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Obtaining a rigorous justification of the kinetic theory of gases from Newton's second law of dynamics for a large number of identical spheres interacting by elastic binary collisions, is a problem formulated by Hilbert in 1900 (Hilbert's 6th problem). In 1975, Lanford demonstrated the validity of the Boltzmann equation over a very short time interval, of the order of a fraction of the average time between two successive collisions experienced by the same particle. This result of Lanford can be interpreted as a kind of law of large numbers as the number of particles tends to infinity. This point of view raises several questions. First, the core of the argument used by Boltzmann to arrive at the equation bearing his name is the assumption that two particles about to collide are statistically almost independent. This suggests to examine the validity of this hypothesis by studying the dynamics of correlations between particles. On the other hand, the interpretation of the Boltzmann equation as a law of large numbers leads to study precisely the fluctuations of the phase space empirical measure of the particle system about its mean (whose evolution is described by the Boltzmann equation). A series of recent papers by T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella answers these various questions and allows going beyond the Boltzmann equation in the understanding of the kinetic theory of gases., Comment: 53 pages, in French language, 3 figures. Bourbaki seminar, November 19th 2022

Quantum resource theories allow us to quantify a useful quantum phenomenon, to develop new protocols for its detection and determine the exact processes that maximize its use for practical tasks. These theories aim at transforming physical phenomena, such as entanglement and quantum coherence, into useful properties for the execution of concrete tasks related to quantum information. In this thesis, we focus on the resource theories of entanglement, discord-like quantum correlations, and quantum coherence, the most intriguing quantum phenomena exploited so far in quantum information theory. We begin by presenting in detail the theoretical tools of these quantum resources, focusing on the most remarkable techniques and computational problems. In this sense, we discuss several mathematical methods that solve some problems related to their quantifications, and some analytical results for bipartite quantum systems are given. We also examine the intrinsic connections between these quantum resources by extracting the links that unite the corresponding measures. In contrast, the revolution of quantum technology has led to a growing interest in quantum metrology, and quantum entanglement has been employed to overcome the classical limit in several quantum estimation protocols. In this work, we analyze the role of quantum correlations beyond entanglement in improving the accuracy of an unknown parameter. According to our results, correlations can be captured using quantum Fisher information, and quantum discord correlations can be exploited to ensure the accuracy of phase estimation protocols. This thesis includes also the contributions on the dynamics of these quantum resources in various models of open quantum systems. Among our objectives, is to study the effects of the environment on these quantum resources and to obtain techniques to protect them from the effects of intrinsic decoherence., Comment: PhD Dissertation, in French language. Successfully defended on September 11, 2021

This report (written in French) is devoted to studying special functions the most used in physics. Special functions are a very broad branch of mathematics, theoretical physics, and mathematical physics. They appeared in the nineteenth century as solutions of equations in mathematical physics, particularly partial differential equations of order two and four. Their knowledge is essential for the proper handling and understanding of current problems in physics. They are also related to the art of scientific computing in physics and mathematics. Special functions are included in many computer algebra software such as Matlab, Mathematica, and Maple, and students are strongly encouraged to take part in this development which has become indispensable for the treatment of almost all current problems in physics. The manuscript contains six chapters: gamma and beta functions, Bessel functions, Fresnel error, and integral function, exponential integral, sine integral, cosine integral, and logarithm integral, orthogonal polynomials, and finally hypergeometric functions., Comment: in French

Metastability appears when a thermodynamic system, such as supercooled water (which is liquid below freezing temperature), lands on the "wrong" side of a phase transition, and remains for a very long time in a state different from its equilibrium state. There exist numerous mathematical models describing this phenomenon, including lattice models with stochastic dynamics. In this text, we will be interested in metastability in parabolic stochastic partial differential equations (SPDEs). Some of these equations are ill-posed, and only thanks to very recent progress in the theory of so-called singular SPDEs does one how to construct solutions, via a renormalisation procedure. The study of metastability in these systems reveals unexpected links with the theory of spectral determinants, including Fredholm and Carleman--Fredholm determinants., Comment: Popularization, in French. 16 pages, 6 figures

Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose another open set $\omega\subset \Omega$. The estimate $0C_{\omega}>0,\exists \omega, \omega\not=\emptyset$, such that $\inf R_{\omega}(u)=0$,and we wish to characterize these two sets. For two patterns we give a sufficient condition, sometimes necessary. As our operator corresponds to a layered media we can give another representation of its spectrum: i.e. a subset of points of $R\times R$ that leads to the suggested partition and others connected results: micro local interpretation, default measures,... Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added., Comment: in French, 34 pages. Rehandled version of arXiv:1911.09947 since the results of section 4.1 was false. This section becomes section 3.1 with new statement and proof. Others modifications: new Theorems, corollaries and lemmas and new numerotation. As a result, we have profoundly changed the Introduction

We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry., Comment: 8 pages, announcement with sketches of the proofs and a summary in French. v2: title slightly changed, small changes in the text, final version. To appeat in Comptes Rendus Mathematique

I present recent results in quantum statistical mechanics, obtained in joint works with Mathieu Lewin and Phan Th{\`a}nh Nam. We consider a certain mean-field limit of the grand-canonical ensemble for a Bose gas at positive temperature. In this limit, the reduced density matrices of the quantum theory converge to their analogues in classical field theory, given by a non-linear Gibbs measure. In particular, we deal with cases where the latter must be defined via a renormalization procedure. The corresponding renormalization at the level of the original quantum grand-canonical model, with non-commuting fields, is one of the important difficulties., Comment: Text in french, intended for the proceedings of the second congress of the french mathematical society (Lille, June 2018)

The goal of this "Habilitation \`a diriger des recherches" is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological recursion has been growing and many results in different fields have been obtained. The first aspect that I will develop deals with the historical domain of the topological recursion: random matrix integrals. I will review the formalism of the topological recursion as well as how it can be used to obtain asymptotic $\frac{1}{N}$ series expansion of various matrix integrals. In particular, a key feature of the topological recursion is that it can recover from the leading order of the asymptotic all sub-leading orders with elementary computations. This method is particularly well known and fruitful in the case of hermitian matrix integrals, but I will also show that the general method can be used to cover integrals with hard edges, integrals over unitary matrices and much more. In the end, I will also briefly mention the generalization to $\beta$-ensembles. In a second chapter, I will review the connection between the topological recursion and the study of integrable systems having a Lax pair representation. Most of the results presented there will be illustrated by the case of the famous six Painlev\'e equations. Though the formalism used in this chapter may look completely disconnected from the previous one, it is well known that the local statistics of eigenvalues in random matrix theory exhibit a universality phenomenon and that the encountered universal systems are precisely driven by some solutions of the Painlev\'{e} equations. As I will show, the connection can be made very explicit with the topological recursion formalism., Comment: Written in French. M\'emoire d'habilitation \`a diriger des recherches. 102 pages including many figures

In the framework of semiclassical resonances, we make more precise the link between polynomial estimates of the extension of the resolvent and propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations., Comment: 7 pages, in French

This article proposes a process to reconstruct a Riemann surface with boundary equipped with a conductivity tensor from its boundary and its Dirichlet-Neumann operator. When initial data comes from a two dimensional real Riemannian oriented surface equipped with a conductivity tensor, this process recover the whole of what can be determined from this data., Comment: in French

The concept of gauge theories progressively emerged during the 20th century while the building of the Standard Model (SM) of particle physics. This work focusses on some aspects of gauge theories. First, a modern interpretation of Maxwell's equations in terms of differential forms motivates a geometric definition of gauge theories as a theory of connections on principle bundles. After presenting some historical facts about the notion of gauge in particle physics, the geometrical construction is used to study the electroweak theory and the SM. Though the SM has not been defaulted by any experiments yet, it still remains theoretically incomplete. The Higgs boson discovered in 2012 at LHC, is a particle about which we do not know lots yet. Some interesting clues about how one should extend the SM could be obtained through the observation of significant deviations between the measured value of some parameter of the Higgs boson, and the one predicted by the SM. The trilinear Higgs coupling is one of those interesting parameters. The last part of this work is a phenomenological study of the impact of variations of this coupling on the distributions one can typically get from the CMS particle detector, at LHC., Comment: in French

We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, Coulomb gases on compact Riemannian manifolds and the usual Gibbs measures in the Euclidean space. Finally, we study the generalization of Fekete points and prove a deterministic version of the Laplace principle known as $\Gamma$-convergence. The approach is partly inspired by the works of Dupuis and co-authors. It is remarkably natural and general compared to the usual strategies for singular Gibbs measures., Comment: 23 pages, abstract also in french, for more details in the proofs see version 1, application to Gaussian polynomials added

