L'objet de ce travail est de généraliser au cas des variétés feuilletées par variétés complexes la théorie des déformations de variétés complexes compactes développée notamment par les travaux de Kodaira et Spencer vers la fi n des années cinquante. Après avoir défni la notion de famille de déformations de variétés feuilletées par variétés complexes compactes, nous avons pu obtenir un analogue des théorèmes de rigidité, de complétude et d'existence dans notre cadre. Les méthodes de démonstration usant de la théorie du potentiel ne sont pas généralisables car les opérateurs différentiels considérés ici ne sont plus elliptiques. On se tourne alors vers des techniques de séries majorantes pour obtenir ces résultats, en particulier pour le théorème d'existence qui généralise la démonstration faite par Forster et Knorr en 1974. / The aim of this work is to generalise the study of deformations of complex manifolds by kodaira and Spencer to the case of manifolds foliated by complex manifolds. After defning the notion of family of deformations of compact manifold foliated by complex manifolds, we prove a theorem of rigidity, one of completeness and one of existence in our framework. We can not apply one potential theory here, so we have to use power series technics.

Les protéines de choc thermique Hsp70 [70 kDa Heat Shock Protein] sont considérées comme des chaperons moléculaires très importants qui assistent au repliement des protéines naissantes ainsi qu’au repliement des protéines dénaturées en condition de stress dans le milieu intracellulaire. Les protéines Hsp70 sont présentes chez tous les organismes, tels que l’humain, la bactérie ou encore la levure et ont pour propriété d’avoir une séquence d’acides aminés hautement conservée entre les différentes espèces. [...] Ainsi la connaissance à l’échelle atomique des structures de la protéine hHsp70 dans ses conformations ouverte et fermée est un prérequis essentiel pour comprendre les interactions entre les deux domaines NBD et SBD de la protéine et pour élucider les mécanismes de communication inter-domaine. Cependant, il n’existe pas de structure expérimentale complète de la protéine hHsp70. Dans cette thèse, nous présentons les structures « tout-atome » des conformations ouverte et fermée de la protéine hHsp70, qui ont été modélisées par homologie à partir de la structure par diffraction des rayons X [DRX] de la protéine Hsp110 de la levure Sacharomyces cerevisae [dans la conformation ouverte] et à partir de la structure résolue par Résonance Magnétique Nucléaire [RMN] de la protéine Hsp70 de la bactérie E. coli [dans la conformation fermée]. Ces deux modèles structuraux de la protéine humaine Hsp70, dans les états ouvert et fermé, ont ensuite été relaxés par dynamique moléculaire non biaisée à la température de 300K en utilisant un solvant explicite sur une échelle de temps respectivement de 2.7 et 0.5 μs. L’hétérogénéité conformationelle de la protéine hHsp70 observée dans les simulations de dynamique moléculaire a été comparée à celle extraite d’expériences de resonance par transfert d’énergie entre fluorophores [FRET pour Förster resonance energy transfer] et de diffraction aux petits angles [SAXS pour Small Angle X-ray Scattering] effectuées sur des protéines homologues à hHsp70. [...] Une fois les structures 3D « tout-atome » résolues, la transition entre la conformation ouverte et la conformation fermée [et vice-versa] des protéines Hsp70 a été étudiée en utilisant deux techniques de simulations numériques : une analyse des modes normaux [Normal Mode Analysis où NMA] de la protéine Hsp70 dans chacune de ces deux conformations et une nouvelle méthode développée au cours de cette thèse, basée sur le concept de paysage d’énergie libre [Free-Energy Landscape où FEL]. [...] Cette étude a également permis d’identifier les sous-domaines et résidus clés qui apparaissent comme jouant un rôle important dans la dynamique conformationelle de la protéine Hsp70 dans l’approximation harmonique. Pour comprendre comment la fixation du nucléotide dans le domaine NBD peut