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Michèle Vergne

Bio: Michèle Vergne is an academic researcher from University of Paris. The author has contributed to research in topics: Equivariant map & Polytope. The author has an hindex of 38, co-authored 157 publications receiving 6866 citations. Previous affiliations of Michèle Vergne include École Polytechnique & Pierre-and-Marie-Curie University.


Papers
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Book
01 Apr 1992
TL;DR: In this article, the authors present a formal solution for the trace of the heat kernel on Euclidean space, and show that the trace can be used to construct a heat kernel of an equivariant vector bundle.
Abstract: 1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.

2,112 citations

Journal ArticleDOI
TL;DR: In this article, the problem of decomposing the tensor products of the harmonic representations into irreducible components to get a series of new unitary irreduceible representations with highest weight vectors of the group G = Mp(n), two-sheeted covering group of the symplectic group, or G = U(p, q).
Abstract: In this paper, we give the answer to the following two intimately related problems. (a) To decompose the tensor products of the harmonic representations into irreducible components to get a series of new unitary irreducible representations with highest weight vectors of the group G= Mp(n), two-sheeted covering group of the symplectic group, or G = U(p, q). (b) To describe the representations of the group GL(n, ~ ) x O(k, ~) (resp. GL(p, C) x GL(q, C) x GL(k, C)) in the space of pluriharmonic polynomials on the space M(n, k; C) of n • k complex matrices (resp. M(p, k; C) x M~q, k; ~)). The second problem arises when we construct an intertwining operator from the tensor product of the harmonic representation into a space of vector-valued holomorphic functions on the associated hermitian symmetric space G/K, or equivalently when we consider highest weight vectors in the tensor products. Some of our motivations are the following: 1) Apart from special cases the unitary dual G of a real semi-simple Lie group is not known. There exist isolated points in G which are not members of discrete or "mock-discrete" series, (for exemple for Sp(2, C) where the unitary dual has been computed by M. Duflo [-18], there are two isolated points in ~, the trivial representation and the odd component of the Segal-Shale-Weil representation) and we are interested to produce series of such representations. 2) We are extending to matrix spaces classical results for harmonic polynomials on N".

468 citations

Journal ArticleDOI
TL;DR: In this article, the conditions générales d'utilisation (http://www.numdam.org/legal.html) implique l'accord avec les conditions generales d’utilisation, i.e., usage commerciale ou impression systématique, constitutive of an infraction pénale.
Abstract: © Bulletin de la S. M. F., 1970, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

399 citations

Journal ArticleDOI
TL;DR: In this paper, a formule explicite (théorème 1-6) based on a méthode inspired by R. Bott and R. Riemans is presented.
Abstract: Introduction, Soit M une variété différentiable, munie d'une action d'un groupe de Lie T. Soit t l'algèbre de Lie de T. Dans [2] nous avons introduit des espaces de cohomologie équivariante H(X), pour X E t, pour lesquels on a une propriété de localisation : supposons M et T compacts ; si μ E H(X) alors fM μ ne dépend que de la restriction de μ à la variété des points fixes du groupe à un paramètre exp tX . Lorsque ces points fixes sont non dégénérés, (donc en nombre fini), nous obtenons une formule explicite (théorème 1-6) par une méthode inspirée de R. Bott [3] : en dehors des points fixes, la forme μ est une forme exacte dv, et on calcule fM μ par la formule de Stokes. (Le cas où la variété des points fixes est de dimension > 0 sera traité dans un autre article) . Lorsque M est une variété symplectique, et que l'action de T est hamiltonienne, un élément remarquable de H(X) est fourni par la forme (non homogéne) JX (o/2i ir), où JX est le moment de X et Q la 2-forme symplectique . Dans [2] nous retrouvions par cette méthode la formule de DuistermaatHeckman [4] pour IM e ~X (a n/ n ! ). Lorsque M est une orbite de la représentation coadjointe du groupe de Lie T, munie de sa structure symplectique canonique, on a J(m) _ pour m E M C t* . Si T est simplement connexe et si M est entière, il y a sur M un f ibré en droites canonique, muni d'une connexion canonique, et la forme o coïncide avec la courbure de cette connexion [6] . Revenons au cas d'une variété M quelconque, munie d'une action à gauche d'un groupe de Lie T. Soit P un f ibré principal de base M, de fibre un groupe de Lie G. Supposons que T opère sur P en préservant une connexion w . Notons l la courbure de w et X * le champ de vecteurs sur P associé à X . Nous définissons une application \"moment\" J : P -> t* ® g par JX(p) = w(X*)(p), pour X dans t et p dans P. Nous considerons la forme (non homogène) sur P, à valeurs dans d'algebre de Lie de G, JX (S2/2ir). A l'aide de cette forme, nous construisons des éléments de H(X) en suivant la méthode de Chern-Weil pour les classes caractéristiques ordinaires (ceci fait l'objet du chapitre 2) . Ces classes caractéristiques équivariantes permettent de transformer des formules de points fixes en intégrales sur M de formes dans H(X). Nous appliquons ceci à une généralisation du théorème 2 de [3], et à la formule de Riemann-Roch .

240 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, and that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the class of Poisson structures on X modulo diffeomorphisms.
Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

2,672 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, which can be interpreted as correlators in topological open string theory.
Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

2,223 citations

Book
01 Jan 1996
TL;DR: A review of the collected works of John Tate can be found in this paper, where the authors present two volumes of the Abel Prize for number theory, Parts I, II, edited by Barry Mazur and Jean-Pierre Serre.
Abstract: This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

2,014 citations

Journal ArticleDOI
TL;DR: In this paper, a mathematical study of the differentiable deformations of the algebras associated with phase space is presented, and deformations invariant under any Lie algebra of "distinguished observables" are studied.

1,564 citations

Journal ArticleDOI
TL;DR: The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action as discussed by the authors, and the heat kernel coefficients are given in terms of several geometric invariants.

1,150 citations