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Bertrand Monthubert

Activité de recherche

Domaines d'intérêt :

Géométrie non commutative, théorie de l'indice, groupoïdes, algèbres d'opérateurs

Publications :

puce B.M. et Victor Nistor, A Topological Index Theorem
for manifolds with corners
, à paraître dans Compositio Mathematica
    Abstract: We define an analytic index and prove a topological index theorem for a non-compact manifold M_0 with poly-cylindrical ends. We prove that an elliptic operator P on M_0 has an invertible perturbation P+R by a lower order operator if and only if its analytic index vanishes. As an application, we determine the K-theory groups of groupoid C*-algebras of manifolds with corners.
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puce B.M. et Victor Nistor, The $K$-groups and the index theory of certain comparison $C^*$-algebras, Noncommutative Geometry and Global Analysis, Contemporary Mathematics 2011, 213-224
    Abstract: We compute the $K$-theory of comparison $C^*$-algebra associated to a manifold with corners. These comparison algebras are an example of the abstract pseudodifferential algebras introduced by Connes and Moscovici \cite{M3}. Our calculation is obtained by showing that the comparison algebras are a homomorphic image of a groupoid $C^*$-algebra. We then prove an index theorem with values in the $K$-theory groups of the comparison algebra.
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puce

J. Aastrup, S. T. Melo, B. M. & E. Schrohe, Boutet de Monvel’s Calculus and Groupoids I, Journal of noncommutative geometry Volume 4, Issue 3, 2010, pp. 313–329

    Abstract: Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra of pseudodifferential operators on some Lie groupoid G? If it could, the kernel of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra of G. While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal isomorphic to the kernel of the principal symbol homomorphism on Boutet de Monvel's algebra.
Preprint disponible sur http://fr.arxiv.org/abs/math.KT/0611336
puce P. Carrillo-Rouse et B. M., An index theorem for manifolds with boundary, C. R. Acad. Sci. Paris, Ser. I 347 (2009)
    Abstract:In Connes (Non Commutative Geometry, 1994, II.5), a proof is given of the Atiyah–Singer index theorem for closed manifolds by using deformation groupoids and appropriate actions of these on RN. Following these ideas, we prove an index theorem for manifolds with boundary.
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puce B.M., Contribution of Noncommutative Geometry to Index
Theory on singular manifolds
, Geometry and topology of manifolds, 221–237, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007
Format PDF 331 KB
puce B.M., Contribution de la Géométrie Non-Commutative à la théorie de l'indice sur les variétés singulières, manuscrit d'Habilitation à diriger des recherches (décembre 2005) Format PDF 311 KB
puce Robert Lauter, B.M. et Victor Nistor, Spectral invariance for certain algebras of pseudodifferential operators, J. de l'Inst. Math. Jussieu 4 (2005), Issue 03, 405-442
    Abstract: We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.
preprint disponible sur 
http://arXiv.org/abs/math/0112091 (format DVI zippé , 64 kB, ou PostScript zippé , 166 kB)
puce
Robert Lauter, B.M. et Victor Nistor, Invariance spectrale des algèbres d'opérateurs pseudodifférentiels, C. R. Math. Acad. Sci. Paris 334 (2002), no. 12, 1095--1099
DVI format , 31 kB, ou PostScript format , 118 kB
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B.M., Groupoids and pseudodifferential calculus on manifolds with corners, Journal of Functional Analysis 199(2003), 243-286
    Abstract: We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calh we define a pseudodifferential calh we define a pseudodifferential calculus. This calculus generalizes the $b$-calculus of R. Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension.
    Using tools from the theory of $C^*$-algebras of groupoids, we also obtain new proofs for the study of $b$-calculus.
format DVI zippé , 267 kB, ou PostScript zippé , 407 kB
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Robert Lauter, B.M. et Victor Nistor, Pseudodifferential analysis on continuous family groupoids, Documenta Math. 5 (2000) 625-655
    Abstract: We study properties and representations of the convolution algebra and the algebra of pseudodifferential operators associated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodifferential operators of order zero completely reduces to the study of the representations of the ideal of regularizing operators. This recovers the usual boundedness theorems for pseudodifferential operators of order zero. We prove a structure theorem for the normure theorem for the norm completions of these algebras associated to groupoids with invariant filtrations. As a consequence, we obtain criteria for certain pseudodifferential operators on certain non-compact manifolds to be compact or Fredholm. We end with a discussion of the significance of these results to the index theory of operators on certain singular spaces. For example, we give a new approach to the question of the existence of spectral sections for operators on coverings of manifolds with boundary. We expect that our results will also play a role in the analysis on more general singular space
format DVI , 129 kB, ou PostScript , 372 kB
puce B.M. et Pierre-Yves Le Gall, K-theory of the indicial algebra of a manifold with corners, K-Theory 23 (2001), no. 2, 105--113
    Abstract: We compute the K-theory groups of the C*-algebra of the groupoid of a manifold with corners, in which the analytic index takes its values.
format DVI , 49 kB, ou PostScript , 244 kB
puce B.M., Groupoids of manifolds with corners and index theory , dans Groupoids in analysis, geometry, and physics (Boulder, CO, 1999), 147--157, Contemp. Math., 282, Amer. Math. Soc., Providence, RI, 2001.
    Abstract: This article is a survey on the relations between pseudodifferential calculus on manifolds with corners and groupoids, based on a contribution to the conference ``Groupoids in Geometry, Analysis and Physics'' held in Boulder, USA in 1999.
format DVI, 41 kB, ou PostScript, 206 kB
puce B.M. : Thèse de doctorat : Groupoïdes et calcul pseudo-différentiel sur les variétés à coins, soutenue le 16 janvier 1998 format dvi.gz , 96 kB, ou ps.gz , 589 kB
puce B.M. : Pseudodifferential calculus on  manifolds with corners and groupoids, Proceedings of the AMS : 127 (1999), no.~10, 2871--2881
    Résumé : Nous construisons un groupoïde différentiable longitudinalement lisse associé à une variété à coins. Le calcul pseudo-différentiel sur ce groupo&iifférentiel sur ce groupoïde coïncide avec le calcul pseudo-différentiel de Melrose (aussi appel&eacutl de Melrose (aussi appelé $b$-calculus). Nous définissons également une algèbre de fonctions à décroissance rapide sur ce groupoïde ; elle contient les noyaux des opérateurs régularisants du (petit) $b$-calculus.

format DVI , 50 kB, ou PostScript , 146 kB
puce B.M. et François Pierrot : Indice analytique et Groupoïdes de Lie  (C.R. Acad. Sci. Paris, Sér. I, 325, No. 2, 193-198 (1997)
    Résumé : Soit $G$ un groupoïde de Lie. Le calcul pseudodifférentiel sur $G$ définit l'indice analytique. Le groupoïde tangent associé à $G$ induit un morphisme de $K$-théorie dont nous prouvons qu'il coïncide avec l'indice analytique. Ce dernier peut donc se définir sans recours au calcul pseudodifférentiel.
format DVI , 29 kB, ou PostScript , 106 kB
puce mon Mémoire de Magistère (format PostScript, 635 kB)
puce
Pendant quelques années, j'ai organisé un séminaire de Philosophie et Sciences à l'Ecole Normale Supérieure , à Paris. Un de nos sujets de réflexion a été le problème du déterminisme. Un peu plus d'information est disponible ...
puce
Et si vous aimez la littérature autant que les maths, vous apprécierez peut-être ces quelques mots ...