engendrer un changement important de conformation de la protéine Hsp70, nous avons réalisé des simulations de dynamique moléculaire tout-atome non biaisée [sur une échelle de temps de 2 μs] de la protéine Hsp70 de la bactérie E. coli [appelée E. coli DnaK], dans trois conditions de nucléotides différentes [liée à l’ATP, liée à l’ADP et sans nucléotide]. [...] Finalement, en combinant l’analyse des modes normaux et du paysage d’énergie libre de la protéine Hsp70, nous avons pu établir une liste de résidus et de structures locales impliqués dans la dynamique conformationelle et dans les mécanismes de communication de la protéine hHsp70. La plupart de ces résidus ont été identifiés expérimentalement comme jouant un rôle crucial dans la communication entre les domaines NBD et le domaine SBD de protéines Hsp70 homologues. Notre étude nous a également permis d’identifier de nouveaux résidus clés. Ces nouveaux résidus pourraient être testés expérimentalement par mutagénèse et leurs positions pourraient être de nouvelles cibles pour la fixation d’inhibiteurs de fonctions biologiques de Hsp70, notamment dans le cas de tumeurs cancéreuses. / The 70 kDa heat shock proteins [Hsp70s] are key molecular chaperones which assist in the correct folding of nascent proteins and refolding of proteins under stress conditions in the intracellular environment. Hsp70s are present in all organisms and are highly conserved between the different species. [...] The conformational dynamics between the two conformations is governed by the ATP binding, ATP hydrolysis and by nucleotide exchange through an allosteric mechanism which is not fully understood.Knowledge of the conformations of hHsp70 at the atomic level is central to understand the interactions between its NBD and SBD. However, no complete structure of hHsp70 is known. In the present thesis, we report two conformations of hHsp70, constructed by homology modeling from the yeast Saccharomyces cerevisiae co-chaperone protein Hsp110 [openconformation] and from the bacteria Escherichia coli Hsp70 [closed conformation]. The open and closed conformations of hHsp70 built by homology were relaxed by using unbiased all-atom molecular dynamics [MD] simulations at 300 K in explicit solvent on a timescale of 2.7 and 0.5 μs, respectively. The conformational heterogeneity of hHsp70 observed in MD simulations was comparedwith those extracted from single-molecule Forster resonance energy transfer [FRET]experiments and to small-angle X-ray scattering [SAXS] data of Hsp70 homologs. [...] In the present thesis, the transitions between the open and closed conformation of Hsp70s were studied by using two different computational methods: the Normal Mode Analysis [NMA] and a new method developed in the present thesis based on the Free-Energy Landscape [FEL] concept.[...] These collective modes provide a mechanistic representation of the communication between the NBD and the SBD and allow us to identify subdomains and residues that appear to have a critical role in the conformational dynamics of Hsp70s in the harmonicapproximation. Second, in order to understand how the nucleotide binding in the NBD of Hsp70 induces a conformational change of the whole protein, we performed unbiased all-atom MD simulations [2 μs] of E. coli Hsp70 [named E. Coli DnaK], in three different nucleotide-binding states [ATPbound,ADP-bound and nucleotide free]. [...] Finally, by combining the NMA and the FEL analysis, we established a list of the local structures and of the residues relevant for the conformational dynamics and for the interdomain communication in hHsp70. Most of these residues could be related to previous experimental evidences of their role in the interdomain communication between the NBD and SBD domains of Hsp70 homologs but other were never identified before. All the relevant residues found in MD could be tested experimentally by mutational analysis and could be crucial locations to dock small peptides and for the design of inhibitors for the cancer therapy

This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.

