# José Adem Chahín

### Quick Info

Tuxpan, Veracruz, Mexico

Mexico City, Mexico

**José Adem**was a Mexican mathematician who made many outstanding contributions to topology. He is remembered today for the "Adem relations" which were introduced in his Princeton Ph.D. thesis of 1952.

### Biography

**José Adem**was the son of Jorge Adem Jerasle (1890-1974) and Almas Chahín Yehel (1903-1986). Jorge Adem was born in Lebanon and his wife Almas was born in Beirut, Lebanon; both had emigrated from Lebanon to Mexico. Jorge was Arabic speaking while his wife Almas spoke both Arabic and French. Jorge Adem was a merchant and, let us note, had a hobby as an angler winning a major angling championship in 1958. The 1930 Census lists the Adem family living at 36 Calle de Humbolt, Tuxpan, Veracruz. The city of Tuxpan (meaning place of the rabbits in the Aztecs language) lies on the Pantepec River and is a major port for Mexico City. The 1930 Census gives the children of Jorge and Almas Adem as: José Adem (aged 8), Julián Adem (aged 6), Antonio Adem (aged 4) and María Elena Adem (aged 1). In fact Julián Jorge Adem was born on 8 January 1924 and Antonio Jorge Adem was born on 17 February 1926. After the 1930 Census, three more girls were born, Labibe Adem (born about 1931), Alicia Adem (born 13 November 1933) and Esbaide Adem (born 1938); all the seven children were born in Tuxpan, Veracruz. Julián Adem became an applied mathematician and geophysicist and has a biography in this archive. Antonio Adem became a businessman. Esbaide Adem became a promoter and pioneer in nuclear physics research in Mexico.

Samuel Gitler writes [7]:-

José Adem grew up in a family where harmony, love and tranquillity always reigned. Since he was a child he started fishing with a great interest that only increased with the years.José Adem attended a primary, secondary and high school in Tuxpan. The secondary and high school had been founded by Professor Manuel C Tello in 1938. From an early age he loved mathematics and showed considerable skill in the subject. A career as a mathematician was not one to really contemplate for a young man in Mexico at this time and in 1941 Adem went to Mexico City where he enrolled to study engineering at the National Autonomous University of Mexico. His choice of engineering was made because that course involved a considerable amount of mathematics. Although following the engineering course, Adem took as many pure mathematics courses as he could.

World War II was beneficial to Adem in perhaps a surprising way. Solomon Lefschetz had a great love for travel and during the 1920s and 1930s made many trips to European countries. The outbreak of World War II in 1939 meant that Europe was no longer possible for travel so Lefschetz looked for other places to visit. Mexico seemed like a good choice and in 1944 he spent the summer in Mexico City at the National Autonomous University of Mexico. It was the first of many visits and he eventually fell into the habit of spending the summer months there every year. His contribution to mathematics in Mexico was, as a result of these visits, of major importance and he helped build a flourishing school there. His contributions were recognised when he was awarded the Order of the Aztec Eagle in 1964. In 1970 Lefschetz wrote [14]:-

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. Conditions were of course quite different from ours, but as I became rapidly fluent in Spanish, it gave me many advantages. Through the years I found quite a number of capable young men, several of whom I directed to Princeton for further advanced training up to the doctorate and later. Among them I may mention Dr José Adem, Chairman of the Department of Mathematics of the newly founded Centro de Estudios Avanzados in Mexico City. My long connection with Mexico has been the occasion of many side trips (especially in connection with meetings of the Mexican Mathematical Society), so that I have a fair acquaintance with that wonderful country.Lefschetz quickly saw that Adem had considerable mathematical talents and suggested that he go to Princeton University to undertake research for a doctorate. He helped Adem get financial support for the trip. Adem completed his B.S. degree in 1945 and then undertook graduate work at the Mathematics Institute from 1946 to 1948. He explained in the speech [10], made in 1960, how he progressed from engineering to topology research at Princeton:-

Although I began my professional studies at the National School of Engineers, it was in the professorships of maestro Alfonso Nápoles Gándara where I discovered my true vocation. In this way, when I finished my studies at the Faculty of Sciences in 1945, I attended the graduate courses and seminars offered at that time as they had been for several years. I remember the great influence that Roberto Vázquez's courses, the Topology seminar organised by Vázquez and Recillas, Enrique Valle's modern Algebra seminar, and Francisco Zubieta's Mathematical Logic seminar had on my formation.As well as attending pure mathematics courses, he also undertook work on applied mathematics between 1946 and 1948 and he published the paper

During the visit that Lefschetz made to the Institute in the summer of 1949, I told him of my desire to go abroad for my doctorate. Days after his return to the United States, I received an offer from Princeton University, which I immediately accepted. In September of that same year, upon my arrival at Princeton, I began my research in algebraic topology under the direction of N E Steenrod. The preparation that I received in Mexico was satisfactory.

