# Georges de Rham

### Born: 10 September 1903 in Roche, Canton Vaud, Switzerland

Died: 9 October 1990 in Lausanne, Switzerland

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**Georges de Rham**'s parents were Léon de Rham (1863-1945) and Marie Louise Du Pasquier (1870-1958). Léon de Rham was born at Orges, Vaud, Switzerland on 26 July 1863 into a family originally from Giez, a small town at the foot of the Jura in the canton of Vaud. He married Marie on 18 April 1893 at Neuchâtel, Switzerland. Marie had been born on 23 April 1870 at Neuchâtel. The family moved to Roche in 1896 where Léon de Rham was an engineer in a large cement factory. Léon and Marie de Rham had six children, all born at Roche: Jacqueline de Rham (born 1896), Marcel de Rham (born 1897), Gilbert de Rham (born 1899), Etienne de Rham (born 1901), Georges de Rham (the subject of this biography, born 1903) and Maurice de Rham (born 1906). This large family lived in a beautiful house in a rural setting. Georges said (see [1], [7] or [8]):-

Georges attended primary school in Roche but attended the secondary school Collège d'Aigle from 1914 to 1919. Aigle being about 7 km from his home, he had to travel there every day by train. Now the years that he attended this school are almost exactly the years of World War I and, because of the war, there was a much reduced train service. To get to school there was only one option, namely the train that left at 6 a.m. Georges said (see [1], [7] or [8]):-I was lucky to be born in the country and there spend my first sixteen years.

There was a flu epidemic in 1918 and de Rham thought that he was fortunate to catch flu [1]:-I did not like school very much and was not a brilliant student. My passion was drawing and painting, watercolours, my dream was to become a painter. Yet I remember one of the many mathematics teachers who made me understand the concept of algebra, which consists of calculating with an unknown x, and the large resulting simplification for solving arithmetic problems. But this teacher did not stay long. Another came, who no doubt loved mathematics, but apparently he liked even more the "Aigle " - the Aigle wine has indeed a good reputation.

On 8 September 1918, de Rham, together with his father and his brothers, ascended the Diablerets massif on a trip from Aigle through the Pillon pass to Anzeinde. His love of the mountains lasted throughout his life. See THIS LINK.During my last year at college, I was a victim of the1918flu, and the doctors ordered that, for my recovery, I had to miss school for several months. I think it was lucky, not only because I could indulge my passion for watercolour, but mostly because then I had to work alone, to catch up on what was done during my absence at school, and then I discovered that working only with the help of a book and sometimes the notes of a friend, you learn a lot better and in a much more profitable way than going to school.

In 1919 the de Rham family moved to Lausanne and Georges had to study at a new school. He attended the Gymnase classique in Lausanne from 1919 until 1921 [11]:-

The Gymnase classique in Lausanne specialised in Greek and Latin and taught little in the way of mathematics. What basic mathematics was studied at the school did not fill him with any enthusiasm and de Rham's interests turned more towards philosophy and literature. He spoke about his time at the Gymnase classique when he was awarded the Prix de la Ville from Lausanne in May 1979 (see [7]):-It is interesting to observe that the family eventually moved into a set of apartments of the well-known Château de Beaulieu, in central Lausanne, which remained Georges de Rham's permanent address until the end of his life; thus '7, avenue des Bergières, Lausanne' became a familiar address to all his many mathematical friends all over the world, many of whom stayed at this address when they came to Lausanne(as evidenced by several letters).

In August 1920, in the summer break between his two years at the Gymnase classique in Lausanne, de Rham ascended the Grand Combin, one of the highest peaks in the Alps, on a trip from Valsorey to Panossière.At the Gymnasium, Maurice Millioud proposed that I make a comparative study between the 'Traité des passions de Descartes' and the 'Psychologie des sentiments de Ribot'. Benjamin Valloton asked me to give a talk on 'La Légende des siècles'Ⓣby Victor Hugo. This work captivated me. And the lessons on Greek by Jean Franel were exciting.

Having graduated from the Gymnasium, despite his newly discovered interest in philosophy and literature, de Rham entered the University of Lausanne in 1921 with the intention of studying chemistry, physics and biology. At this stage he didn't consider studying mathematics, mainly because the way the subject had been taught led him to believe that there was nothing new to be discovered. He began to study mathematics in an attempt to understand questions that arose in the physics he was studying. After five semesters he gave up biology and, in the spring of 1924, turned to mathematics. He said [7]:-

While at university, he continued his mountaineering in the summer vacations making ascents of the greatest difficulty. In the autumn of 1925 he obtained his Licence ès Sciences. The two mathematicians who had influenced him most profoundly were Gustave Dumas (1872-1955), the professor of differential and integral calculus and higher analysis, and Dimitri Mirimanoff (1861-1945). Dumas encouraged de Rham to read the works of Henri Poincaré while Mirimanoff advised him to read books by Émile Borel, René Baire, Henri Lebesgue and Joseph Serret'sI launched myself with passion and I've never regretted it.

