**Jacqueline Ferrand**is also known by the name

**Jacqueline Lelong-Ferrand**which she used during the time she was married to the mathematician Pierre Lelong. Jacqueline was born in Alès, which is in southern France about 70 km from the coast and about 40 km north of Nimes. Her parents were Auguste Ferrand and Jeanne Chambon. Her father, Auguste, was a school teacher of Latin and Greek. Jacqueline attended the lycée de Jeunes Filles in Nimes, a school which was founded in 1832 and celebrated its 50

^{th}anniversary during the time that she studied there. The school moved to Feuchères in Nimes in 1931. Ferrand performed outstandingly in all the subjects she took at the lycée but it was mathematics which was her greatest love and she received the first prize in a national competition in mathematics. In 1936 the École Normale Supérieure changed its admission policy to allow women to compete in the entrance examination for the small number of places. Let us say a little more about the history behind this decision.

Marie-Louise Jacotin (known by her married name of Dubreil-Jacotin) had been ranked second for admission to the École Normale Supérieure, on Rue d'Ulm, in 1926 but, because she was a woman, she had been moved down the list to a position too low for entry. At this time women had to attend the École Normale Supérieure de Jeunes Filles at Sèvres just outside Paris. After arguing her case, and supported by influential friends, she had been given special permission to attend the École on Rue d'Ulm but it continued its general policy of not admitting women students. However, for admission in 1936 the École allowed women to compete for one of about 20 places for mathematics. To get an idea how difficult admission was, it was typical that around 250 students competed for these 20 places. Ferrand performed brilliantly at the École Normale Supérieure and she came first equal in the Agrégation de Mathematiques in 1939. The other candidate who came first equal with Ferrand was Roger Apéry. Jean Dieudonné was an examiner for analysis and he wrote:-

After graduating, Ferrand was appointed to teach at the École Normale Supérieure de Jeunes Filles at Sèvres. She taught there from 1939 to 1943, tutoring for the Agrégation de Mathematiques. However, during these years she undertook deep research on conformal representations. She published three notes, each of three or four pages, in 1941, namelyI was a member of the agrégation jury, for the only time in my life, by the way, and I gave a rather unusual analysis problem. Only two of the papers impressed me with their sense of analysis and precocious maturity very rare among candidates for the agrégation. Those two were by Roger Apéry and Jacqueline Ferrand.

*Sur la représentation conforme au voisinage d'un point frontière*,

*Sur l'itération des fonctions analytiques*, and

*Sur les conditions d'existence d'une dérivée angulaire dans la représentation conforme*. In the following year she published three further short notes and three major papers relating to her doctoral thesis. Two of these major papers, of around 100 pages in total,

*Étude de la correspondence entre les frontières dans la représentation conforme*and

*Étude de la représentation conforme au voisinage de la frontière*formed her doctoral thesis which she defended in June 1942. The president of the jury was Paul Montel, while the examiners on the jury were Arnaud Denjoy and Georges Valiron. In this thesis she investigated the behavior of conformal transformations in the neighborhood of a boundary point. She expressed her thanks to those who had supported her doctoral work in the second of these papers:-

This remarkable work was recognised by the Académie des Sciences when she was awarded their Prix Girbal-Baral in 1943. In the same year she was appointed as an assistant professor at the University of Bordeaux. After two years at Bordeaux, she moved to the University of Caen, again as an assistant professor. On 22 November 1947 she married Pierre Lelong, a mathematics professor in Lille. He had studied at the École Normale Supérieure and had taught at the University of Grenoble before being appointed to the University of Lille. Pierre Lelong's name is attached to several mathematical concepts, for example the Poincaré-Lelong equation, the Lelong-Demailly numbers, Lelong's problem, and the Lelong-Skoda transform. The couple had four children, Jean, Henri, Françoise and Martine. While she was married to Pierre Lelong, Jacqueline Ferrand published under the name Jacqueline Lelong-Ferrand. However, we shall continue to refer to her as Ferrand in this biography. We note at this point that, in 1977, Ferrand separated from her husband and reverted to publishing under the name Jacqueline Ferrand.I have to thank very specially M Denjoy who suggested the subject of this study and has been continually encouraging me. I'm pleased to record here the expression of my gratitude. I extend my deepest gratitude and respect to M Montel who kindly presented my 'Notes' for publication and accepted this Memoir for the 'Annals of the École Normale Supérieure'. M Valiron has constantly given me active sympathy; I thank him sincerely. Finally, I recall that I am indebted to M Wolff who, despite the difficulties of correspondence, kept me informed of the most recent work.

Returning to Ferrand's career, she was promoted to full professor at Caen in 1948 and, later in the same year, she was appointed to the chair of calculus and higher geometry at the University of Lille, filling the chair left vacant when Bertrand Gambier (1879-1954) retired. Her research during these years moved towards potential theory and she produced important papers such as *Sur le principe de Julia-Carathéodory et son extension à l'espace à p-dimensions* (1949) in which she generalised results due to Gaston Julia based on her study of certain potentials of double layers, and *Extension du théorème de Phragmen-Lindelöf-Heins aux fonctions sous-harmoniques dans un cône ou dans un cylindre* (1949) in which she generalised the Phragmen-Lindelöf theorem. In 1955, returning to her earlier interest in conformal representation, she published the book *Représentation conforme et transformations à intégrale de Dirichlet bornée* . Walter Hayman writes in a review [3]:-

In 1954This is an original and stimulating book. ...[It presents]deep theorems, many of which, although having considerable interest have not previously been included in books. The author deserves our thanks for having made them available to us, in an account which is lucid and gives full proofs. At the beginning of each chapter there is a short summary of the material contained in it and at the end a historical note, referring us to an extremely copious bibliography and index. ... This book is advanced in scope and in many instances brings us right up to the boundary of present knowledge. Perhaps only those who already know something of the more orthodox approach to problems of conformal mapping will be able to appreciate the power and originality of the present author's approach. Her book will be essential for anyone interested in the more theoretical aspects of conformal or quasi-conformal mapping.