*An elementary solution of a problem of anisotropic elasticity*(1949). Albert W Sáenz writes in the review [15]:-

The author treats the problem of a semi-infinite anisotropic plate whose axes of symmetry coincide with the Cartesian axes, and which is subjected to a normal concentrated force at the origin. The solution is obtained by elementary methods ...In 1949 Adem set out for Princeton to begin his Ph.D. studies. He crossed the border between Mexico and the United States at San Antonio, Texas, on 21 September 1949. On the Border Crossing form Adem is described as 6 ft $\large\frac{1}{2}\normalsize$in tall with a dark complexion, black hair and brown eyes. He was initially allowed to stay until 25 June 1950 as a student. Alejandro Adem Diaz de Leon writes [1]:-

By the year 1950, the development of mathematics had extended beyond the classical formal areas and was growing rapidly in the interactions between geometry, algebra and analysis. In particular, algebraic methods were introduced to study topological situations. This principle, developed under the tutelage of Lefschetz, among others, represented an exceptional advance for mathematics. Among the most successful exponents of this new philosophy was Norman Steenrod, under whose direction Adem was fortunate to complete his doctoral thesis.He was awarded a John Simon Guggenheim Memorial Foundation Fellowship in 1951 to continue his research. He did outstanding research for his thesis and, before submitting his thesis, wrote up a short paper containing the main results promising to provide full proofs in a later work. The paper giving the results was

*The iteration of the Steenrod squares in algebraic topology*(1952) published by the United States National Academy of Sciences. This work introduces the concept now known as "Adem relations". Today [January 2023]

*MathSciNet*lists over 200 papers with "Adem relations" mentioned in either the title or the review. The 1952 paper begins [3]:-

In this work it is proved that the Steenrod squares on cohomology classes satisfy certain identities. These identities are applied to solve particular homotopy problems. Full details will appear elsewhere.The Guggenheim Fellowship we have just mentioned was only one of several sources of support that Adem was receiving and in this paper he gives the following acknowledgement [3]:-

The research summarised in the present note has been supported at various times by the following institutions: Instituto de Matematicas de la Universidad Nacional and Instituto Nacional de la Investigacion Cientifica, Mexico City, the Higgins Foundation, Princeton University and the Rockefeller and Guggenheim Foundations.In 1952 Adem was awarded a Ph.D. by Princeton University for his 80-page thesis

*Iterations of the squaring operations in algebraic topology*; his thesis advisor had been Norman Steenrod. In 1953 Adem published another summary of results from his thesis in the paper

*Relations on iterated reduced powers*which again was published by the United States National Academy of Sciences. This paper begins [4]:-

In this note we present the generalisation of the relations on iterated squares to the case of iterated cyclic reduced powers of arbitrary prime period p. As in the case p = 2, the new relations are used to solve some particular problems.He notes in the paper [4]:-

I have heard that H Cartan has obtained relations of the same type, using methods quite different from mine.We will give a quote below from Henri Cartan concerning this.

Samuel Gitler writes of Adem's time at Princeton [7]:-

He had a great sense of humour, a noble soul that, together with his privileged intelligence and his enormous desire for knowledge, made him soon establish friendship with a large number of mathematicians from various countries of the world with whom he maintained correspondence throughout his life.Adem remained at Princeton until 1954 when he returned to Mexico City on an Air France plane leaving New York on 21 June. After arriving, he was appointed as a researcher at the Mathematics Institute of the National Autonomous University of Mexico. In July 1954 he attended the 'Second symposium on some mathematical problems that are being studied in Latin America' which was held at the UNESCO Centre for Scientific Cooperation for Latin America in Montevideo, Uruguay. He published an expository paper in the

*Proceedings*discussing results he had already published.

Solomon Lefschetz's 70th birthday was on 3 September 1954 and the volume

*Geometry and topology A symposium in honor of S Lefschetz*(1957) was published. This volume contains a 48-page paper

*The relations on Steenrod powers of cohomology classes*by Adem. Henri Cartan writes in the review [5]:-

The author gives detailed proofs of the relations between iterated Steenrod operations which he had previously announced in ['The iteration of the Steenrod squares in algebraic topology' (1952), 'Relations on iterated reduced powers' (1953)]. In the meantime, the proof by H Cartan appeared [(1955)] which uses a completely different method. Adem adopts the point of view of Steenrod ...In 1956 Adem took over as editor of the

...

