# Roberto Vázquez García

### Quick Info

Mexico City, Mexico

Mexico City, Mexico

**Roberto Vázquez**was a 20th century Mexican mathematician who did outstanding work in topology. He was a pioneer of categorical topology and introduced the concept of a connection category.

### Biography

**Roberto Vázquez Garciá**was the son of Enrique Feliciano Vázquez Espinosa (1879-1941) and Dolores Garcia Vázquez (1886-). Roberto had the siblings Leonila Vázquez García, Raquel Vázquez García and Eugenia Vázquez García.

He began his primary studies in 1924 at the "Galación Gómez" junior school in the Cuauhtémoc district of Mexico City. He continued his studies at the "El Pensador Mexicano" school graduating in 1928. His secondary studies continued at Secondary School number 4 between 1929 and 1931. Certainly Roberto excelled at school, showing particular talents in mathematics. What is less clear is whether he enjoyed his school days. Certainly in later life he said that he regarded his school days as a huge waste of time. He said that state education around 1930 in Mexico was a shameless farce of colossal dimensions and it had improved little over the years. He certainly was not being critical of his teachers for he considered many of them to be excellent but rather it was the system that he was critical of.

He attended the National Preparatory School in 1932 and studied there until 1933. This school, originally a Jesuit College in the centre of Mexico City, had been taken over by the government in 1868 and brought pupils up to the standard required for entering the Universidad Nacional Autónoma de México. At this school he was taught by Alfonso Nápoles Gándara who had been appointed head of mathematics there in 1926 but had spent the years 1930-32 at the Massachusetts Institute of Technology.

In 1934, Vázquez entered the National School of Engineers. Although he was keen to study mathematics, at this time the only way to do this in Mexico was to take an engineering degree. Things were beginning to change, however, at about this time and moves were being made towards introducing degrees in mathematics. José Adem explains in [5] that when Nápoles Gándara returned from studying in the United States, he taught high level mathematics courses:-

The lectures that Nápoles Gándara gave upon his return became the main basis for training a group of young higher mathematics teachers who would later constitute the physical-mathematical faculty of our Faculty of Sciences.José Adem explained how mathematics developed in Mexico providing an environment which allowed Vázquez to train as a mathematician [5]:-

Parallel to teaching, research began to be organised when in 1932, and at the initiative of a group headed by Sotera Prieto, Nápoles Gándara, Jorge Quijano and Mariano Hernández, the Section of Mathematics and Theoretical Physics was formed at the Antonio Alzate National Academy of Sciences. The weekly sessions that were held for several years were one of the first stimuli received by mathematical research in Mexico. Several professors participated in this mathematical seminar, as well as a group of distinguished students of Sotera Prieto and Nápoles Gándara.Both Alberto Barajas and Carlos Graef had, like Vázquez, began their undergraduate studies in the National School of Engineers. They were a couple of years ahead of Vázquez and, in 1934, they both decided to dedicate themselves professionally to mathematics. They were a major influence on the younger Vázquez both in adding to his enthusiasm for mathematics and also for increasing his knowledge of it.

Vázquez began teaching in 1937 at the National Preparatory School, where he taught for nine years, and at the National School of Engineering where he taught for ten years.

It was through the efforts of Nápoles Gándara that, towards the end of 1938, the Faculty of Sciences was created in the Universidad Nacional Autónoma de México. Vázquez moved from the School of Engineers to attend courses at the Faculty of Sciences as soon as it was set up. We quote from Carlos Graef about the mathematical environment at the Universidad Nacional Autonoma de Mexico that Vázquez entered [3]:-

... the Department of Physics and Mathematics was located in the Palacio de Minería, home to the School of Engineers. The neighbourhood in the space of the two establishments was very fruitful. The professors in the Physics and Mathematics department were professors from the School of Engineering who had a special love for exact sciences. Those who were then dedicated to teaching these subjects had to give many hours of class every day to be able to collect a decent remuneration. There was no physics or mathematics institute, and the only income for them was from their teaching work.Vázquez completed his studies in 1939, but the Faculty of Sciences had not at that time received the right to award degrees so Vázquez graduated with the degree of Master of Science in 1941 with the thesis

*Conjuntos Limites*Ⓣ. The first courses he taught at the Faculty of Sciences were

*Theory of Functions of Real Variables*,

*Mathematical Analysis I*, and

*Mathematical Analysis II*. A scholarship from the Rockefeller Foundation allowed him to spend two years at Princeton University. There he took courses given by Hermann Weyl and began undertaking research with the aim of submitting a Ph.D. thesis to Princeton. Solomon Bochner suggested a topic for Vázquez to study and, not surprisingly, he suggesting problems associated with the work he was thinking about at the time. Bochner had published

*Hilbert distances and positive definite functions*in 1941 and was working on the paper

*Completely monotone functions in partially ordered spaces*which was published in 1942. Vázquez's scholarship was only for two years and was not specifically to allow him to obtain a Ph.D., so he had to return to Mexico City in 1943. His intention at the time was, however, to complete work on his thesis and, in due course, to submit it to Princeton.