The Wavelet Finite Element Method (WFEM) involves combining the versatile wavelet analysis with the classical Finite Element Method (FEM) by utilizing the wavelet scaling functions as interpolating functions; providing an alternative to the conventional polynomial interpolation functions used in classical FEM. Wavelet analysis as a tool applied in WFEM has grown in popularity over the past decade and a half and the WFEM has demonstrated potential prowess to overcome some difficulties and limitations of FEM. This is particular for problems with regions of the solution domain where the gradient of the field variables are expected to vary fast or suddenly, leading to higher computational costs and/or inaccurate results. The properties of some of the various wavelet families such as compact support, multiresolution analysis (MRA), vanishing moments and the “two-scale” relations, make the use of wavelets in WFEM advantageous, particularly in the analysis of problems with strong nonlinearities, singularities and material property variations present. The wavelet based finite elements (WFEs) presented in this study, conceptually based on previous works, are constructed using the Daubechies and B-spline wavelet on the interval (BSWI) wavelet families. These two wavelet families possess the desired properties of multiresolution, compact support, the “two scale” relations and vanishing moments. The rod, beam and planar bar WFEs are used to study structural static and dynamic problems (moving load) via numerical examples. The dynamic analysis of functionally graded materials (FGMs) is further carried out through a new modified wavelet based finite element formulation using the Daubechies and BSWI wavelets, tailored for such classes of composite materials that have their properties varying spatially. Consequently, a modified algorithm of the multiscale Daubechies connection coefficients used in the formulation of the FGM elemental matrices and load vectors in wavelet space is presented and implemented in the formulation of the WFEs. The approach allows for the computation of the integral of the products of the Daubechies functions, and/or their derivatives, for different Daubechies function orders. The effects of varying the material distribution of a functionally graded (FG) beam on the natural frequency and dynamic response when subjected to a moving load for different velocity profiles are analysed. The dynamic responses of a FG beam resting on a viscoelastic foundation are also analysed for different material distributions, velocity and viscous damping profiles. The approximate solutions of the WFEM converge to the exact solution when the order and/or multiresolution scale of the WFE are increased. The results demonstrate that the Daubechies and B-spline based WFE solutions are highly accurate and require less number of elements than FEM due to the multiresolution property of WFEM. Furthermore, the applied moving load velocities and viscous damping influence the effects of varying the material distribution of FG beams on the dynamic response. Additional aspects of WFEM such as, the effect of altering the layout of the WFE and selection of the order of wavelet families to analyse static problems, are also presented in this study.

The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. In several particular cases we show that all closed gaps lie on the infinite spectral arc. In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromy-free operators.

The thesis studies homogenization of Dirichlet boundary value problems for divergence type elliptic operators, and the associated boundary layer issues. This type of problems for operators with periodically oscillating coeffcients, and fixed boundary data are by now a classical topic largely due to the celebrated work by Avellaneda and Lin from late 80's. The case when the operator and the Dirichlet boundary data exhibit periodic oscillations simultaneously was a longstanding open problem, and a progress in this direction has been achieved only very recently, in 2012, by Gerard-Varet and Masmoudi who proved a homogenization result for the simultaneously oscillating case with an algebraic rate of convergence in L2. Aimed at understanding the homogenization process of oscillating boundary data, in the first part of the thesis we introduce and develop Fourier-analytic ideas into the study of homogenization of Dirichlet boundary value problems for elliptic operators in divergence form. In smooth and bounded domains, for fixed operator and periodically oscillating boundary data we prove pointwise, as well as Lp convergence results the homogenization problem. We then investigate the optimality (sharpness) of our Lp upper bounds. Next, for the above mentioned simultaneously oscillating problem studied by Gerard-Varet and Masmoudi, we establish optimal Lp bounds for homogenization in some class of operators. For domains with non smooth boundary, we study similar boundary value homogenization problems for scalar equations set in convex polygonal domains. In the vein of smooth boundaries, here as well for problems with fixed operator and oscillating Dirichlet data we prove pointwise, and Lp convergence results, and study the optimality of our Lp bounds. Although the statements are somewhat similar with the smooth setting, challenges for this case are completely different due to a radical change in the geometry of the domain. The second part of the work is concerned with the analysis of boundary layers arising in periodic homogenization. A key difficulty toward the homogenization of Dirichlet problem for elliptic systems in divergence form with periodically oscillating coefficients and boundary condition lies in identification of the limiting Dirichlet data corresponding to the effective problem. This question has been addressed in the aforementioned work by Gerard-Varet and Masmoudi on the way of proving their main homogenization result. Despite the progress in this direction, some very basic questions remain unanswered, for instance the regularity of this effective data on the boundary. This issue is directly linked with the up to the boundary regularity of homogenized solutions, but perhaps more importantly has a potential to cast light on the homogenization process. We initiate the study of this regularity problem, and prove certain Lipschitz continuity result. The work also comprises a study on asymptotic behaviour of solutions to boundary layer systems set in halfspaces. By a new construction we show that depending on the normal direction of the hyperplane, convergence of the solutions toward their tails far away from the boundaries can be arbitrarily slow. This last result, combined with the previous studies gives an almost complete picture of the situation.