*Cours d'algèbre supérieure*Ⓣ [16]:-

In 1925 de Rham was appointed as an assistant to Gustave Dumas. He also taught at the local... de Rham considered moral straightness, altruism and respect as the very reasons of Dumas' success in teaching.

*Collège classique*in the summer of 1926. Encouraged by Dumas to undertake research in topology, he spent two periods, each of 7 months, in Paris between 1926 and 1928. The first of these visits was in November 1926. He did this without any financial support, relying only on his savings and living as cheaply as possible. He attended courses at the Sorbonne, enjoying those by Élie Cartan, Ernest Vessiot, Gaston Julia, Arnaud Denjoy, Émile Picard and others. He also attended courses at the Collège de France, being particularly impressed by those of Jacques Hadamard and Henri Lebesgue. It was Lebesgue who provided encouragement, advice and unfailing support throughout de Rham's work towards his doctorate. While in Paris, de Rham read all the topology books he could find and, importantly for the work he went on to do, he read James Alexander's paper

*Note on two three dimensional manifolds with the same group*. After his second visit to Paris, he returned to Lausanne in the autumn of 1928 and again taught at the

*Collège*and the

*Gymnase*. His first paper,

*Sur la dualité en Analysis situs*Ⓣ, appeared in 1928 in the

*Comptes Rendus*. Another important step in the progress of his research was during this period when he read Élie Cartan's paper

*Sur les nombres de Betti des espaces de groupes clos.*On reading the paper he became very excited realising how he could solve many of the problems he was considering. He continued to correspond with Lebesgue and published further notes in

*Comptes Rendus*. In April 1930 his thesis was complete and he sent a copy of it to Lebesgue who helped him to publish it in Liouville's journal, the

*Journal de mathématiques pures et appliquées*. He spent the winter term of 1930-31 at the University of Göttingen where he met, among others, Pavel Aleksandrov, Richard Courant, Charles Ehresmann, Gustav Herglotz, Andrey Kolmogorov, Edmund Landau, Emmy Noether and Hermann Weyl. He was awarded his doctorate in mathematical sciences from the Faculty of Science, University of Paris, in 1931 after defending his 87-page thesis

*Sur "l'Analysis situs"*des variétés

*à "n" dimensions*Ⓣ on 20 June before a committee consisting of Élie Cartan (who was the president), Paul Montel and Gaston Julia. His thesis [11]:-

The International Congress of Mathematicians was held in Zürich in September 1932 and de Rham attended giving a talk based on material from his thesis. He realised that his work was achieving international recognition when it was mentioned by Élie Cartan in his plenary address to the Congress on... is divided into four chapters; the first gives a good summary with improvements of the theory of finite complexes and their homology; the second chapter discusses intersection theory of chains in a complex; the third chapter introduces the use of multiple integrals over chains in ann-dimensional variety using as integrands differential forms, and the fourth chapter gives several examples of complexes which have the same Betti numbers and the same torsion, but are not equivalent.

*Les espaces riemanniens symétriques*Ⓣ. He was appointed as a lecturer at the University of Lausanne in 1932. In 1935 he attended the First International Congress of Topology in Moscow organized by Pavel Aleksandrov. Many of the leading topologists attended, including Heinz Hopf, Witold Hurewicz, Jacob Nielsen, André Weil and Hassler Whitney. De Rham spoke on Reidemeister torsion and lens spaces. At Lausanne he was promoted to extraordinary professor in April 1936 and to full professor in 1943. He continued to work at Lausanne until he retired and was given an honorary appointment by Lausanne in 1971. However de Rham also held a position at the University of Geneva although he always lived in Lausanne. He was appointed to Geneva as an extraordinary professor in 1936, being promoted to full professor in 1953. He retired from Geneva and was given an honorary position there in 1973.

In addition to these permanent appointments de Rham held a number of visiting professorships. He visited Harvard in 1949-50 and the Institute for Advanced Study at Princeton in 1950. Raoul Bott was at Princeton at this time and he describes de Rham's visit in [6]:-

De Rham visited the Institute for Advanced Study again in 1957-58. He also visited the Tata Institute in Bombay in 1966.De Rham had a subtle charm which drew younger people to him immediately. In those early days at Princeton he would easily mingle with the boisterous postdocs, his exquisite manners contrasting amusingly with our rude ways.

In [6] Raoul Bott describes the context of de Rham's famous theorem:-

The details of the de Rham theorem are given in [6] but as far as this article is concerned it is sufficient to give the 'feel' for the type of theorem as nicely described there:-In some sense the famous theorem that bears his name dominated his mathematical life, as indeed it dominates so much of the mathematical life of this whole century. When I met de Rham in1949at the Institute in Princeton he was lecturing on the Hodge theory in the context of his 'currents'. These are the natural extensions to manifolds of the distributions which had been introduced a few years earlier by Laurent Schwartz and of course it is only in this extended setting that both the de Rham theorem and the Hodge theory become especially complete. The original theorem of de Rham was most probably believed to be true by Poincaré and was certainly conjectured(and even used!)in1928by Élie Cartan. But in1931de Rham set out to give a rigorous proof. The technical problems were considerable at the time, as both the general theory of manifolds and the 'singular theory' were in their early formative stages.