Shortly after Vázquez returned to Mexico City, on 30 June 1943 the Mexican Mathematical Society was founded there. It initially consisted of 131 founding members, one of these being Vázquez. The Instituto Matemáticas of the Universidad Nacional Autónoma de México was founded on 30 June 1942 and Nápoles Gandára was appointed as its first director. Vázquez joined the Institute of Mathematics on his return in 1943 and rapidly became involved in its activities. The Institute of Mathematics began to operate in the Palacio de Minería, in the historic centre of Mexico City. The building also housed the National School of Engineers and the recently founded Faculty of Sciences. The Institute was subdivided into three areas: Pure Mathematics, led by Alberto Barajas and Roberto Vázquez, Applied Mathematics, led by Carlos Graef, and Logic and Fundamentals led by Francisco Zubieta. These four young researchers and the director Alfonso Nápoles Gándara were the only members of the Institute's academic staff in the first few years of its existence.

In 1944, George David Birkhoff made one of his regular visits to the Institute of Mathematics to encourage research activities. In addition to working to complete his thesis, Vázquez began undertaking research on problems proposed by G D Birkhoff. Collaborating with Francisco Zubieta Russi, Vázquez published jointly authored articles such as

*The linear homogeneous continua of George D Birkhoff*(Spanish) and

*Note on the continuum*(Spanish) in the first volume of the

*Bulletin*of the Mexican Mathematical Society in 1944. He also published the single authored article

*Hypersurfaces with width*(Spanish) in the second volume of the

*Bulletin*of the Mexican Mathematical Society in 1945.

In addition to undertaking joint research activities with his colleague Francisco Zubieta Russi from the Institute of Mathematics, Vázquez also did joint work with Javier Barros Sierra, who worked in the National School of Engineering. Javier Barros Sierra (1915-1971) had studied civil engineering at the Universidad Nacional Autónoma de México and taught for many years both at the National Preparatory School and in the National School of Engineering. Looking at another problem posed by G D Birkhoff, Vázquez and Barros published the joint paper

*Theorems on geodesic circles and the Gaussian curvature*(Spanish) in the third volume of the

*Bulletin*of the Mexican Mathematical Society in 1946. You can see a list of papers published by Vázquez at THIS LINK.

Vázquez also wrote, with co-author Javier Barros Sierra, the textbook

*Introducción al cálculo diferencial e integral*Ⓣ which was first published in 1946. A second edition was published in 1958 and, since the book enjoyed a continued popularity, by 1973 a fifth edition had appeared [1]:-

It is worth mentioning about this book that, from its first edition, it became the text par excellence not only for the corresponding subject in the National Preparatory School but also for the first courses of this subject taught in faculties such as Engineering, Chemical Sciences and Sciences.G D Birkhoff died in November 1944 and Solomon Lefschetz took over his role in encouraging research at the Institute of Mathematics in Mexico City. Lefschetz was keen that Vázquez should return to Princeton and present his completed thesis for a Princeton Ph.D. This, however, proved impossible since Vázquez did not have funds to pay the fare and accommodation costs. Partly this was because he had just got married: he had married Isabel Landázuri Prieto on 11 August 1944. Isabel had been born in Mexico City on 14 October 1920 to civil engineer Raúl Landázuri and his wife Teresa Prieto. Roberto and Isabel had three children: Raúl Vázquez Landázuri (born 8 July 1945); Isabel Vázquez Landázuri (born 30 May 1947) and Rocío Vázquez Landázuri.

Unable to afford going to Princeton to complete all the necessary procedures to earn his Ph.D., Vázquez decided to submit his thesis to the Universidad Nacional Autónoma de México which now had the right to award doctorates. The thesis,

*Funciones Definitivamente Positivas en Espacios Parcialmente Ordenados*Ⓣ, already had the approval of Solomon Bochner, so acceptance was straightforward. He was awarded the Ph.D. in 1947 becoming the first person to be awarded a Ph.D. in mathematics from the Universidad Nacional Autónoma de México. Vázquez's research had been in functional analysis and differential geometry but, influenced by Lefschetz's lectures on topology, he became interested in research in that topic. Lefschetz wrote in [7] about his interactions with the mathematicians in Mexico City:-

