# Alberto González Domínguez

### Quick Info

Born
11 April 1904
Buenos Aires, Argentina
Died
14 September 1982
Buenos Aires, Argentina

Summary
Alberto González Domínguez was a major figure in the development of mathematical sciences in Argentina. He worked on analytic functions, the theory of distributions and their application to quantum theory, the representation of functions by Fourier integrals, Laplace integrals and orthogonal series, and on probability theory.

### Biography

Alberto González Domínguez was the son of Julio González. Alberto was born in Argentina to a family from Andalusia in southern Spain. Manuel Balanzat writes [4]:-
Even though by character he was totally Argentine and particularly from Buenos Aires, there were in his behaviour many of the stately traits of his lineage.
González Domínguez attended both primary and secondary schools in Buenos Aires. His first love was not for mathematics but rather he was very fond of languages when he was studying at High School, so after graduating from High School, he entered the Faculty of Philosophy and Letters of the University of Buenos Aires in 1922. His interest was both in classical languages, where he studied Greek, Latin, Hebrew and Sanskrit, and in modern languages such as French, English, Italian and German. He was also interested in philosophy, grammar, lexicography and literature and he continued to maintain these interests throughout his whole life. He graduated with an arts degree in 1928.

During his studies at the University of Buenos Aires, González Domínguez had come into contact with the professor of mathematics Julio Rey Pastor who had come to Argentina from Spain in 1921 to build up mathematics at the University of Buenos Aires. Rey Pastor founded the Mathematics Seminar in 1928. He saw that González Domínguez, despite studying languages, had an interest and a talent for mathematics. With Rey Pastor's encouragement, González Domínguez began to study mathematics in the Faculty of Exact and Natural Sciences of the University of Buenos Aires. He was awarded his Licentiate in Physical Sciences and Mathematics in 1934 but even before this, in 1933, he had been appointed as an assistant in mathematics at the Mathematics Seminar. His first paper was also published before he graduated with his Licentiate, namely Aplicación de la teoría de las integrales singulares a la demostración de una fórmula de Stieltjes .

From 1933 onwards, González Domínguez worked in the Faculty of Exact and Natural Sciences of the University of Buenos Aires. At first, in addition to working as an assistant, he undertook research for a doctorate. He produced some outstanding research over the following years publishing the following papers: Sobre la convergencia débil de funcionales lineales (1936), Una nueva demostración del segundo teorema límite del cálculo de probabilidades (1936), Sur un théorème de M Glivenko (1936), Sur les intégrales de Laplace (1937), Generalización de un teorema de Cantelli (1937), Sobre las series de funciones de Hermite (1938), Sobre la inversión de integrales de Laplace absolutamente convergentes (1938) and Condiciones necesarias y suficientes para que una función sea una integral de Laplace (1938).

The Argentina Mathematical Union was founded in September 1936 when twelve mathematicians met in the Faculty of Exact and Natural Sciences of the University of Buenos Aires. González Domínguez and Rey Pastor were two of the twelve; González Domínguez was elected secretary of the Argentina Mathematical Union and Rey Pastor was elected Vice-President for 1936-38. González Domínguez continued to serve on the Board of the Union from 1938 to 1945.

In June 1939 González Domínguez submitted his doctoral thesis Un nuevo teorema límite del cálculo de probabilidades to the Faculty of Exact and Natural Sciences of the University of Buenos Aires and was awarded the degree on 7 August. He expresses his thanks to Julio Rey Pastor in this thesis:-
We do not want to conclude without expressing the unpayable debt of gratitude that links us to our dear teacher Julio Rey Pastor. Even before I knew him, his excellent books (the only ones in the Spanish language and comparable to the best foreign ones) and the bibliography they contained, had introduced us to modern Mathematics. Then, the years of frequenting his classes and his treatment have only increased that debt, and it is certain that without his teachings, his example, and even his suggestions, that this thesis would not have been written. There are certain debts that can neither be described nor can be repaid. Those kind are ours. That which has already been said is worth, at least, an expression of our deepest gratitude.
While he was studying at the University of Buenos Aires, González Domínguez married Matilde V de Gonzalez who had been born in Buenos Aires on 8 December 1906. They sailed together from Buenos Aires to New York on the ship Brazil arriving on 18 September 1939. González Domínguez had been awarded a John Simon Guggenheim Memorial Fellowship to study with Jacob Tamarkin at Brown University in Providence, Rhode Island, USA. The passenger list gives the following details for González Domínguez: age, 35, height 5 ft 7 in, complexion dark, hair colour brown, eye colour brown. For his wife Matilde V de Gonzalez, the details are: age, 33, height 5 ft 5 in, complexion fair, hair colour blond, eye colour brown. They returned to Buenos Aires in 1941. Alberto and Matilde González Domínguez had a son Alvaro V González Domínguez born in 1943.

