**Yvonne Bruhat**is also known by the name

**Yvonne Fourès-Bruhat**which she used while she was married to the mathematician Léonce Fourès. However, she is best known by the name

**Yvonne Choquet-Bruhat**, that she has used since she married her second husband Gustave Choquet. For simplicity, we will use the name Yvonne Bruhat throughout this biography. Yvonne Bruhat was the daughter of Georges Bruhat (1887-1945) and Berthe Hubert (1892-1972). Georges Bruhat had been ranked first in the Concours d'agrégation in physics in 1909. He was awarded his doctorate in physics before the outbreak of World War I in 1914, was mobilised during the war, and then appointed to the University of Lille in 1919. He was promoted to professor of physics at Lille in 1921. He had married Berthe Hubert who had had been ranked first in the Concours d'agrégation in arts in 1917. She went on to become a professor of philosophy. Georges and Berthe Bruhat had two children, a daughter Yvonne (the subject of this biography) born in 1923 and a son François Bruhat (who also has a biography in this archive) born in 1929. In 1927, when Yvonne was under four years old, her father was appointed to the Faculty of Science in Paris so she was brought up in that city.

Yvonne Bruhat undertook her secondary school education in Paris. In 1941, she entered the Concours General, a competition to determine the best pupils in the whole of France, and she won the silver medal for physics. Of course, by this time France was occupied by German troops. World War II had begun on 1 September 1939 when German forces entered Poland. On the following day, Britain, France and several other countries, declared war on Germany but over the following months France was not involved in any fighting, but spent time trying to build defences to protect the country from an invasion by Germany. The war changed dramatically for France on 10 May 1940 when the German army crossed the Dutch and Belgium borders and by June France had surrendered and fighting ended. Georges Bruhat had been appointed as deputy director of the École Normale Supérieure in 1935 and was appointed to the chair of physics there in 1941, the previous holder of the chair having lost his position through a decision of the Vichy government. In 1943 Yvonne Bruhat began her studies at the École Normale Supérieure de Jeunes Filles at Sèvres just outside Paris. This university for women was the partner institution of the École Normale Supérieure, on Rue d'Ulm. University studies in occupied Paris was difficult enough but tragedy struck the Bruhat family in 1944 [1]:-

Baillou, mentioned in this quote, is Jean Baillou who was the general secretary of the École Normale Supérieure. Yvonne was in the middle of her studies when the tragedy happened. Speaking years later of this extremely difficult time she said [10]:-When the Gestapo arrested a student who was absent from Rue d'Ulm during the night of August41944^{th},, they undertook a systematic search of the premises. Roger Apéry, who had been making false identity papers in his room, hurriedly burned all the compromising documents. The Gestapo took Mrs Bruhat and Mrs Baillou hostage, to exchange them for their husbands the next day; the cleaning woman, not realizing the implications, was complaining loudly about the ashes left in Apéry's room. Bruhat and Baillou were deported, and Bruhat perished at Buchenwald.

She dedicated the bookI was very protected, even too much, so that I could continue my studies. In a way[my father]influenced me a lot, but then he was more interested in his son. He did not think that his daughter would become a leading scientist. He thought that I would be a good mother for my family, a high school teacher.

*General Relativity and the Einstein Equations*(2009) she wrote many years later to:-

Georges Bruhat's books include the four volumeGeorges Bruhat,1887-1944, who gave me the love of physics by his kindness and explanations, and after his death by his books. Berthe Bruhat, born Hubert,1892-1972, who always encouraged me to try to understand some of the world and to pursue scientific research.

*Cours de physique générale*Ⓣ. Volume 1 is

*Electricity*(1924), Volume 2 is

*Thermodynamics*(1926), Volume 3 is

*Optics*(1930) and Volume 4 is

*Mechanics*(1934). Let us add at this point that Yvonne's mother, Berthe Hubert Bruhat, published

*Traite de pedagogie generale*Ⓣ in 1946.

Yvonne Bruhat graduated from the École Normale Supérieure de Jeunes Filles in 1946 and undertook research advised by André Lichnerowicz. From 1946 to 1949 was an assistant at the École Normale Supérieure. In 1949-51, she was first a Research Assistant, then promoted to Research Associate, at the Centre National de la Recherche Scientifique. She was awarded her doctorate in 1951 for her thesis *Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires* Ⓣ. However, she had many papers published even before submitting her thesis. They were all published under the name Yvonne Fourès-Bruhat, since she had married the mathematician Léonce Guy Fourès who, like Bruhat, was undertaking research in Paris. He was awarded his doctorate in 1951 for his thesis *Sur quelques points de la théorie des surfaces de Riemann* Ⓣ.