The calculations [by Adem] are delicate, and carried out with great care; in general, the exposition of this difficult subject is made with great clarity and precision. ... Topological applications of the theory are promised for future work.

*Bulletin*of the Mexican Mathematical Society. He published many of his papers in that journal [7]:-

With his great mathematical prestige and his hard editorial work, he managed to make it a journal of recognised international prestige, where many articles by mathematicians from various countries around the world have been published, including of course Mexican mathematicians. He kept it at that level until his death.Adem 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. At the Symposium, Adem lectured and published a 37-page paper in the conference

*Proceedings*. Guy Hirsch writes in a review of the paper [8]:-

The paper describes secondary cohomology operations associated with the relations between iterated Steenrod squares which the author proved in an earlier paper ['Algebraic geometry and topology, A symposium in honor of S Lefschetz, 1957']; essentially the same method is used here, of which an outline is given (with less details than in the former paper) in the first part of this paper.From his Immigration records, we know that Adem lived at Eugenia 408, Mexico City. This is just outside the city centre in the south west part of the city. This is on the right side for the University which is on the south west edge of the city. His travel records show many trips to the United States in addition to his time as a postgraduate student at Princeton. For example: he flew from Mexico City to New York arriving 22 November 1955; he flew from Mexico City to San Antonio, Texas arriving 22 April 1957; he flew from Mexico City to Chicago arriving 15 July 1958 on his way to stay at Fine Hall, Princeton, and he flew from Mexico City to New York arriving 24 April 1961 on his way to stay at Fine Hall, Princeton.

On 14 June 1960 he became a member of El Colegio Nacional which is a public institution dedicated to the dissemination of scientific, artistic and humanistic culture. Today it has 37 members. Its headquarters are at Donceles 104, in the Historic Centre of Mexico City. The citation for his award reads [10]:-

He discovered universal formulas, the "Adem Relations", in which the algebraic nature associated with each geometric object is established. In the field of algebraic topology he worked on the iteration of Steenrod's squares in their relation and application to geometry. He is the author of 'Algebraic Geometry and Topology' (1957), 'Lecture Notes in Mathematics' (1970), among other works. He co-founded the Department of Mathematics at Cinvestav. He was awarded the National Prize for Sciences and Arts 1967. He entered El Colegio Nacional on 14 June 1960.His admittance speech to El Colegio Nacional begins as follows [10]:-

I arrive today with a deep emotion to this welcoming House. My election as a Member of the National College has been the most pleasant and profound of surprises. Surprise not only for me, but also for my mathematical colleagues, with whom I have been so actively and fruitfully linked in recent years. The nature of our work apparently closes to us some doors more accessible to other specialists, but this does not mean that there is not a decided interest among us in everything related to culture in general and to the development of our country. The great honour that is now being given to me, I take as both an award and an incentive to a team of scientific workers of which I am a part.For an English translation of his full admittance speech on the history of the mathematical movement in Mexico, see THIS LINK.

Samuel Gitler was a Mexican mathematician who, like Adem, went to Princeton to study for a Ph.D. which he was awarded in 1960. He writes [7]:-

I remember that before I returned to Mexico, Norman Steenrod, also my thesis advisor, told me that I was very lucky to go to work with José Adem, because I would learn from him how to do mathematics and how to write it. A fundamental part of mathematical research is how to write the results already obtained, because when you do, you find generalisations and many, many times, better results. Each article José Adem wrote went through many drafts, always searching for the best expression. I had the opportunity to collaborate in several of them and I learned a lot from this experience. His articles are a model of how to write mathematics; express the results in a precise way and in the demonstrations look for clarity and elegance.Adem helped in the formation of the Centre for Research and Advanced Studies of the National Polytechnic Institute, assisting the director Arturo Rosenblueth. In 1961 Adem became the director of the mathematics department of the Centre for Research and Advanced Studies of the National Polytechnic Institute, a position he held until 1973. He taught at both the National Autonomous University of Mexico and the Higher School of Physics and Mathematics of the National Polytechnic Institute. On a number of occasions he was offered the position of director of the Centre for Research and Advanced Studies which he turned down. He was also offered national roles like Undersecretary of Public Education which he also turned down. He refused such roles in the belief that were he to accept he would become a mediocre researcher and a poor administrator. His passion was mathematical research and he felt that it was in this area that he could make the greatest contribution to his country.