At Brown University, González Domínguez worked on the Fourier integral, Hille functions and, more generally, on integral equations. He published two articles relating to his visit to Brown University, namely La Universidad de Brown (1940) and Algunos aspectos de la vida universitaria norteamericana (1941). For his outstanding achievements, he was awarded the Second National Science Award by Argentina in 1940. In 1946 he was promoted to full professor and, in the following year, he began his first spell as president of the Argentina Mathematical Union serving during 1947-49. He attended the International Congress of Mathematicians held at Harvard University, USA, from 30 August to 6 September 1950. He was accompanied by his wife Matilde and his son Alvaro. He became a full member of the National Academy of Exact, Physical and Natural Sciences in 1950.

González Domínguez made two research trips to Paris in the 1950s. During 1954-55 he worked on the theory of distributions at the Laurent Schwartz Seminar and on applications to quantum physics at the Louis de Broglie Seminar. His second visit to Paris was in 1958-59 when again he worked on the theory of distributions at the Laurent Schwartz Seminar and also lectured at the Henri Poincaré Institute. During this visit he gave lectures at the Department of Mathematics at the University of Montpellier. In addition to his visits to Paris, he was a visiting professor at the university of Chicago, the University of Illinois Chicago, the University of California Berkeley and the University of Hawaii.

In addition to his work at the University of Buenos Aires, González Domínguez had other appointments which increased his influence on the development of mathematics in Argentina. He was appointed as a member of the board of directors of the National Atomic Energy Commission of Argentina in October 1955 and later became Head of the Mathematics Department of the Commission. The Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) was founded in 1958 as a agency of the Argentine government to coordinate scientific research in Argentina. González Domínguez was a researcher at CONICET from 1961 onwards. He also served as Professor of Mathematical Analysis at the Higher Technical School of the Ministry of the Army and Professor of Analysis at the Military College of the Nation.

Let us look at reminiscences about González Domínguez from a colleague and from a student. First we quote from the theoretical physicist and philosopher Mario Bunge (1919-2020) who became a professor at the University of Buenos Aires in the 1960s [6]:-
I also befriended Alberto González Domínguez, an analyst interested in physics and engineering, open to new ideas and ever ready to help with some mathematical problem or other and offer us a cup of coffee. He introduced us to cybernetics, game theory, and the theory of distributions.
Next we quote from Calixto Pedro Calderón who was born on 29 December 1939 in Mendoza, Argentina. He was twenty years younger than his half-brother Alberto Pedro Calderón. Calixto Calderón tells us not only about González Domínguez but also paints a picture of life in the Department of Mathematics at the University of Buenos Aires in the 1960s [7]:-
My life in Mathematics began when I transferred from The University of Cuyo, San Juan, to the University of Buenos Aires in 1961. My brother Alberto Calderón helped me economically and morally for the jump. Upon my arrival to Buenos Aires, Dr Alberto González Donnínguez (1904-1982) helped and oriented me with the change. The first subject I took in the Mathematics Department, School of Exact Sciences, was Funciones Reales I, first course on Lebesgue Integration. Prof Evelio Oklander was the instructor.