Nous montrons que les \'equations du rep\`ere mobile des surfaces de Bonnet conduisent \`a une paire de Lax matricielle isomonodromique d'ordre deux pour la sixi\`eme \'equation de Painlev\'e. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painlev\'e equation., Comment: in French. Formula (14) added. Two paragraphs added p 5-6. Old ref 1 removed, ref 7 added. To appear, Comptes rendus Acad. sciences Paris

In this thesis manuscript we explore different facets of random tensor models. These models have been introduced to mimic the incredible successes of random matrix models in physics, mathematics and combinatorics. After giving a very short introduction to few aspects of random matrix models and recalling a physical motivation called Group Field Theory, we start exploring the world of random tensor models and its relation to geometry, quantum gravity and combinatorics. We first define these models in a natural way and discuss their geometry and combinatorics. After these first explorations we start generalizing random matrix methods to random tensors in order to describes the mathematical and physical properties of random tensor models, at least in some specific cases., Comment: PhD Manuscript, 163 pages, based on several precedent results. Contains some unpublished results. Summary and Acknowledgments in French. Main body of the manuscript in English

We study in this paper a new family of stable algebraic symplectic vector bundles of rank $ 2n $ on the complex projective space $P^{2n+1}$ whose classical null correlation bundles belongs. We show that these bundles are invariant under a miniversal deformation. We also study the sufficient cohomological conditions for a symplectic vector bundle on a projective variety to be stable., Comment: 32 pages, in French

The kinetic theory of Maxwell and Boltzmann has been the subject of major scientific controversies. The alleged incompatibility between the reversible nature of the equations of classical mechanics and the increase of entropy, which, in the kinetic theory of gases, is a mathematical property of the Boltzmann equation known as the H Theorem, was one of the most basic arguments against the validity of kinetic theory. About a century later, in 1974, O. Lanford proposed a strategy to prove that the Boltzmann equation describes a particular asymptotic limit of Newton's equations of classical mechanics for a system made of a large number N of spherical particles interacting by elastic collisions. A recent work by I. Gallagher, L. Saint-Raymond and B. Texier completes Lanford's proof and extends it to the case where the interaction between particles is given by a short range potential. In a later article, T. Bodineau, I. Gallagher and L. Saint-Raymond study the dynamics of a tagged particle among N other identical particles in the same asymptotic regime, and prove the validity of the linear Boltzmann equation on arbitrary large periods of time as N tends to infinity. Using classical results on the asymptotic theory of the linear Boltzmann equation, the same authors prove that the stochastic process known as the "Brownian motion" describes a particular limit of the deterministic dynamics of interacting particles., Comment: 39 pages, in French. Bourbaki Seminar, March 2014

These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the systems in question, exploiting the indistinguishablity of particles. The focus is on quantum aspects, particularly the mean-field approximation for the ground state of a large system of bosons, in connection with Bose-Einstein condensation: structure of reduced density matrices of a large bosonic system, localization methods in Fock space, derivation of Hartree and non-linear Schr\"odinger effective energy functionals., Comment: These are the lectures notes from my cours Peccot at the Coll\`ege de France, given in March-April 2014. They are in french for the moment, some english translation should be available soon(er or later). Two appendices contain each an unpublished result obtained in collaboration with Mathieu Lewin. Further small corrections and a few more references in this version.

The subject of the thesis is the construction of a perturbative quantum theory of interacting fields on a curved space-time, following a point of view pioneered by Stueckelberg and Bogoliubov and developed by Epstein-Glaser on the flat Minkowski space-time. In 2000, a breakthrough was done by Brunetti and Fredenhagen who were able to extend the Epstein-Glaser theory by exploiting the point of view developed by Radzikowski to define quantum states on a curved space-time in terms of wave-front sets. These results were further extended by Fredenhagen, Brunetti, Hollands, Wald, Rejzner, etc. to Yang-Mills fields and the gravitation. However, even for theories without gauge invariance, many mathematical details were left unexplored and unquestioned. Our task was precisely to derive fully rigorously this theory in the case there is no gauge invariance. We propose in our work a complete review of the result, solving numerous questions, adding many new results around this program and, eventually, giving more precise details on the counterterms and ambiguities in the renormalization process, and a deeper understanding of the geometry of the wave front set of the n-point functions., Comment: added Acknowledgements in French and corrected bibliography. arXiv admin note: text overlap with arXiv:1008.0129 by other authors