Bruhat's first four papers were published in 1948: *Sur une expression intrinsèque du théorème de Gauss en relativité générale* Ⓣ; (with André Lichnerowicz) *Sur un théorème global de réduction des ds*^{2}* statiques généraux d'Einstein* Ⓣ; *Sur l'intégration du problème des conditions initiales en mécanique relativiste* Ⓣ; and *Théorème global sur les ds*^{2}* extérieurs généraux d'Einstein* Ⓣ. Further papers followed: *Théorème d'existence pour les équations de la gravitation einsteinienne dans le cas non analytique* Ⓣ (1950); *Un théorème d'existence sur les systèmes d'équations aux dérivées partielles quasi linéaires* Ⓣ (1950); and *Théorèmes d'existence et d'unicité pour les équations de la théorie unitaire de Jordan-Thiry* Ⓣ (1951). In 1952, in addition to two short papers, she published a major paper in *Acta mathematica* with the same title as her thesis [2]:-

Bruhat spent part of the academic year 1951-1952 and the academic year 1952-53 in the United States, at the Institute for Advanced Study in Princeton. Her husband, Léonce Fourès, was also a member of the Institute for Advanced Study in Princeton during the same period. Returning to France she was appointed to the Faculty of Science in Marseille in 1953. Fourès was also appointed to Marseille. Bruhat and her husband were soon back at Princeton spending part of the academic year 1955-56 at the Institute for Advanced Study. She remained on the faculty in Marseille for five years before being appointed to the University of Reims in 1958. She only held this position for one year before being appointed as a full professor at Faculty of Science in Paris in 1960. On 16 May 1961 she married her second husband, Gustave Alfred Arthur Choquet (1915-2006). He was a mathematician who had obtained his doctorate in 1946 and, from 1950, had been a professor at the University of Paris. Here is an extract from his obituary [5]:-One has to face difficult nonlinear partial differential equations in Einstein's theory of gravity. She carried out this task at a new level of mathematical rigour that she herself created. This led to a far-reaching enrichment of both mathematics and physics. ... Of very great significance is that Yvonne Bruhat's analysis enabled her to prove rigorously for the first time local-in-time existence and uniqueness of solutions of the Einstein equations.

Bruhat was appointed to the Chair of Mechanics at the University of Paris VI, the Pierre and Marie Curie University, where her husband was also a professor. This University was established in 1971 when the University of Paris was divided into several separate universities. The Pierre and Marie Curie University was largely the old Faculty of Sciences, on the Jussieu Campus in the Latin Quarter of Paris. She held this chair until she retired in 1992, becoming professor emeritus at that time.He published more than160mathematical articles and11books. He made important contributions to a variety of fields: topology, measure theory, descriptive set theory, potential theory, and functional analysis. Two themes deserve particular mention. His monumental1953-4paper 'Theory of capacities' contains among many riches his celebrated capacitability theorem. A second great theme is his theory of integral representations in compact convex sets and weakly complete cones, now usually known as Choquet theory, which launched a huge development. The work on capacities and that on integral representations have both found many applications in analysis and probability. His personal distinction was allied in a remarkable way to great kindness and humanity. He was an inspiring teacher who gave generous encouragement and support to pupils and younger colleagues.

The Institut des Hautes Études Scientifiques gives the following summary of Bruhat's contributions [4]:-

MathSciNet lists 241 publications by Yvonne Choquet-Bruhat (some of these are under the name Yvonne Fourès-Bruhat. Given this incredible publication record, it is impossible to give a good indication here of its contents. Let us single out a few very significant papers.Yvonne Choquet-Bruhat is an internationally recognized mathematician. Her research area is located on the border between mathematics and physics. Her research covers a very wide range of knowledge from the first mathematical proof for the existence of solutions of Einstein's relativistic theory of gravitation to the study of the conversion of electromagnetic waves into gravitational waves(or the reverse)in the vicinity of a black hole. Her research has created new mathematical methods that have provided a solid foundation for the study of several physical theories: the theory of general relativity, relativistic hydrodynamics, non-Abelian gauge theory, the theory of supergravity, etc. She introduced some new formulations of Einstein's theory of gravitation which has led to spectacular recent progress in numerical relativity, including the calculation of gravitational waves emitted during the collapse and merger of two black holes. These results are of great importance for giant interferometric detectors of gravitational waves such as VIRGO(Franco-Italian)or LIGO(American).

*Ondes Asymptotiques et Approchees pour des Systemes d'Equations aux Derivees Partielles non Lineaires*Ⓣ, published in 1969, gives a method for constructing asymptotic and approximate wave solutions about a given solution for nonlinear systems of partial differential equations.

*Global Solutions of the Problem of Constraints on a Closed Manifold*, published in 1973, shows that the existence of global solutions of the constraint equations of general relativity on a closed manifold depend on subtle properties of the manifold. She gave the first deep mathematical study of supergravity in

*Causalite des Theories de Supergravite*Ⓣ (1984). After this very brief look at some of her highly significant papers, let us look at some of the books that she has written.