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 Gonzalez Dominguez, 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.
...
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. He was a personality larger than life, a prodigy child that spoke 14 languages and was well versed in Philosophy, Literature and History. He would frequently cite portions of French and Spanish classic literature. One of his favourites was Baudelaire. However, the most important part of his personality was his generosity and selflessness. He had been a student of Dr Julio Rey Pastor, and after his graduation he went on a Guggenheim fellowship to Brown University, USA, to study under David Tamarkin. Nevertheless, the most discernible influence on him 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.
Let us look briefly at some of González Domínguez's publications. He published Sobre algunos puntos de la teoría matemática de los circuitos lineales (1950) which was awarded the First National Science Award. LeRoy MacColl begins the review [12] of this paper as follows:-
Much of the mathematical theory of linear circuits, as it is found in present day papers and treatises, is based upon the assumption that the circuits consist of finite collections of lumped elements. On the other hand, many of the relations obtained in the theory hold under much broader conditions. Thus there has existed some uncertainty as to just what assumptions are required for the deeper parts of the theory. The chief purpose of this paper is to clarify this point.
Many of González Domínguez's papers arose from lectures he gave at conferences. For example, he gave the lecture Distribuciones y funciones analíticas at the 'Symposium on some mathematical problems that are being studied in Latin America' held at the UNESCO Scientific Cooperation Centre for Latin America in Montevideo, Uruguay in December 1951. Irving Segal writes [15]:-
The author gives a rigorous treatment of a number of singular functions employed in quantum electrodynamics, via the Schwartz distribution theory. Some of the formulas of Schwinger, Feynman and Dyson are obtained more simply in their present rigorous formulation.
González Domínguez also lectured at the 'Second Symposium on some mathematical problems that are being studied in Latin America' which was held at the UNESCO Scientific Cooperation Centre for Latin America in July 1954. He gave the lecture Sobre algunas integrales divergentes de la Electrodinámica Cuántica . Freeman Dyson writes [10]:-
A discussion of some divergent integrals which occur in the perturbation-theory treatment of quantum electrodynamics. It is shown that these integrals can be systematically made finite by interpreting them as integrals over distributions in the sense of Laurent Schwartz.
In 1959 González Domínguez published the book Propiedades en el contorno de funciones analíticas [3]:-
According to the author, his principal objective in this book is to give an exposition of some results of V P Potapov who, in a paper in 1955, generalised various classical results of function theory to the case of matrices of holomorphic functions. This objective is accomplished in a brief and clear manner in the two final chapters and two appendices of this book. It is likely, however, that the book's greatest usefulness will be found in its clear and interesting account of the classical boundary behaviour and representation theorems associated with such names as Fatou, Herglotz, Plessner, Julia, Carathéodory and Rolf Nevanlinna.
In 1964, in collaboration with Carlos Bollini and Juan Jose Giambiagi, González Domínguez wrote the paper Analytic regularization and the divergences of quantum field theories. The authors give the following Abstract [5]:-
We present a method of analytic regularization with which any element of the S-matrix becomes an analytic function of a complex parameter. The usual divergences appear simply as poles at the physical value of the parameter. The subtraction of these poles leads to the usual finite parts. A simple example is discussed, in which the mathematical justification for these subtractions is given. In the consideration of this example we discuss the causal Green's functions of the iterated D'Alembertian. They are constructed from the retarded and advanced solutions introduced by M Riesz. The application to the self-energy of the electron is explicitly given. An heuristic deduction is then used to convert the problem of the evaluation of the self-energy into the problem of solving a differential equation. The self-energy integral is a formal solution of the latter equation, the finite part (with the pole subtracted) being a rigorous solution.
González Domínguez retired in 1971 and the Revista de la Unión Matemática Argentina published the 448-page volume 25 containing papers to honour his 65th birthday. The volume begins with a Dedicatoria [9]:-
The performance of Professor González Domínguez in the Argentine Mathematical Union has been continuous and fundamental: founding member, several times President and always the soul of its Directive Commissions. The creation and subsistence of the Argentine Mathematical Union and its Journal, although it has always had the encouragement of many, has been the tenacious and persevering work of a few, among whom Professor González Domínguez occupies an exceptional place.