**Jacques Chapelon (1884-1973) retired from his positions in Paris which provided vacant positions for which Ferrand applied. She was appointed to the Sorbonne during 1955-56 but before taking up her position there she visited the Institute for Advanced Study at Princeton along with her husband and children [1]:-**

While she was at the Institute for Advanced Study at Princeton, Ferrand worked on ideas which she announced inIn1956[Jacqueline Ferrand]and her husband, Pierre Lelong, also professor at the University of Paris, were in residence at the Institute for Advanced Study, carrying on research on Riemann manifolds and functions of several complex variables, respectively. In visiting the United States, the Lelongs did not interrupt their family life, for they brought to Princeton with them their three children, Jean Henri, and Françoise, aged seven, five, and four at the time.(Another daughter, Martine, was born to the Lelongs in January1958).

*Application of Hilbert Space Methods to Lie Groups Acting on a Differentiable Manifold*(1957).

On returning from Princeton, Ferrand took up her chair at the University of Paris, a position she continued to hold until she retired in 1984. During these years when she taught in Paris, she published a number of textbooks aimed at undergraduate teaching based on courses she had delivered. For example H W Guggenheimer, reviewing Ferrand's book *Géométrie différentielle* (1963), writes:-

She published her textbook on advanced calculusThis is a textbook for the geometry part of the course "Mathématiques II" according to the new French undergraduate program. This explains the fact that a textbook of differential geometry contains a chapter on non-euclidean geometry and one on the axiomatics of euclidean geometry, the first with a minimum, the second without any use of differential methods. As a whole the book is a striking example of the success of the reform of French university instruction and the decisive modernization of the curriculum.

*Cours d'analyse*(1968-70) in three volumes. Volume I covered multivariable differential calculus, with a little differential geometry. Volume II covered series, elementary functions of a complex variable, and elementary measure and integration. Volume III covered multivariable integral calculus, further topics in functions of a complex variable, Fourier series and ordinary differential equations. In 1971 she published, in collaboration with Jean-Marie Arnaudiès, the first volume of

*Cours de mathématiques*which covered algebra. The second volume, published in 1974, covered analysis (multivariable differential calculus and one-variable integral calculus) while the third volume, published in the following year covered geometry with applications to mechanics. The fourth and final volume was published in 1974 and covered ordinary differential equations, multivariable integral calculus and holomorphic functions. Ferrand continued to publish books aimed at undergraduate teaching, for example

*Exercices résolus d'analyse*(1977) which has the very convincing approach that the best way to learn mathematics is to solve problems. Having produced these books intended for students, Ferrand then published

*Les fondements de la géométrie*(1985) which was a text on the foundations of geometry intended for teachers of mathematics.

We must not give the impression that after a brilliant start to her research career, Ferrand became more interested in teaching mathematics than in research. Far from it, she continued to make some remarkable research contributions. In 1964 André Lichnerowicz posed a famous conjecture on differential geometry in his paper *Sur les transformations conformes d'une variété riemannienne compacte* . Over the following years some partial results were achieved as mathematicians attacked the problem. However, in 1969 Ferrand gave a complete proof of the conjecture in her paper *Transformations conformes et quasiconformes des variétés riemanniennes; application à la démonstration d'une conjecture de A Lichnerowicz* . She continued to produce further work on the topic in papers such as *Transformations conformes et quasi-conformes des variétés riemanniennes compactes (démonstration de la conjecture de A Lichnerowicz)* (1971). This led to her being an invited speaker in the Section on 'Differential Geometry and Analysis on Manifolds' at the International Congress of Mathematicians held in Vancouver in August 1974. She gave a talk entitled *Problèmes de géométrie conforme* which was published in the *Proceedings* of the Congress in 1975. However, still more remarkable results followed and she published further generalisations of the Lichnerowicz problem in *Action du groupe conforme sur une variété riemannienne* (1994) and *The action of conformal transformations on Riemannian manifolds* (1996). It is worth noting that these remarkable research contribution were done when Ferrand was approaching 80 years of age.

Let us end this biography by quoting Pansu's conclusion from [4]:-

Jacqueline Ferrand's work has had a significant influence in several branches of mathematics. However, they are little known in France. She has not sought to establish a school. As she says modestly, she hesitated to lead young people on tracks she considered insufficiently promising. Jacqueline Ferrand has undertaken some collaborations abroad, especially in Finland, where she enjoys high esteem. However, her intellectual path has been mostly solitary. The value of her work was not fully recognized except when her current mathematical achievements became known, which occurred in1942at the time of her thesis, in1969with the problem of Lichnerowicz and again in1996.

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