Influential Mathematicians: Émile Borel
Hello quantum enthusiasts! For those who recently joined the community, this is our pre-course newsletter series aimed at highlighting mathematicians that have directly or indirectly contributed to the mathematical formalism of quantum mechanics. It is our humble attempt of taking any fear of mathematics out of your mind by showing that mathematicians are ordinary humans too!
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Since I’m currently on a short mini-break in southern France, today I'll highlight Émile Borel, another French mathematician!:)
Who was Émile Borel?
The School of Mathematics and Statistics University of St Andrew has a pretty comprehensive bio on Émile Borel that I've summarised for you below.
Émile Borel's father was Honoré Borel who was a Protestant minister. Honoré Borel himself was the son of a craftsman from Montauben, the capital of Tarn-et-Garonne in the Midi-Pyrénées region. Émil's mother was Émilie Teissié-Solier who was the daughter of a rich wool merchant from Saint Affrique. Émilie's father was opposed to her marrying a minister, wanting her to marry a rich landowner from the district. As a result he disinherited her and bad feeling existed between Emilie and her father. At the time Émile was born, his parents were living in a fine eighteenth century house in which Honoré Borel had a school for the children of Protestant families in the neighbourhood. Honoré and Émilie had two daughters who were much older than Émile, being sixteen and fourteen years old when he was born. Honoré Borel was an intelligent man and for the first years of his son's life he educated him at home.
When Émile was eleven years old he left home in Saint Affrique and went to live with his eldest sister, Madame Lebeau, who was married to a minister from Montauben. This meant that he was able to attend the Lycée at Montauben. There he showed extraordinary talents and, several years later, went to Paris as a bursar to the Collège Sainte-Barbe. While studying there, he attended courses at the Lycée Louis-le-Grand in preparation for taking the examinations to enter the École Polytechnique and the École Normale. He had the distinction of being ranked first in both these examinations and could choose either of the two universities.Family friends urged Borel to enter the École Polytechnique, which was considered the more prestigious establishment, but he had other ideas.
The family friends were merchants and industrialists, and their advice was to aim at an occupation where he would make a lot of money. Borel was advised that a degree from the École Polytechnique would give him the best opportunities for a job in industry or business. He, however, had very strong views that he wanted a job in science, particularly in mathematics, and he was sure that the best opportunities for such a position would be achieved by studying at the École Normale. He did have the support of his father for his decision, entering in 1889. We should mention a significant factor in his decision. While at the Lycée Louis-le-Grand he had met Gaston Darboux's son who was a pupil there. The two became firm friends and through the Darboux family Borel came to know leading mathematicians of the day, in particular Émile Picard whom he later remarked was a major influence on him at this time.
While studying at university he undertook military service with the engineers at Montpellier for a while but he seemed to be able to do this without too much disruption to his mathematical work. He published his first two mathematical papers in 1890. These were fairly minor works but still remarkable for a young undergraduate student just starting his studies. In 1892 he was awarded the agrégation in mathematics and in the same year his sixth paper, which is his first major memoir, was published. One year later he was awarded his doctorate after submitting his thesis ‘Sur quelques points de la théorie des fonctions’.Later Borel spoke about the mathematicians who had influenced him most in his early years mentioning, among others, Camille Jordan, Émile Picard, Paul Appell, Édouard Goursat, Paul Painlevé and Marcel Brillouin. At almost exactly the same time that he was receiving his doctorate, when still only 22 years of age, Borel was appointed Maître de Conférence at the University of Lille. He spent three years at Lille during which time he published 22 papers, with several others in the hands of the printer when he left. It was not only the quantity that is amazing but also the remarkable quality of the work that he produced. He returned to Paris in January 1897 when appointed Maître de Conférence at the École Normale Supérieure. Later in 1897 he was joint secretary at the first International Congress of Mathematicians held in Zürich from 9 August to 11 August.
Borel achieved much over the next years, both in his career and in the outstanding mathematics which he produced. From 1899 to 1902 he taught at the Collège de France and was reserve for the Cours Peccot. He was appointed examiner for entry to the École Navale in 1900, holding this position for ten years. He was awarded the Grand Prix of the Academy of Sciences in 1898, he was awarded the Poncelet Prize in 1901, he received the Vaillant Prize in 1904, and the Petit d'Ormcy Prize in 1905. Also in 1905 he was elected president of the French Mathematical Society.
In 1909 Borel was appointed to a chair of Theory of Functions created specially for him at the Sorbonne and he went on to hold this professorship until 1941. The year 1910 saw him attain the prestigious position of Deputy Director of the École Normale Supérieure, on the death of Jules Tannery, which he held for ten years. He later said that he regarded this appointment as the greatest honour he had received up to that time.
In 1918 he received the Croix de Guerre for his efforts for his country during World War I. However returning to his position at the École Normale after the war ended proved emotionally very difficult for him. Half of his students had been killed, and his own adopted son had also been killed in the war. He resigned as Deputy Director of the École Normale in 1920 and this period marks a change in direction in both his career and in his research interests. In 1921 he was elected to the Académie des Sciences, becoming its vice-president in 1933 and its president in 1934. Also in 1921 he was appointed to succeed Joseph Boussinesq in the chair of Probability and Mathematical Physics and, in the following year, he founded the Institut de Statistique de l'Université de Paris (Institute of Statistics at the University of Paris). In 1928, with financial support from Rockefeller and Rothschild, he set up the Institut Henri Poincaré (the Centre Émile Borel is now part of the Institute) and he ran the Institute for thirty years.
Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics. Borel created the first effective theory of the measure of sets of points. This work, along with that of two other French mathematicians, René Baire and Henri Lebesgue, marked the beginning of the modern theory of functions of a real variable. Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899. After 1905 he became interested in the theory of probability.
Quantum Formalism Influence
Although Borel did not directly contribute to the actual quantum formalism, his work in key branches of mathematics such as Measure Theory played an important role in the development of the quantum formalism. For example, loosely speaking, the set containing the possible measurement outcomes of a quantum operator e.g Hamiltonian, is a Borel measurable set! We’ll indeed cover Measure Theory in the advanced modules where we’ll also touch the Lebesgue integral (see our post on Henri Lebesgue).
That’s it for today! I’ll end with the reading materials related to Émile Borel:
Author: Bambordé Baldé, co-founder at Zaiku Group. Please feel free to connect with us on social media below!;)