Yvonne Choquet-Bruhat

Yvonne Choquet-Bruhat (French: [ivɔn ʃɔkɛ bʁy.a] ^ⓘ; born 29 December 1923) is a French mathematician and physicist. She has made seminal contributions to the study of Einstein's general theory of relativity, by showing that the Einstein 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.^[1]

She was the first woman to be elected to the

From 1949 to 1951 she was a research assistant at the French National Centre for Scientific Research, as a result of which she received her doctorate.^[4]

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 Marseilles, precipitating her early departure from the Institute. In the same year, she published the local existence and uniqueness of solutions to the vacuum

In 1947, she married fellow mathematician Léonce Fourès. Their daughter Michelle is now (as of 2016) an

At the Universite Pierre et Marie Curie 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.^[6] 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.^[7]

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 Comité international de relativité générale et gravitation ("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

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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 (M, g, k) in which M is a three-dimensional
  smooth manifold
  , g is a smooth Riemannian metric on M, and k is a smooth (0,2)-tensor field on M.
• Given an initial data set (M, g, k), a development of (M, g, k) is a four-dimensional
  Lorentzian manifold (M, g) together with a smooth embedding f : M → M and a smooth unit normal vector field along f such that f ^*g = g and such that the second fundamental form
  of f, relative to the given normal vector field, is k.

• An initial data set (M, g, k) satisfies the vacuum constraint equations, or is said to be a vacuum initial data set, if the following two equations are satisfied:

${\displaystyle {\begin{aligned}R^{g}-|k|_{g}^{2}+(\operatorname {tr} ^{g}k)^{2}&=0\\\operatorname {div} ^{g}k-d(\operatorname {tr} ^{g}k)&=0.\end{aligned}}}$

Here R^g denotes the scalar curvature of g.

One of Choquet-Bruhat's seminal 1952 results states the following:

Every vacuum initial data set (M, g, k) has a development f : M → (M, g) such that g has zero Ricci curvature, and such that every inextendible timelike curve in the Lorentzian manifold (M, g) intersects f(M) exactly once.

Briefly, this can be summarized as saying that (M, g) is a vacuum spacetime for which f(M) is a Cauchy surface. Such a development is called a globally hyperbolic vacuum development. Choquet-Bruhat also proved a uniqueness theorem:

Given any two globally hyperbolic vacuum developments f₁ : M → (M₁, g₁) and f₂ : M → (M₂, g₂) of the same vacuum initial data set, there is an open subset U₁ of M₁ containing f₁(M) and an open subset U₂ of M₂ containing f₁(M), together with an isometry i : (U₁, g₁) → (U₂, g₂) such that i(f₁(p)) = f₂(p) for all p in M.

In a slightly imprecise form, this says: given any embedded spacelike hypersurface M of a Ricci-flat Lorentzian manifold M, the geometry of M near M is fully determined by the submanifold geometry of M.

In an article written with

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 (ℝ³, g, k) is a vacuum initial data set with g and k 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

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.
  MR53338
• Choquet-Bruhat, Yvonne; Geroch, Robert. Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14 (1969), 329–335.
  MR0250640

• 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.
  ISBN 0-444-86017-7
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile. Analysis, manifolds and physics. Part II. North-Holland Publishing Co., Amsterdam, 1989. xii+449 pp.
  ISBN 0-444-87071-7
• 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.
  ISBN 978-0-19-923072-3
• 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.
  ISBN 88-7088-223-3
• 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.
  ISBN 978-0-19-966645-4, 978-0-19-966646-1
• 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.

• 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.
  ISBN 978-981-3231-62-7

• Médaille d'Argent du Centre National de la Recherche Scientifique, 1958
• Prix Henri de Parville of the Académie des Sciences, 1963
• Member (since 1965), Comite International de Relativite Generale et Gravitation (President 1980-1983) [8]
• 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.^[9]

References

External links

Wikimedia Commons has media related to Yvonne Choquet-Bruhat.

• Contributions of 20th Century Women to Physics'
• "Yvonne Choquet-Bruhat", Biographies of Women Mathematicians, Agnes Scott College
• Christina Sormani; C. Denson Hill; Paweł Nurowski;
  ISSN 1088-9477
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