In this work we consider PT-symmetric perturbations of a self-adjoint semi-classical Schr\"odinger operator on the real axis in the case of a simple potential well. We assume that the potential is analytic and show that the eigenvalues remain real under the perturbation. Dans ce travail, on consid\`ere des perturbations PT -sym\'etriques d'un op\'erateur de Schr\"odinger semi-classique auto-adjoint sur la droite r\'eel dans le cas d'un puits de potentiel simple. On suppose que le potentiel soit analytique et on montre que les valeurs propres restent r\'eelles sous la perturbation., Comment: in French

In the context of elastic wave propagation in damaged solids, an analytical approach for scattering of antiplane waves by two-dimensional periodic arrays of cracks is developed. Before considering the study of arrays of cracks, the scattering of an antiplane wave by a flat crack is first studied. Then, using the representation formula for the scattered displacement by a flat and by considering the periodicity condition of the crack-spacing, a boundary integral equation is obtained for the crack face displacement of the reference crack. Numerical results for the reflection and transmission coefficients are presented as functions of the crack-spacing, the frequency of excitation, and the angle of incidence. Finally, the propagation of antiplane waves by two-dimensional periodic arrays of cracks is studied. Despite the use of a finite number of linear arrays, one recognizes the effects of band-pass filtering or band rejection characteristics of the transmission spectra of a periodic medium. Effects due to a disorder in the periodicity are also analysed., Comment: 21 pages (including 9 figures), in French

We consider two interacting connected graded Hopf algebras, the former being a comodule-coalgebra on the latter. We show how to define analogues of Connes-Kreimer's renormalization group and Beta function, when the graduation operator is replaced by any biderivation coming from an infinitesimal character of the second Hopf algebra., Comment: 20 pages, article in French

These notes form an introduction to Lie algebras and group theory. Most of the material can be found in many works by various authors given in the list of references. The reader is referred to such works for more detail., Comment: 87 pages, latex. Notes for a course on Lie symmetries for students of high energy physics, in French. Submitted version

We present a study of constrained mechanical systems and of their quantisation, emphasising the importance of the role played by Poisson brackets in the study of gauge theories., Comment: 13 pages, in French, to appear in "Simeon-Denis Poisson: les mathematiques au service de la science", ed. Yvette Kosmann-Schwarzbach, Editions de l'Ecole Polytechnique 2012

This paper reviews the recent mathematical progresses made on the study of the orbital stability properties for the gravitational Vlasov-Poisson system. We present in details the paper of Lemou, M\'ehats and Rapha\"el (Inventiones 2011) and we review also the previous works by Dolbeault, Guo, Hadzic, Lin, Rein, S\'anchez, Soler, Wan, Wolansky. We also include a discussion of the history of this topic and the pioneering works by physicists like Antonov, Lynden-Bell and Aly. This is the text of a Bourbaki seminar given in november 2011, Comment: 45 pages, in french

Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently: it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications. Keywords: random matrices, random maps, integrable systems, algebraic geometry, loop equations, topological recursion, Hurwitz numbers, Gromov-Witten invariants, Tracy-Widom laws, beta ensemble, large N asymptotics in random matrix theory., Comment: PhD thesis, in French. 300 pages, 7Mo

We study the reduction of 11- dimensional M-theory to (3 +1) dimensions with four conserved supersymmetric charges by following the work of B. Acharya and E. Witten [arXiv:hep-th/0109152]. We first review the various 10D superstrings models, their compactifications to lower dimensions, string/string dualities and links with M-theory. Then, we study compactifications of M-theory first on K3 with ADE singularities; and second on both regular and singular real 7-dimensional G2 manifolds. Others features, such as non abelian 4D quiver gauge models, are also discussed., Comment: 19 pages in double-column format, in French, Latex