We consider a quantum system whose unperturbed form consists of a self-adjoint Δ-operator on a 2-dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential/point scatterer at some select point ρ. The perturbed self-adjoint operator is constructed rigorously by means of self-adjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary. Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the delta-perturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength. Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10]. In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10]. Chapter 4 deals with the construction of non-singular rank-one perturbations for which the eigenvalues and eigenfunctions approximate those of the delta-perturbed operator. This is approached by means of direct analysis of the construction and formulae for the rank-one-perturbed eigenvalues and eigenfunctions, by comparison that of the delta-perturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of non-singular rank-one perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the delta-perturbed operator. Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rank-one perturbations approaching the delta potential within the setting of a whole Euclidean space Rⁿ, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every Eℓ(0,∞). Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rank-one perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12]. Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a full-density subsequence of the full perturbed eigenbasis. Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general delta-perturbed system whose classical dynamics satisfies the particular key property.

Understanding the phenomenon of wave scattering by random media is a ubiquitous problem that has instigated extensive research in the field. This thesis focuses on wave scattering by quasiperiodic media as an alternative approach to provide insight into the effects of structural aperiodicity on the propagation of the waves. Quasiperiodic structures are aperiodic yet ordered so have attributes that make them beneficial to explore. Quasiperiodic lattices are also used to model the atomic structures of quasicrystals; materials that have been found to have a multitude of applications due to their unusual characteristics. The research in this thesis is motivated by both the mathematical and physical benefits of quasiperiodic structures and aims to bring together the two important and distinct fields of research: waves in heterogeneous media and quasiperiodic lattices. A review of the past literature in the area has highlighted research that would be beneficial to the applied mathematics community. Thus, particular attention is paid towards developing rigorous mathematical algorithms for the construction of several quasiperiodic lattices of interest and further investigation is made into the development of periodic structures that can be used to model quasiperiodic media. By employing established methods in multiple scattering new techniques are developed to predict and approximate wave propagation through finite and infinite arrays of isotropic scatterers with quasiperiodic distributions. Recursive formulae are derived that can be used to calculate rapidly the propagation through one- and two-dimensional arrays with a one-dimensional Fibonacci chain distribution. These formulae are applied, in addition to existing tools for two-dimensional multiple scattering, to form comparisons between the propagation in one- and two-dimensional quasiperiodic structures and their periodic approximations. The quasiperiodic distributions under consideration are governed by the Fibonacci, the square Fibonacci and the Penrose lattices. Finally, novel formulae are derived that allow the calculation of Bloch-type waves, and their properties, in infinite periodic structures that can approximate the properties of waves in large, or infinite, quasiperiodic media.

Cette thèse porte sur l’étude des solutions périodiques du problème des N-tourbillons à vorticité positive. Ce problème, formulé par Helmholtz il y a plus de 160 ans, possède une histoire très riche et reste un domaine de recherche très actif. Pour un nombre quelconque de tourbillons et sans contrainte sur les vorticités, ce système n’est pas intégrable au sens de Liouville : on ne peut trouver de solution périodique non triviale par des méthodes explicites. Dans cette thèse, à l’aide de méthodes variationnelles, nous prouvons l’existence d’une infinité de solutions périodiques non triviales pour un système de N tourbillons à vorticités positives. De plus, lorsque les vorticités sont des nombres rationnels positifs, nous montrons qu’il n’existe qu’un nombre fini de niveaux d’énergie sur lesquels un équilibre relatif pourrait exister. Enfin, pour un système de N-tourbillons identiques, nous montrons qu’il existe une infinité de chorégraphies simples. / This thesis focuses on the study of the periodic solutions of the N-vortex problem of positive vorticity. This problem was formulated by Helmholtz more than 160 years ago and remains an active research field. For an undetermined number of vortices and general vorticities the system is not Liouville integrable and periodic solutions cannot be determined explicitly, except for relative equilibria. By using variational methods, we prove the existence of infinitely many non-trivial periodic solutions for arbitrary N and arbitrary positive vorticities. Moreover, when the vorticities are positive rational numbers, we show that there exists only finitely many energy levels on which there might exist a relative equilibrium. Finally, for the identical N-vortex problem, we show that there exists infinitely many simple choreographies.

