He attended secondary education at the Plaza del Monasterio de Mahón Institute and passed the state exam, with which this cycle ended, in 1944 at the age of sixteen.

At the end of high school, he began to prepare for the entrance exam at the School of Civil Engineers, Canals and Ports, and while preparing it, he discovered a taste for mathematics. In 1949 he enrolled at the University of Barcelona to continue his studies in this discipline, before he had been ill with tuberculosis for two years. In 1954 he graduated in Science (Exact section) with an extraordinary award.

Between 1956 and 1958 he completed his training at the Hamburg Mathematical Seminary under the direction of Professors Ernst Witt (1911-1991), a German algebraist disciple of Emmy A. Noether, who was the one who proposed the subject of his doctoral thesis, and Wilhelm Blaschke (1885-1962), Austrian mathematician specializing in differential and integral geometry. During this stay in Germany, he received a scholarship from the Higher Council for Scientific Research and the Deutscher Akademischer Austauschdienst (German Academic Exchange Service). In 1959 he did a three-month stay at the Zurich Polytechnic Institute where he worked on issues of differential geometry with Beno Eckmann (1917-2008), a Swiss mathematician who was a disciple of Heinz Hopf. In 1960 he received a doctorate from the University of Barcelona with the thesis entitled “Sobre la parte p-fundamental del grupo de Brauer”, sponsored by Dr. Joan Augé and published that same year in Collectanea Mathematica.

His entire professional life, from 1955 until his retirement at the age of seventy, was spent at the University of Barcelona. At the age of thirty-three he obtained the chair of Metric Geometry and Differential Geometry at this university. He was the youngest of a generation of professors who formed all the mathematicians’ classes in Catalonia for almost twenty years, among which we must remember the Francesc d’Assís Sales Vallès (Terrassa 1914 – Barcelona 2005), Enrique Linés Escardó (Logroño 1914 – Madrid 1988), Joan Augé Farreras (Barcelona 1919 – Barcelona 1993), Josep Teixidor Batlle (Llers 1920 – Llers 1998), Juan José de Orús Navarro (Barcelona 1921 – Barcelona 2005), Rafael Aguiló Fuster (Palma de Mallorca 1923 – Barcelona 1995), Joaquín M. Cascante Dávila (Barcelona 1925 – Barcelona 1998), Juan B. Sancho Guimerá (Morella 1926 – Salamanca 2011) and Rafael Mallol Balmaña (Barcelona 1926 – Barcelona 1988). Some of them had been Dr. Vaquer’s professors during his undergraduate degree. Throughout all those years the Mathematical Seminar of the University of Barcelona invited some renowned foreign mathematicians to give lectures that would serve as a stimulus and inspiration to younger mathematicians. We would highlight those given by professors Ernst Witt and Jean A. E. Dieudonné (1906-1992), the latter founding member of the Nicolas Bourbaki collective, a group that had a notable influence on the teaching of mathematics in Catalonia during the second middle of the 20th century.

We must remember, appreciate and thank the effort made by the teachers of this time to keep the publication of Collectanea Mathematica alive. It is the oldest mathematics magazine in Spain that had been founded in 1948 by Dr. José María Orts Aracil (Paterna 1891 – Barcelona 1968) in order to be able to exchange with the mathematical publications of other centers and, in this way, provide to the Mathematical Seminar of a reference library. Dr. Vaquer participated in this task as Editorial Secretary for more than 15 years.

He was the director of two theses: Joan Girbau Badó, “Algunos resultados sobre cohomología de las variedades Kählerianas”, University of Barcelona (1971), and Carlos Currás Bosch, “Sobre inmersiones isométricas de variedades riemannianas en espacios euclídeos”, University of Barcelona (1977).

Concerned from the beginning about the quality of teaching, a concern that accompanied him all his life, in 1960 he published with Dr. Teixidor an introductory book to mathematics popularly known as the “Teixidor-Vaquer”. For years it was the reference text for first-year science majors, the then-called selective course. Eleven editions of the Mathematics Course of doctors J. Teixidor and J. Vaquer were made between 1960 and 1976.

To his teaching activity at the university must be added the organization, also for many years, of the Mathematical Olympiad, aimed at high school students and professional development training cycles.

Committed to the country and its language, from the 1965-66 academic year, during the Franco regime, he taught Differential Geometry classes for the 5th year in Catalan. He was the only professor at the Faculty who at this time dared to breach the obligation to teach in Spanish.

Despite not being to his liking administrative and management activity, he was Dean of the Faculty of Mathematics (currently Mathematics and Computer Science) of the University of Barcelona between 1976 and 1978. In 1978 he was elected associate member of the Institut d’Estudis Catalans and in 1989 he became a permanent member attached to the Science and Technology Section. For four years, from 1991 to 1995, he was president of the Catalan Mathematical Society.

The biography of Dr. Vaquer would be manifestly incomplete without highlighting his goodness, and also the human and social activity that, together with his wife Mercè Guilemany Bas, he always did, helping everyone in need and especially the most disadvantaged.

He died in Barcelona on March 24, 2020, the year of the pandemic, at the age of 91.

## References

[1] J. Vaquer.

*Sobre el producto ordinal de dos conjuntos fijos*. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid, 54(2), 1960.[2] J. Vaquer. *Sobre la parte p-fundamental del grupo de Brauer*. Collectanea Mathematica, 12(2) 103-129, 1960.

[3] J. Vaquer. *Geometría diferencial de las familias de planos*. Collectanea Mathematica, 13(1-2) 169-182, 1961.

[4] J. Teixidor, J. Vaquer. *Curso de Matemáticas*. First edition 1965.

[5] A. Reventós, S. Xambó. *Entrevista al doctor Josep Vaquer i Timoner*. SCM/Notícies No. 11, 1999.

## Acknowledgements

This biographical note has been prepared by Dr. Gerard Gómez Muntané, Professor of Applied Mathematics at the University of Barcelona.