We develop in this article the principal constructive arguments used in quantum field theory, limiting us to bosonic theories, for which there does not exist any recent general presentation. The article is primarily written for mathematicians or mathematical physicists knowing the basic arguments of quantum field theory, and desiring to discover a general framework in which they can be made rigorous. It also provides a glimpse of a recent series of articles \cite{MagUnt1,MagUnt2} whose aim is to give a constructive definition of rough paths and fractionary stochastic calculus., Comment: 44 pages, 4 figures. Paper in French, but with an essential French-English glossary

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point of view. First, the thesis will focus on the study of the merging of two intervals of the eigenvalues density near a singular point. Specifically, we will show why this special limit gives universal equations from the Painlev\'e II hierarchy of integrable systems theory. Then, following the approach of (bi) orthogonal polynomials introduced by Mehta to compute partition functions, we will find Riemann-Hilbert and isomonodromic problems connected to matrix models, making the link with the theory of Jimbo, Miwa and Ueno. In particular, we will describe how the hermitian two-matrix models provide a degenerate case of Jimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore, the loop equations method, with its central notions of spectral curve and topological expansion, will lead to the symplectic invariants of algebraic geometry recently proposed by Eynard and Orantin. This last point will be generalized to the case of non-hermitian matrix models (arbitrary $\beta$) paving the way to "quantum algebraic geometry" and to the generalization of symplectic invariants to "quantum curves". Finally, this set up will be applied to combinatorics in the context of topological string theory, with the explicit computation of an hermitian random matrix model enumerating the Gromov-Witten invariants of a toric Calabi-Yau threefold., Comment: PhD thesis in French

These are notes of lectures given at the Third School of Theoretical Physics in Jijel (Algeria, September 2009). The subject of these notes is differential geometry, complex and quaternionic structures with applications to theoretical physics. Concerning the physical applications, they contain several aspects of Penrose transformation in the Riemannian context (Euclidean signature) and various formulations of the Yang-Mills and Einstein equations among which several are unusual ones., Comment: 118 pages, in French

We review some facts about bivariant Chern characters in Noncommutative Geometry and anomaly formulas in Quantum Field Theory., Comment: Habilitation thesis, in French

This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to Differential Geometry considering some left invariant structures on Lie groups : contact and symplectic structures, Generalized complex structures on real Lie Group. We present also the notion of riemannian and pseudo-riemannian G-symmetric spaces., Comment: In French

We study thermodynamic geometry of certain black holes and black branes with and without generalized uncertainty principle or stringy $ \alpha^{\prime} $-corrections to the entropy. From this perspective, we analyze Ruppenier geometry of Reissner-Nordstr\"om black holes and show that it is well defined and corresponds to a non-interacting statistical system. We investigate that the Weinhold geometry of dilatonic black holes is regular everywhere and that of large mass Reissner-Nordstr\"om black holes in the Poincar\'e patch of $ AdS_4 $ contains certain narrow range of thermodynamically unstable regions in the statespace. We obtain that the generalized uncertainty principle corrected Ruppenier geometry of Reissner-Nordstr\"om black holes correspond to a non-interacting statistical system unlike the magnetically charged black holes. We show that the stringy $ \alpha^{\prime} $-corrections do not introduce singularity in the statespace geometry of non-supersymmetric extremal black holes in $ D= 4 $. Interestingly, the degree of scalar curvature and that of the determinant of this Ruppenier geometry can be written as an integer multiple of the order of $ \alpha^{\prime} $-correction. We further show that the statespace geometry of Gauss- Bonnet corrected supersymmetric extremal black holes in $ D=4 $ as well as non-extremal $D_1D_5$ and $D_2D_6NS_5$ black branes in $ D=10 $ is regular everywhere. Furthermore, the thermodynamic geometry of four dimensional rotating Kerr-Newman extremal black holes in Einstein-Maxwell theory is everywhere ill-defined and that of the Kaluza-Klein black holes in Einstein-Maxwell theory or the one arrising from heterotic string compactification is ill-defined only at the points of the ergo-branch., Comment: 489 pages, Latex, French, English Abstract

