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Person: De Groot, Johannes

Johannes de Groot was a Dutch mathematician who worked in topology.

Mathematical Profile (Excerpt):

• In 1928 Johannes de Groot Sr. was appointed professor of Hebrew linguistics and Israelite archeology at the University of Groningen, succeeding Franz M Th Bohl.
• In the 17th century they had settled in the Het Bildt region which had been reclaimed from the Middelsee (or Bordine).
• De Groot Sr. was interested in mathematics but had decided to study theology and literature at the University of Groningen.
• It was his mathematics teacher J Scholten at the Willem Lodewijk Gymnasium who first awakened his interest in mathematics which led to him deciding to study mathematics at university.
• Among his mathematics lecturers were Johannes Gaultherus van der Corput (1890-1975) and Gerrit Schaake (1892-1945).
• Van der Corput's area of research was analytic number theory and he was a professor at Groningen from 1923 to 1946.
• He then moved to Amsterdam where he was the cofounder, with David van Dantzig and Jurjen Ferdinand Koksma, of the research and service institution, the Mathematisch Centrum, in February 1946.
• Schaake had studied at Amsterdam and, advised by Hendrik de Vries, was awarded his Ph.D. in 1922 for the thesis Afbeeldingen van figuren op de punten eener lineaire ruimte Ⓣ(Images of figures on the points of a linear space).
• After graduating with his first degree, de Groot went on to study for his doctorate, advised by Schaake.
• He began to undertake research on algebraic geometry also looking at certain problems in algebra.
• However, he moved towards topology and was awarded a Ph.D. in 1942 for the 102-page thesis entitled Topologische Studien.
• Indeed, he had chosen that topic independently - in all his studies he has been his own mentor.
• Of course, de Groot's years studying at university had been made difficult by World War II.
• The Netherlands was under German occupation for the following years and this was certainly the case when de Groot completed his university studies.
• The position in 1942 was that although under German occupation, the country's administration attempted to keep things operating as best as they could in very difficult circumstances.
• De Groot became a secondary school teacher of mathematics, first at Coevorden and then at The Hague.
• Let us look at some papers he published in 1942, the year his Ph.D. was awarded.
• In 1942 he published On the extension of continuous functions in which he proves that it is impossible to extend every continuous (bounded) real function on a nonclosed subset of a metric space to a continuous (bounded) real function on that space.
• As is well known, any field can be obtained from its prime field by a succession of simple transcendental extensions followed by a succession of algebraic extensions.
• The authors show that a field with a nontrivial non-Archimedean valuation is separable if and only if it is possible to choose the (transfinite) sequence of transcendental extensions in such a way that there is only a denumerable number of elements which are completely transcendental over the field that they extend.
• Using a result of Zippin (1935) it is proved that the space of such a field can be made compact by adjoining a denumerable number of points.
• There were two universities in Amsterdam, the University of Amsterdam (founded 1632) and the Free (Vrije) University (founded 1880).
• The Mathematical Centre, however, was an independent institution not attached to either of these universities.
• We mentioned above that Van der Corput, one of de Groot's influential teachers at university, had been the co-founder of the Mathematical Centre in Amsterdam in February 1946.
• The following year, de Groot was appointed a lecturer in mathematics at the University of Amsterdam.
• Then, in 1948, he was appointed professor of mathematics at the Technological University of Delft.
• actively participating in and in many instances decisively influencing its research activity.
• Although he worked in Amsterdam for the rest of his career, he made many visits to the United States.
• He visited Purdue University (September 1959-September 1960), Washington University at St. Louis (September 1963-February 1964), the University of Florida at Gainesville (August 1966-March 1967), and the University of South Florida in Tampa (December 1971-March 1972).
• In fact, from November 1967 he was appointed graduate research professor at the University of Florida at Gainesville and, from that time on, divided his time between this position and his professorship in Amsterdam.
• We mentioned above that before de Groot began researching in topology he was interested in algebra.
• In fact, later in his career he returned to his interest in algebra when he undertook some research in group theory.
• Later de Groot worked on set-theoretic topology.
• In his last few years, de Groot also began to do research in infinite dimensional topology and in the topology of manifolds.
• The book by J M Aarts and T Nishiura, Dimension and extensions (1993), discusses a long-standing problem of de Groot.
• The main conjecture made by him, which went back to his 1942 thesis, had been solved not long before this book was written.
• Their common focus is a long-standing problem of Johannes de Groot, the main conjecture of which was recently resolved.
• As is true of many important conjectures, a wide range of mathematical investigations had developed.
• These investigations have been grouped into the two extension problems under discussion.
• The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension.
• This minimal dimension was called the compactness deficiency of a space.
• Early success in 1942 lead de Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space.
• The compactness degree of de Groot was defined by the replacement of the empty space with compact spaces in the initial step of the definition of the small inductive dimension.
• This trait was his strength, where the riches came from his grasp of simple ideas without much background knowledge, which made it possible for him to lead others to work together and to encourage them.
• An important feature of de Groot's career was the number of Ph.D. students that he supervised who later became university lecturers.
• De Groot received many honours, perhaps the most prestigious of which was his election in 1969 to the Royal Dutch Academy of Sciences.
• For many years de Groot suffered prolonged periods of poor health but, despite this, his death was sudden and unexpected.

Born 7 May 1914, Garrelsweer, The Netherlands. Died 11 September 1972, Rotterdam, The Netherlands.

View full biography at MacTutor

Tags relevant for this person:

Origin Netherlands, Topology

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@J-J-O'Connor

@E-F-Robertson

References

Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive