Équipe de Géométrie et Analyse globale séminaire, Groupe de travail Chirurgie et spineurs harmoniques -- L'invariant de Bauer-Furuta.
CRDM : Bibliothèque du Laboratoire de Mathématiques et de la Fédération.

Ateliers Math.en.Jeans. Présentation 2011 -- 2010 -- plaque de chocolat
sujets 2009, 2010, 2011,

Conseil scientifique de l' Insmi, place des femmes dans les mathématiques françaises, une présentation de Ch. Kassel à la réunion des DU du 29/03/2012.

SMF, Gazette des Mathématiciens

L'association SLR

Adresse electronique : colette.anne 'at' univ-nantes.fr
Adresse personnelle : 24, avenue de l'Eperonnière, 44000 Nantes.

" Proposition LXX. L'homme libre qui vit parmi les ignorants s'applique, autant qu'il peut, à éviter leurs bienfaits" Spinoza, L'éthique.

geometrie plane hyperboliqueEscher Disque de Poincaré

biographie de mathématiciennes
petite expo Jean Leray
Groupe de travail Chirurgie et spineurs harmoniques -- L'invariant de Bauer-Furuta

archives

Cristaux
Math_en_jeans 2011
Math_en_jeans 2010 -- plaque de chocolat
La lettre du president CPCN du 10/10/2007.
compte-rendu (en ps) de la table ronde sur "Les filles dans les écoles d'ingénieurs", tenue pendant le colloque "Femmes et Mathématiques" (9/10 Novembre 2001), publié dans la Gazette des Mathématiciens, Avril 2002.

(Pré)publications

• Partial collapsing and the spectrum of the Hodge Laplacian (Juillet 2010), arXiv:1007.2949[math.DG]

  en collaboration avec Junya Takahashi

  Résumé. Nous calculons la limite du spectre de l'opérateur de Hodge-Laplace sur les formes différentielles dans le cas d'éffondrement d'une partie d'une variété. Ce calcule généralise le travail précédent sur les sommes connexes puisque la sphère qui sert de joint entre la partie stable et celle effondrée est remplacée par une sous-variété quelconque. Ce résultat apporte un nouvel éclairage aux questions de blowing up conical singularities introduites par Mazzeo et Rowlett.

• P-spectrum and collapsing of connected sums, calculus of the limit (Juillet 2008-09), arXiv:0807.0760v3[math.DG] à paraître dans Trans. AMS.

  en collaboration avec Junya Takahashi

  Abstract The goal of the paper is to calculate the limit spectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.

• Gaps in the differential forms spectrum on cyclic coverings (Août 2007), arXiv:0708.3981, paru dans Math. Z. 262 n°1 (2009), 57-90.

  en collaboration avec Gilles Carron et Olaf Post

  Abstract We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $\Sigma$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-\Sigma$ is a fundamental domain of the covering. If the cohomology group $H^{n/2}(\Sigma)$ is trivial, we can construct for each $N \in \N$ a metric $g=g_N$ on $M$, such that the Hodge-de Rham operator on the covering $(X,g)$ has at least $N$ gaps in its (essential) spectrum. If $H^{n/2}(\Sigma) \ne 0$, the same statement holds true for the Hodge-de Rham operators on $p$-forms provided $p \notin \{n/2,n/2+1\}$.

• Une définition topologique du fibré de Maslov, (Janvier 2003), prépublication de Nantes 2003-09-01, version anglaise parue dans dans Cubo journal, Vol. 8, n.1, Avril 2006.

  L'indice de Maslov apparait comme le terme de phase lorsque l'on veut définir le symbole d'un Opérateur Intégral de Fourier (OIF). Ce symbole se voit alors comme une section du fibré de Maslov construit sur une sous variété Lagrangienne de $T^\ast X$. Dans son article historique [Acta,1971] Hormander propose une construction de ce fibré en termes de co-cycles et essaie de faire le lien avec une construction strictement topologique présentée par Arnol'd [Func.Ana,1967] à l'origine pour un appendice du livre de Maslov. Ce lien n'est établi que pour des sous-variétés Lagrangienne de $T^\ast \R^n$. Je propose ici une construction pour les sous-variétés Lagrangiennes de $T^\ast X$, X variété lisse.
  Texte en Postscript (comprimé).

• Bohr-Sommerfeld conditions for several commuting Hamiltonians, (Juillet 2002), paru dans Cubo journal, Vol. 6 n.2 (2004).

  en Collaboration avec Anne-Marie Charbonnel.

  Abstract : The goal of this paper is to find the quantization conditions of Bohr-Sommerfeld of several quantum Hamiltonians ${ Q_1(h), ...,Q_k(h)}$ acting on $ { {\R}^n}$, depending on a small parameter h, and which commute to each other. That is we determine, around a regular energy level $E_0\in \R^k$ the principal term of the asymptotics in $h$ of eigenvalues $\lambda_j(h),\, 1\leq j\leq k$ of the operators $Q_j(h)$ that are associated to a common eigenfunction. Thus we localize the so-called joint spectrum of the operators. Under the assumption that the classical Hamiltonian flow of the joint principal symbol $q_0$ is {\it periodic with constant periods on the one energy level} $q_0^{-1}(E_0)$, we prove that the part of the joint spectrum lying in a small neighbourhood of $E_0$ is localized near a lattice of size $h$ determined in terms of actions and Maslov indices. The multiplicity of the spectrum is also determined.
  Texte en Postscript (comprimé)

• Perturbation of several commuting h-pseudodifferential operators (Juin 2006).

  en Collaboration avec Anne-Marie Charbonnel.

  Abstract : We consider $k$ pseudodifferential operators $\Q{1},\ldots,\Q{k}$, acting on $\R^n$ commuting together, and depending on a small parameter $h$. Under the assumption that the classical Hamiltonian flow of the joint principal symbol $q_0$ is periodic with constant period on one given energy level $q_0^{-1}(E_0)$, we have shown, in Bohr-Sommerfeld conditions for several commuting Hamiltonians, that the joint spectrum of these operators lying in a $h$-depending \nbd $I(h)$ of $E_0$ is localized near a lattice. This paper followed the now classical method initiated by Helffer and Robert, and then by Charbonnel for several operators, which is based on the FIO theory. We investigate in this present article the geometric method proposed by Colin de Verdi\`ere to Dozias for one operator acting on $\R^n$ and obtain a theorem of perturbation.
  Texte en Postscript (comprimé).

• A shift between Dirichlet and Neumann Spectrum for generalized linear elasticity (Février 1997). Paru in Asymptotic Analysis 19 (1999) 297-316.

Abstract : defining the equation of linear elasticity on a general riemannian manifold with boundary, we prove a formula relating the counting functions of the Neumann and the Dirichlet problem to the counting function of the Dirichlet2Neumann operator. Namely the difference of the two counting functions at a equals the number of negative eigenvalues of the Dirichlet to Neumann operator related to the resolvant at a. With this formula we can show that this difference is always bigger than one in the homogeneous case (i.e. when the Lamé functions are constant) for bounded domains of symmetric spaces of non compact type and rank bigger than 2, for instance the euclidean space, and for rank 1 if the dimension of the nilpotent part is less than a constant depending on the Lamé coefficients.
Texte en Postscript (comprimé).

• Bornes sur la multiplicité (Mars 1992).


Résumé : en dimension 2, à l'inverse des dimensions plus grandes, la multiplicité d'une valeur propre d'un opérateur de Schroedinger est bornée par la topologie de la surface. La meilleure borne connue à ce jour est due à Nadirashvili, elle n'est en général pas optimale. Ce résultat nécessite l'utilisation de théorèmes importants de l'analyse comme le théorème de Courant,et aussi le théorème de Cheng que l'on peut considérer comme un théorème sur la "détermination finie" ; ce théorème qui devrait permettre d'étudier les lieux d'annulation d'une fonction propre n'est véritablement efficace qu'en dimension deux. On a aussi besoin d'un résultat topologique concernant un graphe plongé dans une surface, il s'énonce simplement avec la proposition 2.2.
Bruno Sévennec a par la suite apporté une légère amélioration aux résultats présentés ici.
Texte en Postscript (comprimé).

• Correction a la publication des Annales Scient. de l'ENS (Lettre, Avril 1999).

Texte en Postscript.

• Anné, Colette : A note on the generalized Dumbbell problem.

Proc. Am. Math. Soc. 123, No.8, 2595-2599 (1995).
Langue: English
The author has obtained some interesting results for the calculation of the asymptotics of the small eigenvalues and corresponding eigenfunctions for the Laplace operator with Neumann boundary conditions on a domain that is obtained by adding several thin channels between given bounded domains. Rassias (Athens) from ZentralBlatt.
Mots clés: Dumbbell problem; Hilbert space; quadratic form; spectrum; symmetric matrix; asymptotics; small eigenvalues; eigenfunctions; Laplace operator
Class. math.: 58G18
Texte en Postscript comprimé.

• Anné, Colette ; Colbois, Bruno : Spectre du Laplacien agissant sur les p-formes différentielles et écrasement d'anses. (Spectrum of the Laplacian acting on the differential p-forms and down breaking handles).

Math. Ann. 303, No.3, 545-573 (1995).
Langue: French
Class. math.: 53C20 Riemannian manifolds (global), 58G25 Spectral problems of PDE on manifolds, etc.
Texte en Postscript comprimé.
• Anné, Colette : Laplaciens en interaction. (Laplacians in interaction).

Manuscr. Math. 83, No.1, 59-74 (1994).
Langue: French
The author studies the asymptotics of the eigenfunctions of the Laplacian on a compact Riemannian manifold in the situation of adding handles. It is shown that resonance between the two limit manifolds can only occur when the difference of their dimension is one or two. J. Marschall (Feldafing) from ZentralBlatt.
Mots clés: asymptotics; eigenfunctions; Laplacian; Riemannian manifold; adding handles
Class. math.: 58G18, 58G25. Texte en Postscript comprimé.
• Anné, Colette; Colbois, Bruno : Operateur de Hodge-Laplace sur des varietes compactes privees d'un nombre fini de boules. (Hodge-Laplace operator on compact manifolds deprived of a finite number of balls.).

J. Funct. Anal. 115, No.1, 190-211 (1993).
Langue: French
For a compact, orientable, connected Riemannian manifold M with boundary bd(M), the Laplace-Hodge operator (or rather the square root D) is considered. Estimates for the boundary behaviour of forms are applied to study the situation where M has been perturbed to M(t)=M- (N balls of radius t) ; t less than the radius of injectivity of M. In particular the aim is to compare spectra and eigenforms for D on M and M(t) and the possible convergence of these quantities as t tends to 0.
Various types of topologically distinct boundary conditions, all guaranteeing the ellipticity and self-adjointness of the operator D, are considered. The Friedrichs' inequality, required by the ellipticity of the boundary value problem, involves a constant C which depends critically on the curvature of bd(M) (in fact one might have C to infinity). As a preliminary for the rest of the work this dependence is controlled in a very precise way. Then, and in a quite technical way, it is proven that in fact eigenvalues and eigenspaces of the Laplace-Hodge operator on M(t) do converge to those on M as t tends to 0, with the -- on topological grounds -- expected exception of forms of degree (dim(M) - 1). The latter simply becoming singular in a neighbourhood of bd(M) as t tends to 0. S.I. Andersson (Goeteborg) from ZentralBlatt.
Mots clés: convergence; harmonic forms; Riemannian manifold; Laplace-Hodge operator; eigenforms; eigenvalues
Class. math.: 58C40, 35P05, 58G20. Texte en Postscript comprimé.
• Anné, Colette : Perturbation du Laplacien de Hodge par excision de petites boules. (Perturbation of the Hodge-Laplace operator by deletion of small balls).