Of course de Rham produced much in the way of important mathematics in addition to the 'de Rham theorem'. He gave a reducibility theorem for Riemann spaces which is fundamental in the development of Riemannian geometry. He also worked on Reidemeister torsion and his work on this topic was the beginning of rapid developments. He wrote a number of very important books: (with Kunihiko Kodaira)The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years.

*Harmonic Integrals*(1950);

*Variétés différentiables. Formes, courants, formes harmoniques*Ⓣ (1955); (with S Maumary and M A Kervaire)

*Torsion et type simple d'homotopie*Ⓣ (1967); and

*Lectures on introduction to algebraic topology*(1969).

For brief extracts from reviews of these books see THIS LINK.

As a teacher, de Rham had a superb reputation:-

We now give two descriptions of de Rham's character, the first by Bott and the second by Chandrasekhar:-Essentiality, linearity of thought and calmness made up the elegance of his lectures and seminars, where, according to Henri Cartan, he was able to "suggest a lot of things in few words." His approach to the audience was always full of benevolence and care. ... "Never forget that you must love your pupils", was his favourite advice to his assistants.

The second description:-... de Rham had a subtle charm which drew younger people to him immediately. In those early days at Princeton he would easily mingle with the boisterous postdocs, his exquisite manners contrasting amusingly with our rude ways. He was always lean and one could feel the steel in his sinews, but he never boasted of his mountaineering exploits and it was only at second hand that the daredevil in him became apparent...

De Rham received many honours. He was secretary/treasurer of the Swiss Mathematical Society in 1940-42, vice-president in 1942-44, and president in 1944-45. He was President of the International Mathematical Union from 1963 to 1966 and, in this capacity, was president of the International Congress of Mathematicians held in Moscow in August 1966. He was elected a member of the Accademia dei Lincei (1962), the Göttingen Academy of Sciences (1974), and the Académie des Sciences of the Institute of France. He received honorary degrees from the universities of Strasbourg, Genoble, Lyon, and l'École Polytechnique Fédérale Zürich. He received the Prize of the Marcel Benoist Foundation (1965) and of the City of Lausanne (1979). An international colloquium was held in Geneva in March 1969 to honour de Rham. An address on this occasion stated:-Tough as steel in his adherence to principle, resilient, placable, self-less and generous beyond the dictates of fashion, steadfast in friendship, but not at the price of reason, de Rham strides the world of mathematics a happy warrior.

There are many different aspects of de Rham's career, combining his passion for mathematics and for mountaineering. We give some details of de Rham as a mountaineer at THIS LINK.... he did as much as any one man could do to bring mathematicians together, young and old, classical and modern, from the East and West. He found a special joy in the spectacle of younger colleagues adding to the old heritage, and did as much as he could to encourage them.

Let us quote from [16] where these different aspects of de Rham are beautifully illustrated and brought together:-

Let us end this biography by quoting de Rham's advice to his students (see [16]):-According to the testimonies of those who have known Georges de Rham, it seems hard to separate, in his personality and his public engagement, the loyal colleague from the generous friend, the passionate researcher from the fatherly teacher, the brilliant scientist from the brave alpinist. He regarded proving a theorem, delivering a lecture or reaching the top of a mountain as a personal endeavour, requiring a full involvement of the deepest and most precious faculties of the individual, in the name of some kind of transcendental beauty. On the occasion of the centenary of de Rham's birth, his former doctoral student Oscar Burlet remembers that he once explained to him: "For alpinism is not only a physical exercise, but it also a task for the mind, and it allows one to create a marvellous harmony between nature, soul and body." Climbing higher to attain a larger view could be considered as a metaphor for de Rham's constant attempt to advance in abstraction, to gain generality by looking at single problems and objects from above. The development set off by his first proof of the theorem bearing his name moved exactly in the direction of embracing specific questions in a more and more general framework. His scientific approach is based on the comparison of structures, which, unlike in the so-called 'abstract nonsense', should not be treated as empty buildings, but employed as containers that enable us to grasp different particular cases at the same time.

The courses, the books should only be suggestions and inspirations to work: a mathematician must judge on his own account, he must be critical and must not admit anything which he has not clearly recognized himself as well-founded.

**Article by:** *J J O'Connor* and *E F Robertson*

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**List of References** (20 books/articles)

**Mathematicians born in the same country**

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**Honours awarded to Georges de Rham**

(Click the link below for those honoured in this way)

1. | President of the Swiss Mathematical Society | 1944-45 |

2. | BMC plenary speaker | 1956 |

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