the Institute of Science in Santa Fe, Argentina in 1983 and was on the faculty at Oakton College in Des Plaines, Illinois from 1988 onwards.

In [8] Calixto Calderón's first wife is given as Graciela Gnavi and it states that they had three children: Graciela, Sara, and Rodrigo. This must refer to Graciela Delia Gnavi who was a Ph.D. student at the University of Buenos Aires when Calixto worked there in 1970-72. Her thesis advisor was Alberto González Domínguez and her thesis was

*Factorización de operadores característicos de puertos de Hilbert y síntesis de 2N-puertos reactivos de matriz de transferencia prefijada*Ⓣ (1974). The reference [8] states that Calixto and Graciela were divorced. We have not found any collaborative evidence for Calixto Calderón's first marriage and we note that Graciela Gnavi continues to write papers until 1988 giving her address as the University of Buenos Aires. Calixto Calderón then married the medical practitioner Cathie E Dunal (born September 1952) with whom he had three children: Cayetana (born about 1986), Ruy Gaspar (born about 1989), and Alvar (born about 1991).

Calixto Calderón retired from the University of Illinois Chicago in 2003 but continued to teach at Oakton College in Des Plaines. He writes in [4]:-

After my retirement. I remained in Chicago. I contacted the analysts at De Paul University ... I had the chance to participate actively in their Analysis Seminar.In 2012 a conference in honour of Calixto Calderón was organised at the Department of Mathematics and Actuarial Sciences of Roosevelt University, 16-18 November. The references [1], [2], [3], [4] are articles arising from talks given at that conference published in the book

*Special Functions, Partial Differential Equations, and Harmonic Analysis. In Honor of Calixto P Calderón*(Springer International Publishing, 2014) edited by A M Stokolos, C Georgakis and W Urbina. We end this biography by quoting from the Preface to that book [11]:-

Calixto P Calderón, like many Argentine mathematicians of his generation, was introduced and mentored into Harmonic Analysis and Partial Differential Equations at the University of Buenos Aires by Alberto González Domínguez, his teacher and Ph.D. advisor, whose mentor was David Tamarkin of Brown University. Moreover, Calixto P Calderón proved very early in his career to be an influential teacher and mentor for Luis A Caffarelli, a world authority on Free Boundary Value Problems in Partial Differential Equations and recent recipient of the prestigious Israel's Wolf Prize.Calixto Calderón continues to undertake research and publish papers. Four joint papers appearing after 2014 are:

The papers presented by his friends or colleagues and some of his former doctoral students at the conference in honour of Calixto P Calderón at Roosevelt University were indicative to a large extent of the wide scope and depth of his research work and scholarship in pure and applied mathematics. Furthermore, the friendly and festive atmosphere among the participants at the conference from the USA and Latin America reflected very much his wonderful charismatic personality and humility. A memorable moment was the scene where the president of Roosevelt University, while addressing the participants briefly, asked those in the audience who were Calixto P Calderón's students to raise their hands, and Luis A Caffarelli's hand went up!

Calixto P Calderón's numerous research publications include: Classical Fourier series and Orthogonal Expansions, Calderón-Zygmund Theory of Singular Integrals, Ordinary and Partial Differential Equations especially the Navier-Stokes p.d.e, Probability Theory, Mathematical Models Applied to Biology and Medicine, and History of Mathematics and Science. Clearly, his work in pure mathematics was influenced by Alberto González Domínguez, Alberto P Calderón - his brother, Antoni Zygmund, and several other collaborators and colleagues. Whereas his considerable research work in Mathematical Biology and Medicine might have been the result of the influence he received while growing up from his father, Dr Pedro J Calderón, an accomplished physician trained in Medicine in Buenos Aires and Paris. One can't help but note that, likewise, the father of Aristotle, Nicomachus, who did experimental research in Biology and Botany, although he was Plato's greatest student of pure philosophy, was also a medical physician and, in fact, the personal physician of King Philip of Macedon.

Except for the appointments at the University of Cuyo, the University of Buenos Aires, and the University of Minnesota, Calixto P Calderón spent most of his distinguished academic career at the University of Illinois at Chicago (U.I.C.). He taught a variety of courses ranging from Harmonic Analysis and Partial Differential Equations to Probability Theory and Statistics, and had several Ph.D. students. His strong interest in probability and statistics may be attributed in part to his Statistics teacher - Roque Carranza at the University of Buenos Aires. He wrote two papers on the subject including a paper on Kolmogorov's strong law of large numbers for pair-wise independent random variables. The themes of the thesis of his Ph.D. students, like his research, ranged from Harmonic Analysis to the history of Spanish mathematics and related science. During his tenure at U.I.C., Calixto P Calderón was an active participant and invited speaker at seminars in the Chicago area, including the Calderón-Zygmund Seminar at the University of Chicago, and the analysis seminars at U.I.C., and at DePaul University. Furthermore, Calixto P Calderón was an excellent and eloquent lecturer, and remained actively engaged in issues related to mathematics education at U.I.C.

*Some non standard applications of the Laplace method*(2014);

*Potential operators with mixed homogeneity*(2017);

*Some new results on H summability of Fourier series*(2019); and

*A characterization of the Lorentz space $L(p, r)$ in terms of Orlicz type classes*(2021).

### References (show)

- E Afenya, An Appraisal of Calixto Calderón's Work in Mathematical Biology, in
*A M Stokolos, C Georgakis and W Urbina (eds.), Special Functions, Partial Differential Equations, and Harmonic Analysis. In Honor of Calixto P Calderón*(Springer International Publishing, 2014), 15-24. - A Bellow, The Calderón Brothers, a Happy Mathematical Relation, in
*A M Stokolos, C Georgakis and W Urbina (eds.), Special Functions, Partial Differential Equations, and Harmonic Analysis. In Honor of Calixto P Calderón*(Springer International Publishing, 2014) - L Caffarelli, Calixto Calderón As I Knew Him, in
*A M Stokolos, C Georgakis and W Urbina (eds.), Special Functions, Partial Differential Equations, and Harmonic Analysis. In Honor of Calixto P Calderón*(Springer International Publishing, 2014), 13-14. - C P Calderón, Remembrances and Silhouettes, in
- Calderón, Calixto Pedro, in
*Biography and Genealogy Master Index*(Gale Research Company, Detroit, 2011). - Calderón, Calixto Pedro, in
*American Men and Women of Science. The physical and biological sciences*(R R Bowker, New Providence, 1982). - Calderón, Calixto Pedro, in Who's Who among Hispanic Americans (Gale Research Company, Detroit, 1994).
- Calixto Pedro Calderon, educator, mathematician,
*prabook.com*(2021).

https://prabook.com/web/calixto_pedro.calderon/3374193 - D E Newton, Calixto Pedro Calderón, in
*Latinos in Science, Math, and Professions*(Facts on File Incorporated, 2014), 42. - 2009 Steele Prizes,
*Notices Amer. Math. Soc.***56**(4) (2009), 488-491. - A M Stokolos, C Georgakis and W Urbina (eds.), Preface,
*Special Functions, Partial Differential Equations, and Harmonic Analysis. In Honor of Calixto P Calderón*(Springer International Publishing, 2014), vii-viii.

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Written by J J O'Connor and E F Robertson

Last Update November 2022

Last Update November 2022