Semin. Theor. Spectrale Geom., Chambery-Grenoble 10, Annee 1991-1992, 85-92 (1992).
Langue: French
Let (M,g) be a connected, compact, oriented Riemannian manifold of dimension bigger than 3 and t a positive real number more less than the radius of injectivity of M. Let M(t) be the manifold obtained from $M$ by deletion of a finite number of balls of radius t. The author considers the Hodge-Laplace operator on M(t) and establishes some results about the convergence of its spectrum.
This note is an advance of a paper of the author written jointly with {\it B. Colbois} [``Operateur de Hodge-Laplace sur des varietes compactes privees d'un nombre fini de boules'', Preprint]. E. Outerelo (Madrid).
Mots clés: Hodge-Laplace operator; Riemannian manifold; harmonic forms
Class. math.: 58G25.
• Anné, Colette : Majoration de multiplicité pour l'opérateur de Schrödinger. (Upper bound for the multiplicity for the Schrödinger operator).

Semin. Theor. Spectrale Geom., Chambery-Grenoble 8, Annee 1989-1990, 53-62 (1990).
Langue: French
Mots clés: multiplicity for the Schroedinger operator.
Class. math.: 47F05, 35J10.
Texte en pdf
• Anné, Colette : Fonctions propres sur des varietes avec des anses fines, application a la multiplicite. (Eigenfunctions on manifolds with thin handles, the multiplicity mapping).

Commun. Partial Differ. Equations 15, No.11, 1617-1630 (1990).
Langue: French
The author proves some results on the behaviour of the eigenfunctions of the Laplace operator under a singular perturbation obtained by adding a thin handle to a compact manifold. D.Robert (Nantes) From ZentralBlatt.
Mots clés: multiplicity; adding a thin handle
Class. math.: 35P15, 35J05, 58G03.
• Anné, Colette : Fonctions propres sur des varietes avec des anses fines, application a la multiplicite. (Eigenfunctions on manifolds with thin handles and applications to the multiplicity).

Semin. Theor. Spectrale Geom. 7, Annee 1988-1989, 123-133 (1989).
Langue: French
This paper gives an answer to the following perturbation problem in spectral geometry: how do the spectral data of a Riemann 2-manifold react when a thin (in metric sense) handle is attached? Is there any control upon this reaction in terms of the perturbation data?
To describe the main result consider M the original Riemann 2-manifold, D its Laplacian. Let M(t(p,q)$ be the new Riemann manifold with a handle of thickness t attached at the points p,q of M and denote by D(t) its Laplacian. Suppose a is an eigenvalue of order m of D and E is the corresponding eigenspace.
The main result is the following: if either f(p)=f(q)for all f in E or f(p)=-f(q) f in E then for t sufficiently small we can glue cleverly a t-handle at p,q such that a is an eigenvalue of order m+1 of D(t).
L. Nicolaescu
Mots clés: multiplicity; perturbation; spectral geometry; Riemann 2-manifold; eigenspace
Class. math.: 58G25.
• Anné, Colette : Principe de Dirichlet pour les formes differentielles. (Dirichlet principle for differential forms).

Bull. Soc. Math. Fr. 117, No.4, 445-450 (1989).
Langue: French
Let (M,g) be a Riemannian manifold with a boundary N. The study of tensor fields on M with some properties on N is an important problem in differential geometry. The aim of this paper is to prove the following result: Let (M,g) be a Riemannian manifold with boundary N, if w is a harmonic p-form on M with the property w=0 on N, then w is zero on the whole manifold. G. Tsagas
Mots clés: Riemannian manifold with boundary; harmonic p-form
Class. math.: 53C20, 58E20.
• Anné, Colette : Spectre du Laplacien et ecrasement d'anses. (Spectrum of the Laplacian and down breaking handles).

Ann. Sci. Ec. Norm. Super., IV. Ser. 20, 271-280 (1987).
Langue: French
Abstract. We show, by the study of the convergence of the spectrum of the Laplacian in the case of a manifold X with handles breaking down to a manifold of lower dimension Y with boundary in X, that the suitable limiting operator is the Laplacian with Dirichlet boundary conditions on Y.
Mots clés: spectrum of the Laplacian; Dirichlet boundary conditions
Class. math.: 58G25.

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