In 1944 I joined as a part-time connection the Instituto de Mathematicas at the National University of Mexico. This continued until 1966. At the Instituto I was as free as under my Princeton professorship. I conducted seminars in topology and differential equations, gave a couple of times a "volunteer" course on "general mathematical concepts" directed at beginners and, thanks to a good working library, was able to continue research.Although Vázquez began research in topology, he did not at that time have Mexican colleagues working on the topic so he worked alone and only published one paper on topology in the ten years between 1947 to 1956. We know he was working on the topic since, in 1951, he presented the paper

*Three notes on cohomology groups*to the Mexican Scientific Congress. He spent 1953-54 as a visiting professor at Princeton University where he increased his knowledge of topology. The International Congress of Mathematicians was held in Amsterdam, The Netherlands, from 2 September to 9 September 1954. The Instituto de Mathematicas in Mexico City was represented at the Congress by José Adem Chahín, Alfonso Nápoles y Gándara, Roberto Vázquez García and Emilio Lluis Riera. Emilio Lluis Riera (1925-2020) had a bachelor's and master's degree from the Universidad Nacional Autónoma de México and then worked for a Ph.D. at the new Ciudad Universitaria Mexico.

Vázquez was one of eight organisers of the 'International Symposium on Algebraic Topology' held in Mexico City at the National Autonomous University of Mexico in August 1956 in memory of Witold Hurewicz. The other seven organisers were José Adem, Alberto Barajas, Solomon Lefschetz, Emilio Lluis, Alfonso Nápoles Gándara, Félix Recillas and Guillermo Torres. The papers at this Symposium were given by the world's leading topologists, including: Henri Cartan, Samuel Eilenberg, Jean-Pierre Serre, René Thom, Donald Spencer, Michael Atiyah, Shiing-shen Chern, John Milnor, Friedrich Hirzebruch, Norman Steenrod, Ioan James, Henry Whitehead, Edwin Spanier, Peter Hilton, Raoul Bott, Hans Samelson, Hassler Whitney and James Eells. This meeting with lectures by the world's leading topologists presented a remarkable opportunity to Vázquez.

In 1959 he travelled to France as an Exchange Researcher, remaining there until February 1960. While there he attended the Henri Cartan Seminar at the Henri Poincaré Institute. The

*Anales del Instituto de Matemáticas*began publication in 1961 with Vázquez as a member of the Editorial Board from Volume 1. He remained on the Editorial Board until 1984 when he was still an editor of Volume 24.

We quote from [8] concerning a major change in the direction of Vázquez's research, beginning in the 1960s:-

In the 1960s, there was a shift in his research work, prompted, among other things, by a seminar organised at the Institute of Mathematics, in which José Adem and Félix Recillas participated, and in which the homological algebra book of Cartan and Eilenberg was studied. It was then that Vázquez began to study problems about projective modules and flat modules, and began to head towards the branch of categorical topology, of which he is one of the creators ... It is in this branch that his most important production is framed, especially highlighting the categorical conceptualisation of the connectedness property, which extends from geometric objects to arbitrary objects in a connection category. Of importance at this stage was his collaboration with the brilliant mathematician Graciela Salicrup, who died as a result of a tragic accident. She wrote her undergraduate thesis with Dr Vázquez and later began a joint research project with him, part of which gave rise to the thesis with which she obtained her doctorate. Their joint research continued for several years, yielding important results in connection theory.Horst Herrlich (1937-2015), also a pioneer of categorical topology, was Professor of Mathematics at the University of Bremen from 1971. He writes about the joint work of Vázquez and Graciela Salicrup explaining that they [4]:-

... were among the first mathematicians who realised that certain topological concepts and constructions were best understood in the light of category theory. ... they analysed topological problems by means of categorical terms and methods, in many cases having first to develop the yet missing categorical machinery. For nearly a decade they produced roughly one joint paper per year, evincing a high sensitivity for suitable problems, and demonstrating great originality and skill. Their best papers of this period ... are prominent achievements in the newly emerging field of categorical topology. Unfortunately they published their results in a journal neither well known nor easily accessible in the non-Spanish speaking mathematical world. Thus, their work which otherwise would have had a considerable impact on categorical topologists - remained for a long time almost unknown, and several of their results had to be recovered independently by later investigators.In fact Vázquez wrote eleven papers between 1970 and 1979, all being joint with Graciela Salicrup. Their joint paper

*Connection categories*was published in 1972.

