Yvonne Choquet-Bruhat

Yvonne Choquet-Bruhat (Template:IPA; 29 December 1923 – 11 February 2025) was a French mathematician and physicist. She made seminal contributions to the study of general relativity, by showing that the Einstein field equations can be put into the form of an initial value problem which is well-posed. In 2015, her breakthrough paper was listed by the journal Classical and Quantum Gravity as one of thirteen 'milestone' results in the study of general relativity, across the hundred years in which it had been studied.

Choquet-Bruhat was the first woman to be elected to the French Academy of Sciences and was a Grand Officer of the Legion of Honour.

Early life

Yvonne Bruhat was born in Lille on 29 December 1923. Her mother was the philosophy professor Berthe Hubert and her father was the physicist Georges Bruhat, who died in 1945 in the concentration camp Oranienburg-Sachsenhausen. Her brother François Bruhat also became a mathematician, making notable contributions to the study of algebraic groups.

Bruhat undertook her secondary school education in Paris earning her Baccalauréat in 1941. She received the prestigious Concours Général national competition, winning the silver medal for physics. From 1943 to 1946 she studied at the École normale supérieure de jeunes filles in Paris, and from 1946 was a teaching assistant there and undertook research advised by André Lichnerowicz.

Career

From 1949 to 1951 Bruhat was a research assistant at the French National Centre for Scientific Research, as a result of which she received her doctorate.

In 1951, she became a postdoctoral researcher at the Institute for Advanced Study in Princeton, New Jersey. Her supervisor, Jean Leray, suggested that she study the dynamics of the Einstein field equations. He also introduced her to Albert Einstein, whom she consulted with a few times further during her time at the institute.

In 1952, Bruhat and her husband were both offered jobs at Marseille, precipitating her early departure from the institute. In the same year, she published the local existence and uniqueness of solutions to the vacuum Einstein equations, her most renowned achievement. Her work proves the well-posedness of the Einstein equations, and started the study of dynamics in general relativity.

In 1958, she was awarded the CNRS Silver Medal. From 1958 to 1959 she taught at the University of Reims. In 1960 she became a professor at the Université Pierre-et-Marie-Curie (UPMC) in Paris, and remained professor or professor emeritus until her retirement in 1992.

Choquet-Bruhat with Gustave Choquet at the University of California, Berkeley, in 1974.

At the Template:Lang she continued to make significant contributions to mathematical physics, notably in general relativity, supergravity, and the non-Abelian gauge theories of the standard model. Her work in 1981 with Demetrios Christodoulou showed the existence of global solutions of the Yang–Mills, Higgs, and spinor field equations in 3+1 Dimensions. Additionally in 1984 she made perhaps the first study by a mathematician of supergravity with results that can be extended to the currently important model in D=11 dimensions.

In 1978 Yvonne Choquet-Bruhat was elected a correspondent to the Academy of Sciences and on 14 May 1979 became the first woman to be elected a full member. From 1980 to 1983 she was President of the Template:Lang ("International committee on general relativity and gravitation"). In 1985 she was elected to the American Academy of Arts and Sciences. In 1986 she was chosen to deliver the prestigious Noether Lecture by the Association for Women in Mathematics.

Technical research contributions

Choquet-Bruhat's best-known research deals with the mathematical nature of the initial data formulation of general relativity. A summary of results can be phrased purely in terms of standard differential geometric objects.

• An initial data set is a triplet Template:Math in which Template:Mvar is a three-dimensional smooth manifold, Template:Mvar is a smooth Riemannian metric on Template:Mvar, and Template:Mvar is a smooth (0,2)-tensor field on Template:Mvar.
• Given an initial data set Template:Math, a development of Template:Math is a four-dimensional Lorentzian manifold Template:Math together with a smooth embedding Template:Math and a smooth unit normal vector field along Template:Mvar such that Template:Math and such that the second fundamental form of Template:Mvar, relative to the given normal vector field, is Template:Mvar.

In this sense, an initial data set can be viewed as the prescription of the submanifold geometry of an embedded spacelike hypersurface in a Lorentzian manifold.

• An initial data set Template:Math satisfies the vacuum constraint equations, or is said to be a vacuum initial data set, if the following two equations are satisfied:

$\begin{array}{cc}{R}^g-|k{|}_g^2+({\mathrm{tr}}^{g}k{)}^2& =0\\ {\mathrm{div}}^{g}k-d({\mathrm{tr}}^{g}k)& =0.\end{array}$

Here Template:Math denotes the scalar curvature of Template:Mvar.

One of Choquet-Bruhat's seminal 1952 results states the following: Template:Blockquote Briefly, this can be summarized as saying that Template:Math is a vacuum spacetime for which Template:Math is a Cauchy surface. Such a development is called a globally hyperbolic vacuum development. Choquet-Bruhat also proved a uniqueness theorem: Template:Blockquote In a slightly imprecise form, this says: given any embedded spacelike hypersurface Template:Mvar of a Ricci-flat Lorentzian manifold Template:Math, the geometry of Template:Math near Template:Mvar is fully determined by the submanifold geometry of Template:Mvar.