Very high energy physics needs a coherent description of the four fundamental forces. Non-commutative geometry is a promising mathematical framework which already allowed to unify the general relativity and the standard model, at the classical level, thanks to the spectral action principle. Quantum field theories on non-commutative spaces is a first step towards the quantification of such a model. These theories can't be obtained simply by writing usual field theory on non-commutative spaces. Such attempts exhibit indeed a new type of divergencies, called ultraviolet/infrared mixing, which prevents renormalisability. H. Grosse and R. Wulkenhaar showed, with an example, that a modification of the propagator may restore renormalisability. This thesis aims at studying the generalization of such a method. We studied two different models which allowed to specify certain aspects of non-commutative field theory. In x space, the major technical difficulty is due to oscillations in the interaction part. We generalized the results of T. Filk in order to exploit such oscillations at best. We were then able to distinguish between two mixings, renormalizable or not. We also bring the notion of orientability to light : the orientable non-commutative Gross-Neveu model is renormalizable without any modification of its propagator. The adaptation of multi-scale analysis to the matrix basis emphasized the importance of dual graphs and represents a first step towards a formulation of field theory independent of the underlying space., Comment: PhD thesis, 164 pages. In French. Also available at http://tel.archives-ouvertes.fr/tel-00118044

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate all the previous curvatures. We then examine the curvature term which appears in the classical Weitzenb\"ock formula. We also study the positivity properties of the $p$-curvatures, the second Gauss-Bonnet-Weyl curvature, the Einstein curvature and the isotropic curvature., Comment: In french, extracted from habilitation thesis at Montpellier II University (France), 50 pages

We study the stability of Koszul and Gorentein properties for the semi-cross product of homogeneous algebras. Nous \'etudions la conservation des propri\'et\'es de Koszul et de Gorenstein pour le produit semi-crois\'e des alg\`ebres homog\`enes., Comment: 4 pages in French

In the theory of approximation there are some problems on approximation of compacts in functional spaces by nonlinear families : first we deal with the polynomial case, and then we consider the analytic case. We demonstrate a negative result in which we claim that an analytic familie of functions with $N$ parameters can not approach the compact $\Lambda_l(I^s)$ closer than of order $(N\log N)^{\frac{l}{s}}$, when $N$ increases. As applied to an inverse problem in Sturm-Liouville theory, this assertion provides an answer to a question about the best possible reconstruction of the negative potential $Q$ with $m+1$ integrable derivatives, from its eigenvalues and characteristic values of the equation $-y''+\omega^2Qy=\lambda y$, when $\omega$ increases : we show that it is impossible to get an analytic approximating formula with precision better than of order $(\omega\log\omega)^{-(m+1)}$. Moreover there is from Henkin-Novikova formulas which are almost optimal., Comment: 54 pages, in French

Endomorphisms algebras can replace the concept of principal fiber bundle. Gauge theories are reformulated within this algebraic framework and further generalized to unify ordinary connections and Higgs fields. A 'noncommutative Maxwell' model is built starting from non trivial fiber bundles thus requiring the development of the notion of Riemannian structure. The tools involved in the study of associative algebras are presented and an algebraic method to characterize the usual Chern-Weil morphism is proposed. Then, current results on symmetric fiber bundles are generalized to noncommutative connections and an extension of the Witten ansatz is given. Finally, a generalization of the Born-Infeld action for noncommutative connections is proposed. The corresponding Lagrangians are non-polynomial and the existence of solitonic solutions is shown on several examples, Comment: in french, Ph.D thesis, 157 pages, see http://tel.ccsd.cnrs.fr/tel-00010487

The aim of this thesis is to study the isopectral deformations from the point of view of Alain Connes' noncommutative geometry. This class of quantum spaces constituts a curved space generalisation of Moyal planes and noncommutative tori. First of all, we look at the construction of non-unital spectral triples, for which we propose modified axioms. We then check that Moyal planes fit into this axiomatic framework, and give the keypoints for the construction of non-unital spectral triples from generic non-compact isospectral deformations. To this end, numerous analytical tools on non-compact Riemannian manifolds are developped. Thanks to Dixmier traces computations, we show that their spectral and classical dimensions coincide. In a second time, we study certain features of quantum fields theory on curved isospectral deformations, with a particular view on the ultraviolet infrared mixing phenomenon. We show its intrinsic nature for all such quantum spaces (compacts or not, periodic or not deformations), and we study its consequences on the renormalisability. In particular, the behaviour of Green functions of the planar and non-planar sectors is understood in term of on- and off-diagonal heat kernel contributions. We also see new or inner manifestations of the UV/IR mixing, related to the geometric properties of those quantum spaces and to the arithmetic nature of the deformation parameters., Comment: In french, 151 pages