Vázquez lectured on topology for his whole career after he began research on the topic. Of course he taught other topics but it was his beautifully presented lectures on topology, together with his work as a thesis advisor, which was the main influence in building a strong Mexican research group in the subject. His lectures were very formal affairs, well organised and presented. One of his students wrote [12]:-

The truth is that he never dictated an agenda to us or pointed out long-term objectives. From the first day that I found myself in the student-teacher relationship with him, I learned topology; precisely the definition of this illustrated with examples. When he took us on trust, sometimes the most he would say to us when starting a topic was: "we're going to be in this as far as we go."He appeared to his students as rather humourless but his friends commented on his keen sense of humour and a certain irony in his vision of the world around him. His life seemed completely dominated by mathematics but he enjoyed baroque music and had a great love of animals.

Alfonso Nápoles Gándara was the first director of the Institute of Mathematics, then in 1966 Vázquez replaced him holding this position until 1972. In 1984 Vázquez retired and was made an Emeritus Researcher. He continued to undertake research and had five papers published between 1985 and 1992. He never stopped working even in his final illness and while at home, hoping his health would improve, he wrote six manuscripts. His writing had deteriorated by the time he wrote the last two suggesting that by this time his hand must have trembled. His mind, however, remained clear and the mathematics looked good quality.

On 25 February 1994 the Mexican Mathematical Society organised a meeting to pay a tribute to Vázquez [2]:-

Many of us attended with the hope of seeing him, but his delicate health prevented him from going and his daughter Isabel attended in his place. A toast was made in his honour, music was played (there was a flute, and there was also a guitar), a pretty young lady sang old songs in French, speeches were made ... We never saw him again.Let us note that his excellent contributions were recognised with the award of the "Luis Elizondo" Scientific Prize in 1986:-

The Tecnológico de Monterrey Luis Elizondo Award for a Humanistic Outlook was established in 1967 with a personal contribution from Don Luis to Tecnológico de Monterrey. The award recognises prestigious researchers who have worked for the benefit of Mexico, as well as people and institutions who have devoted their humanitarian, fraternal work to their fellow human beings, have performed outstanding scientific and technological activity and carry out their philanthropic work in the Mexican community, all of whom have profiles similar to that of the Nuevo León philanthropist, Don Luis Elizondo.He was also awarded the prize for Exact Science Research by the National Autonomous University of Mexico in 1989.

Finally let us note that in some ways the career of Roberto Vázquez was different from that of most university professors. He wrote an outstanding undergraduate level textbook when he was around 30 years of age, yet despite its continuing popularity and many editions, he never wrote another undergraduate textbook. It is also unusual that his most important research contributions were made when he was in his late 50s and early 60s.

### References (show)

- Breve Semblanza de Roberto Vázquez,
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto/352-breve-semblanza-de-roberto-vazquez - Elegía al Dr Roberto Vázquez,
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto/353-elegia-al-dr-roberto-vazquez - C Graef Fernández, Palabras Pronunciadas por el Dr Carlos Graef Fernandez Durante el Homenaje que le Rindio, La Sociedad Mexicana De Fisica, in
*J L Fernández Chapou and A Mondragón Ballesteros (eds.),**Carlos Graef Fernández: Obra Científica*(Universidad Autónoma Metropolitana, Mexico, 1993), 680-693. - H Herrlich, Graciela Salicrup - her mathematical work, in H Herrlich and C Prieto (eds.),
*Categorical topology. The complete work of Graciela Salicrup*(Sociedad Matemática Mexicana, México, 1988), 1-18. - José Adem: Ciencias Exactas. Mathematico,
*El Colegio Nacional*.

https://colnal.mx/integrantes/jose-adem/ - Lecciones de Topología (del Dr Roberto Vázquez),
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto/351-lecciones-de-topologia-del-dr-roberto-vazquez - S Lefschetz, Reminiscences of a Mathematical Immigrant in the United States,
*The American Mathematical Monthly***77**(4) (1970), 344-350. - C Prieto, Roberto Vázquez Garciá, in
*Nuestros Maestros*(Universidad Nacional Autónoma de México, 1992).

https://www.matmor.unam.mx/~muciray/smm/60/Vazquez.html - C Prieto, Semblanza del Dr Roberto Vázquez Garciá,
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto/355-semblanza-del-dr-roberto-vazquez-garcia - Roberto Vázquez García: padre de la Topología en México,
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto/350-roberto-vazquez-garcia-padre-de-la-topologia-en-mexico - Roberto Vázquez Garciá,
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto - Un curso de Roberto Vázquez,
*Matemáticos en México*,*Instituto de Matemáticas de la UNAM.*

https://paginas.matem.unam.mx/matematicos/matematicos-r-z/matematicos-v/vazquez-g-roberto/354-un-curso-de-roberto-vazquez

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Written by J J O'Connor and E F Robertson

Last Update June 2023

Last Update June 2023