In an article written with Robert Geroch in 1969, Choquet-Bruhat fully clarified the nature of uniqueness. With a two-page argument in point-set topology using Zorn's lemma, they showed that Choquet-Bruhat's above existence and uniqueness theorems automatically imply a global uniqueness theorem: Template:Blockquote It is now common to study such developments. For instance, the well-known theorem of Demetrios Christodoulou and Sergiu Klainerman on stability of Minkowski space asserts that if Template:Math is a vacuum initial data set with Template:Mvar and Template:Mvar sufficiently close to zero (in a certain precise form), then its maximal globally hyperbolic vacuum development is geodesically complete and geometrically close to Minkowski space.

Choquet-Bruhat's proof makes use of a clever choice of coordinates, the wave coordinates (which are the Lorentzian equivalent to the harmonic coordinates), in which the Einstein equations become a system of hyperbolic partial differential equations, for which well-posedness results can be applied.

Personal life and death

Choquet-Bruhat in 2006

In 1947, she married fellow mathematician Léonce Fourès. Their daughter Michelle is (as of 2016) an ecologist. Her doctoral work and early research is under the name Yvonne Fourès-Bruhat. In 1960, Bruhat and Fourès divorced, she later married the mathematician Gustave Choquet and changed her last name to Choquet-Bruhat. She and Choquet had two children; her son, Daniel Choquet, is a neuroscientist and her daughter, Geneviève, is a medical doctor.

Choquet-Bruhat died on 11 February 2025 in Merignac, France (33700), at the age of 101.

Major publications

Articles

• Fourès-Bruhat, Y. Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires. Acta Math. 88 (1952), 141–225. Template:Doi Template:Bibcode Template:Zbl Template:MR
• Choquet-Bruhat, Yvonne; Geroch, Robert. Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14 (1969), 329–335. doi:10.1007/BF01645389 Template:MR

Survey articles

• Bruhat, Yvonne. The Cauchy problem. Gravitation: An introduction to current research, pp. 130–168, Wiley, New York, 1962.
• Choquet-Bruhat, Yvonne; York, James W. Jr. The Cauchy problem. General relativity and gravitation, Vol. 1, pp. 99–172, Plenum, New York-London, 1980.
• Choquet-Bruhat, Yvonne. Positive-energy theorems. Relativity, groups and topology, II (Les Houches, 1983), 739–785, North-Holland, Amsterdam, 1984.
• Choquet-Bruhat, Yvonne. Results and open problems in mathematical general relativity. Milan J. Math. 75 (2007), 273–289.
• Choquet-Bruhat, Yvonne. Beginnings of the Cauchy problem for Einstein's field equations. Surveys in differential geometry 2015. One hundred years of general relativity, 1–16, Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

Technical books

• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile; Dillard-Bleick, Margaret. Analysis, manifolds and physics. Second edition. North-Holland Publishing Co., Amsterdam-New York, 1982. xx+630 pp. Template:ISBN
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile. Analysis, manifolds and physics. Part II. North-Holland Publishing Co., Amsterdam, 1989. xii+449 pp. Template:ISBN
• Choquet-Bruhat, Y. Distributions. (French) Théorie et problèmes. Masson et Cie, Éditeurs, Paris, 1973. x+232 pp.
• Choquet-Bruhat, Yvonne. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. Template:ISBN
• Choquet-Bruhat, Y. Géométrie différentielle et systèmes extérieurs. Préface de A. Lichnerowicz. Monographies Universitaires de Mathématiques, No. 28 Dunod, Paris 1968 xvii+328 pp.
• Choquet-Bruhat, Yvonne. Graded bundles and supermanifolds. Monographs and Textbooks in Physical Science. Lecture Notes, 12. Bibliopolis, Naples, 1989. xii+94 pp. Template:ISBN
• Choquet-Bruhat, Yvonne. Introduction to general relativity, black holes, and cosmology. With a foreword by Thibault Damour. Oxford University Press, Oxford, 2015. xx+279 pp. Template:ISBN
• Choquet-Bruhat, Y. Problems and solutions in mathematical physics. Translated from the French by C. Peltzer. Translation editor, J.J. Brandstatter Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1967 x+315 pp.

Popular book

• Choquet-Bruhat, Yvonne. A lady mathematician in this strange universe: memoirs. Translated from the 2016 French original. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. x+351 pp. Template:ISBN

Awards

• Médaille d'Argent du Centre National de la Recherche Scientifique, 1958
• Prix Henri de Parville of the Académie des Sciences, 1963
• Member (since 1965), Template:Lang (President 1980–1983)
• Member, Académie des Sciences, Paris (elected 1979)
• Elected to the American Academy of Arts and Sciences 1985
• Association for Women in Mathematics Noether Lecturer, 1986
• Commandeur de la Légion d'honneur, 1997
• Dannie Heineman Prize for Mathematical Physics, 2003
• She was elevated to the 'Grand Officier' and 'Grand Croix' dignities in the Légion d'Honneur in 2008.

External links

• Contributions of 20th Century Women to Physics'
• "Yvonne Choquet-Bruhat", Biographies of Women Mathematicians, Agnes Scott College
• Template:Cite journal
• English translation of 1952 paper, Existence theorem for certain systems of nonlinear partial differential equations

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