This preprint is the introduction of my habilitation thesis for Paris7 university. It is a sumary of a collection of works on the 2 matrix model. In an introduction, 3 different and unequivalent definitions of matrix models are given (convergent model, model with fixed filling fractions on contours, and formal model). Then, a sumary of properties of differential systems satisfied by biorthogonal polynomials, in particular spectral duality and Riemann-Hilbert problem. Then, a section on loop equations and algebraic geometry formulation of the large N expansion. Then, a conjecture for the asymptotics of biorthogonal polynomials., Comment: in french, latex, 80 pages

This paper is devoted to the description of our recent results on the spectral behavior of one-dimensional adiabatic quasi-periodic Schrodinger operators. The specific operator we study is a slow periodic perturbation of an incommensurate periodic Schrodinger operator, and we are interested in energies where the perturbation creates a strong interaction between two consecutive bands of the background periodic operator. We describe the location of the spectrum and its nature and discuss the various new resonance phenomena due to the interaction of the spectral bands of the unperturbed periodic operator., Comment: Cet article est en francais / This paper is in french

We study non-linear evolution equations with periodic initial conditions. In particular, we use the graph method introduced by Galavotti to prove the existence of global solution of Hamiltonian perturbation of KdV without any restriction on the size of the initial condition, in contrast with previous results on this subject., Comment: 67 pages, in french

P.Lecomte has proposed to take into account the covariant derivatives used to build ordering prescriptions for the naturality of transformation properties and has conjectured that there exists an natural ordering prescription for differential operators of any orders between density bundles which in addition is invariant under projective changes of the covariant derivatives. We prove this conjecture by constructing a projectively invariant lift of a torsion-free connexion to a torsion-free connexion on (the positive part of) the total space of the bundle of all $a$-densities for nonzero $a$, by lifting the symbols in a projectively invariant way (they turn out to be in bijection to the space of all $\real^+$-equivariant divergence-free symmetric tensor fields on the total space), and by using the standard ordering procedure (`all the covariant derivatives to the right') on the total space. For Ricci-flat manifolds we show that this ordering prescription coincides --with the appropiate replacements-- with an explicit formula in $\real^m$ obtained by Duval, Lecomte and Ovsienko., Comment: 49 pages, no figures, paper in French, Latex2e

We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (\Omega,\omega), such that \Omega is a symplectic form and \omega is a 3-differential form which satisfies \omega\wedge\Omega=0 and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) Calabi-Yau structure and we study its integrability from the point of view of Monge-Ampere operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge-Ampere equation in the dimension 4. We study from this point of view the example of the Stenzel metric on T*S^3., Comment: 18 pages, in french

In contrast to a widely held opinion, the three-body problem has an analytic solution. This solution was found in 1909 by Sundman. The basic ideas behind it and its history are presented. ----- Contrairement \`a une opinion largement r\'epandue, le probl\`eme des trois corps poss\`ede une solution analytique. Cette solution fut d\'ecouverte en 1909 par Sundman. Nous pr\'esentons dans cet article les id\'ees de base et l'histoire de cette solution. ----- Entgegen einer weitverbreiteten Meinung ist das Dreik\"orperproblem analytisch l\"osbar. Diese L\"osung wurde 1909 von Sundman gefunden. Die ihr zugrundeliegenden Ideen und ihre Geschichte werden in einfacher Form dargestellt., Comment: Latex, 26 pages, with 3 figures, text in French

This PhD thesis aims at describing the applications of noncommutative geometry to particle physics and quantum field theory. It includes a brief survey of the basic principles and definitions of noncommutative geometry such as spectral triples, differential calculus or gauge theories on projective modules. Within this framework, The index theorem and cyclic homology are applied to the probe of topologically non trivial properties of Chern-Simons and Yang-Mills action functionals in dimension 3 and 4. Following earlier work on the standard model in noncommutative geometry, it also provides a general study of all Yang-Mills-Higgs models based on space-times structures which are the product of usual space-time by a discrete space, with emphasis on the physical properties of the resulting theories. Finally, the last part is devoted to the development of classical and quantum gauge theory on the noncommutative torus in any dimension. It includes a discussion of topological properties and symmetries of the action as well as its perturbative quantization., Comment: Latex, 219 pages including 23 eps figures, the author's PhD thesis (in french)