Although he refused to take on major administrative roles, he participated in many ways in the mathematical life of Mexico. He was a member of the National Institute of Scientific Research from 1961 to 1970, and when it became the National Council of Science and Technology he served as an advisor from 1971 to 1976. The Universidad Autónoma Metropolitana was founded in Mexico City in 1974 and Adem served on its Board of Directors from its founding until 1982. He was also a member of the Board of Directors of the National System of Researchers from 1984 to 1988 and a member of the Advisory Council of Sciences of the Presidency of the Republic from 1988. He was a member of the International Committee of the Latin American School of Mathematics from 1968.

In 1981 Adem was 60 years old and a conference was organised to celebrate the occasion. The conference proceedings [6] contains papers by both leading Mexican and leading international topologists. Authors include Samuel Gitler, Vic Snaith, Frank Adams and Edwin Spanier.

Adem died on 14 February 1991 at the age of 69. He continued to write papers to the end of his life, his last

*Generalized Hurwitz maps of the type $S \times V \rightarrow W$*completed only days before his death. Also shortly before his death he attended the Ph.D. viva of one of his students.

Let us end with quoting from Samuel Gitler who knew Adem very well [7]:-

He was an excellent teacher in the classroom. He had a special talent for explaining very complicated topics in a very understandable way, always with relevant examples that made understanding a concept much easier.

...

José Adem had a photographic memory and great intelligence. He was a man of science who tried to understand the results in other areas of knowledge. His engineering training enabled him to understand many problems in physics. He had great pleasure in painting and literature. Conversing with him on any subject was always one of my most happy pleasures. I always came out richer.

...

José Adem leaves us a scientific tradition and goals to achieve; do top-notch research in Mexico and publish it. Only by having first class science can we aspire to raise the levels of education to a higher level and thus be able to face the challenge that is presented to us to improve our quality as human beings and as Mexicans.

### References (show)

- A Adem Diaz de Leon, Semblanza Biográfica de José Adem,
*Centro de Ciencias Matemáticas, National Autonomous University of Mexico*.

https://www.matmor.unam.mx/~muciray/smm/60/adem.html - A Adem Diaz de Leon, Semblanza Biográfica de José Adem,
*Bulletin of the Mexican Mathematical Society*(2)**37**(1-2) (1992), XIII-XIV. - J Adem, The iteration of the Steenrod squares in algebraic topology,
*Proceedings of the National Academy of Sciences of the United States of America***38**(1952), 720-726. - J Adem, Relations on iterated reduced powers,
*Proceedings of the National Academy of Sciences of the United States of America***39**(1953), 636-638. - H Cartan, Review: The relations on Steenrod powers of cohomology classes, by José Adem,
*Mathematical Reviews*MR0085502**(19,50c)**. - S Gitler (ed.), Symposium on Algebraic Topology in honor of José Adem. Papers from the Symposium held in Oaxtepec, August 10-17, 1981,
*Contemporary Mathematics***12**(American Mathematical Society, Providence, R.I., 1982). - S Gitler, Homenaje al Doctor José Adem y al Maestro Rufino Tamayo, el 8 de julio de 1992,
*Matemáticos en México*.

https://paginas.matem.unam.mx/matematicos/matematicos-a-g/matematicos-a/jose-adem/154-homenaje-a-jose-adem - G Hirsch, Review: Second order cohomology operations associated with Steenrod squares (Spanish),
*Mathematical Reviews*MR0103462**(21 #2231)**. - José Adem,
*Colegio Nacional*.

https://web.archive.org/web/20070105165733/http:/www.colegionacional.org.mx/AdemJo0.htm - José Adem: Ciencias Exactas. Mathematico,
*El Colegio Nacional*.

https://colnal.mx/integrantes/jose-adem/ - José Adem, in Jay Kinsbruner (ed.),
*Encyclopedia of Latin American History and Culture*(Charles Scribner's Sons, Detroit, 2008). - José Adem, in
*Latin American Lives*(Macmillan Library Reference, New York, 1996-98). - José Adem,
*John Simon Guggenheim Memorial Foundation*.

https://www.gf.org/fellows/all-fellows/jose-adem/ - S Lefschetz, Reminiscences of a Mathematical Immigrant in the United States,
*The American Mathematical Monthly***77**(4) (1970), 344-350. - A W Sáenz, Review: An elementary solution of a problem of anisotropic elasticity, by José Adem,
*Mathematical Reviews*MR0040157**(12,652c)**. - J Silva Herzog, Adem Chahín, José (1921-1991),
*encyclopedia.com*.

https://www.encyclopedia.com/humanities/encyclopedias-almanacs-transcripts-and-maps/adem-chahin-jose-1921-1991 - J Silva Herzog,
*Biografías de amigos y conocidos*(Cuadernos Americanos, Mexico, 1980).

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Written by J J O'Connor and E F Robertson

Last Update June 2023

Last Update June 2023