From any angle you look at it, Argentine Mathematics has revolved in the last fifty years around the work of Rey Pastor and his main collaborator and continuator González Domínguez. Nothing is born spontaneously and behind the realities achieved and the promising perspectives that are glimpsed for mathematics in the country, the solicitous and careful hand is perceived, the protective shadow of Professor González Domínguez, always present to iron out rough edges, proclaiming compromise in the relational life, firmly maintaining the seriousness and level of scientific work, putting order in value judgments and putting people and things in their place, subtle and delicate work that can only be done when you have sufficient scientific authority and an inexhaustible effective capacity to give love by the hand full and to absorb ingratitude without fainting, although not without pain.
We have mentioned above some honours given to González Domínguez for his outstanding contributions. Here are a few more: the Abraham Mibashan Prize awarded by the Consejo Nacional de Investigaciones Científicas y Técnicas (1960); the Ordre des Palmes académiques awarded by the French Government (1964); the Concejo Deliberante de la Ciudad de Buenos Aires (1969); the Laurel de Plata awarded by the Ateneo Rotario (1975); and honorary membership of the Argentina Mathematical Union (1976).

After he retired from the University of Buenos Aires, González Domínguez took on other roles. In particular he was the first director of the Argentine Institute of Mathematics, a position he held from 1973 until his death in 1982. He remained an active researcher until his death. In fact his last paper, a joint work with Osvaldo Capri, On operator-valued analytic functions with positive real part whose logarithm belongs to a $C_{p}$ class, was published in 1984, two years after his death.

Let us end with the following tribute by Manuel Balanzat [4]:-
To finish we will point out that the key characteristics of his personality were kindness and generosity, qualities that he lavished on his friends and also on those who were not so; he knew how to forgive ingratitude and forget grievances. In short, González Domínguez was a wise man and a good man who dedicated his life to doing science and promoting the development of mathematics in Argentina. As every authentic master wanted to be surpassed by his disciples, their triumphs were his greatest satisfaction; all of them know how to continue his work and take Argentine mathematics to higher and higher levels.

### References (show)

1. Alberto González Domínguez, John Simon Guggenheim Memorial Foundation.
https://www.gf.org/fellows/all-fellows/alberto-gonzalez-dominguez/#
2. Alberto González Domínguez, DBpedia.
3. A A Armendáriz, Review: Propiedades en el contorno de funciones analíticas, by Alberto González Domínguez, Mathematical reviews MR0190339 (32 #7752).
4. M Balanzat, Un hombre bueno y un matemático sabio, Revista de la Unión Mathemática Argentina 30 (3-4) (1982-83), 245-247.
5. C G Bollini, J J Giambiagi and A González Domínguez, Analytic regularization and the divergences of quantum field theories, Nuovo Cimento (10) 31 (1964), 550-561.
6. M Bunge, Between Two Worlds. Memoirs of a Philosopher-Scientist (Springer International Publishing, 2016).
7. C P Calderón, Remembrances and Silhouettes, in Alexander M Stokolos, Constantine Georgakis and Wilfredo Urbina (eds.), Special Functions, Partial Differential Equations, and Harmonic Analysis (Springer International Publishing, 2014), 1-6.
8. A P Calderón, Alberto González Domínguez: Discipulo y continuador de la obra de Julio Rey Pastor en Argentina, LLULL 12 (22) (1989), 181-186.
9. Dedicatoria, Revista de la Unión Matemática Argentina 25 (1-2) (1970-71), v.
10. F J Dyson, Review: On some divergent integrals of quantum electrodynamics (Spanish), by Alberto González Domínguez, Mathematical reviews MR0072748 (17,331f).
11. La obra científica y docente del Dr A González Domínguez, Revista de la Unión Matemática Argentina 25 (1-2) (1970-71), vii-xx.
12. L A MacColl, Review: On some points of the mathematical theory of linear circuits (Spanish), by Alberto González Domínguez, Mathematical reviews MR0041000 (12,778h).
13. D Sarason, Review: O N Capri and A González Domínguez, On operator-valued analytic functions with positive real part whose logarithm belongs to a C_{p} class, Mathematical reviews MR0743073 (85g:47019).
14. L A Santaló, Los primeros 60 anos de la U.M.A., Revista de la Unión Matemática Argentina 43 (2001), 1-187.
15. I E Segal, Review: Distributions and analytic functions (Spanish), by Alberto González Domínguez, Mathematical reviews MR0052588 (14,642l).
16. S E Trione, Dr Alberto González Domínguez (1904-1982), Biographical sketch, gonzalezdominguez.com (September 2005).
https://web.archive.org/web/20130724175527/http:/gonzalezdominguez.com.ar/bsketcheng.htm