In 1963 she published *Recueil de problèmes de mathématiques à l'usage des physiciens* Ⓣ which was translated into English as *Problems and solutions in mathematical physics* (1967). It consists of problems and their solutions required for certificate examinations in the mathematical methods of physics. In 1968 she published *Géométrie différentielle et systèmes extérieurs* Ⓣ. J D Zund writes in a review:-

Bruhat gave courses at the University of Paris to students taking the Master of Mathematics degree which prepared them for the practical use of distributions in the partial differential equations of theoretical physics. These courses became the basis for her French textThis is an extremely elegant account of the methods of differential geometry and exterior differential systems, which as André Lichnerowicz says in his preface, remains faithful to the spirit of Élie Cartan. The book is very attractive and beautifully written. ...[It]is to be most enthusiastically recommended to both pure and applied mathematicians, and it will be of particular value to theoretical physicists who desire a glimpse into the fascinating field of modern differential geometry.

*Distributions*(1973). A collaboration with Cecile DeWitt-Morette and Margaret Dillard-Bleick led to the major three-author work

*Analysis, manifolds and physics*(1977). Rudolph Schmid, reviewing the second edition which appeared in 1982, wrote [8]:-

Bruhat and Cecile DeWitt-Morette produced a companion volume in 1989. J E Marsden writes in a review:-The success of the first edition of this excellent book encouraged the authors to write an enlarged and updated second edition. They have succeeded in gathering in one volume the mathematical infrastructure of modern mathematical physics, which includes the theories of differentiable manifolds and global analysis, Riemannian and Kählerian geometry, Lie groups, fibre bundles and their connections, characteristic classes and index theorems, distributions, and partial differential equations. ... In my opinion, this book accomplishes its purpose well, being a reference which can be used as a text in an advanced physical mathematics course.

We must mention two further important books which were single-author works by Bruhat. First there wasThe original book was well received in the community and is still a valuable resource today. It covers an amazing variety of topics that are of use to a large number of mathematicians, mathematical physicists, and mathematical engineers. The original book already had a number of interesting applications, such as the Schrödinger equation, soap bubbles, electromagnetism, shocks, gravity, Hamiltonian systems, monopoles, spinors, degree theory applied to PDE, Wiener measure, etc. However, many of these were contained in extended problem format and tended to be a bit sketchy because of that, and also for lack of space, given the breadth of topics that the authors wanted covered in a coherent whole. This companion volume extends the applications to a more detailed discussion and to more sophisticated and interesting ones as well. To illustrate, here is a topic from each chapter: the Fierz identity, the stress-energy tensor, Stiefel manifolds, the Weyl Heisenberg group, Calabi-Yau spaces, Chern-Simons classes, and Wightman and Schwinger functions.

*Graded bundles and supermanifolds*(1989) which was reviewed by Alice Rogers who writes:-

Her latest book,These lecture notes give a pedagogical account of the basic concepts in the differential geometry of graded manifolds and supermanifolds. The style is clear and readable. Both graded manifolds and supermanifolds extend classical geometry to include anticommuting variables, an approach which is motivated by the desire to extend the geometric techniques used in theoretical physics to systems which include fermions. The formalism is particularly appropriate for supersymmetric systems, which have symmetries which mix bosonic and fermionic degrees of freedom, as it treats these two kinds of degrees of freedom in a similar manner.

*General relativity and the Einstein equations*was published in 2009. Hans-Peter Künzle begins a review by writing:-

Bruhat had a daughter Michelle with her first husband, Léonce Fourès, and a son Daniel and a daughter Geneviève with her second husband Gustave Choquet. Daniel Choquet studied bioengineering in Paris and obtained a doctorate from the Pierre and Marie Curie University in 1988. He has made important advances in nanoscopic imaging and the organization of receptors in neurons, earning him election to the Academy of Sciences and several prizes.This massive book covers in detail the areas of classical mathematical relativity theory to which the author has made major contributions during her long and active research career. As the title indicates, the emphasis is on the mathematical properties of the Einstein equations, in particular the local and global existence theorems of the initial value problem. But the inclusion of some introductory chapters on Lorentzian manifolds, special relativity and kinetic theory makes it an up-to-date reference work or even a textbook for an advanced graduate course.

For her contributions to mathematical physics, Bruhat has received many honours. In 1958 she was awarded the Silver Medal by the Centre National de la Recherche Scientifique. She was awarded the Henri de Parville Prize by the Academy of Sciences in 1963. She was elected a corresponding member of the Academy of Sciences on 13 March 1978 and, on 14 May 1979, she became a full member becoming the first woman full member. In 1985 she was elected to the American Academy of Arts and Sciences. She was made a Commander of the Légion d'Honneur on 22 October 1997. In 2003, jointly with James York, Cornell University, she received the Dannie Heineman Prize for Mathematical Physics:-

On 11 July 2008 she was elevated to the two 'dignities' in the Légion d'Honneur, namely 'Grand Officier' and 'Grand Croix', the highest level of honour.... for their separate as well as joint work in proving the existence and uniqueness of solutions to Einstein's gravitational field equations so as to improve numerical solution procedures with relevance to realistic physical solutions.

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

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