L’Analyse Non Standard (l’ANS) est un formalisme mathématique particulier inventé dans les années 1960 par le mathématicien A. Robinson. Ce formalisme permet de renouer avec les infinitésimaux de Leibniz qui avaient été abandonnés au XIXème siècle pour satisfaire aux exigences nouvelles de la « rigueur ». Sa pertinence a été contestée par divers mathématiciens parmi les plus grands et a donné naissance à une polémique dans les milieux mathématiques français ; les partisans de l’ANS en sont sortis vaincus et ne se sont plus guère exprimés après 1995. Un quart de siècle plus tard l’ANS est considérée au plan international comme une pratique tout à fait légitime et certains mathématiciens, à leur tour parmi les plus grands, en préconisent l’usage.Pourquoi cette mauvaise réception d’idées nouvelles en mathématiques dans un pays réputé pour son excellence dans ce domaine ?Il est normal que des idées révolutionnaires, voire simplement nouvelles, rencontrent de la résistance et suscitent un débat. Toutefois on observe que ce débat qui commençait à prendre de l’importance au début des années 1980 a été étouffé dans les années 1990 par ceux qui avaient en charge les institutions de la communauté mathématique. Pourquoi ce refus du débat ?La thèse soutenue est que, à cette époque, une des fonctions que l’idéologie dominante assigne aux mathématiques est de dire le vrai ; par exemple les théories économiques libérales prétendent à la scientificité parce que fortement mathématisées. Ne dit-on pas c’est mathématique pour affirmer d’une chose qu’elle est inéluctable. Une dispute trop visible sur la nature de la rigueur mathématique aurait risqué de brouiller cette image du mathématicien. Dans le même ordre d’idées, à la même époque, la communauté mathématique (et plus généralement scientifique) avait refusé de débattre avec un de ses membres les plus brillants, A. Grothendieck, de la responsabilité sociale du savant.Cette question de la réception de l’ ANS illustre la thèse bien connue que si une science se développe en partie pour résoudre des problèmes qu’elle se pose à elle même, ici donner un statut logique irréprochable à la pratique des infinitésimaux, cette motivation interne ne suffit pas à elle seule à expliquer tous les aspects de son développement. Les savants doivent tenir compte de la société dans laquelle ils vivent. Il est intéressant de faire ce constat dans le domaine des mathématiques dites pures, c’est à dire qui se prétendent en dehors de toute contrainte et ne travailler que pour l’honneur de l’esprit humain, pour reprendre la célèbre formule de Jacobi. / Non Standard Analysis (ANS) is a particular mathematical formalism invented in the 1960s by the mathematician A. Robinson. This formalism allows to reconnect with the infinitesimals of Leibniz which had been abandoned in the nineteenth century to satisfy the new requirements of rigor. Its relevance has been challenged by various mathematicians among the greatest and gave birth to a controversy in the French mathematical circles ; the supporters of the ANS came out defeated and hardly spoke after 1995. A quarter of a century later, ANS is considered internationally as a perfectly legitimate practice and some mathematicians, including famous ones, advocate its use.Why this bad reception of new ideas in a country renowned for its excellence in the field of mathematical research?It is natural for revolutionaries, or simply news ideas, to be at the origin of resistance and debate. However, we observe that this debate, which was starting and gaining importance in the early 1980’s, was stifled by those who were in charge of the institutions of the mathematical community. Why this refusal of debate?My thesis is that, at this time, one of the functions that the dominant ideology assigned to mathematics was to « say the truth »; for example liberal economic theories claim to scientificity because they are highly mathematized. It is commun to say « it is mathematical » to say that something is unavoidable. A dispute too visible about the nature of mathematical rigor could blur this image of the mathematician. In the same vein, at the same time, the mathematical (and more generally scientific) community had refused to debate with one of its most brilliant members, A. Grothendieck, on the social responsibility of the scientist.This question of the reception of the ANS illustrates the well-known thesis that if a science develops partly to solve problems that it poses to itself, in our case to give an irreproachable logical status to the practice of infinitesimals, this internal motivation is not enough on its own to explain all aspects of its development. Scholars must consider the society in which they live. It is interesting to make this observation in the so-called field of pure mathematics, which claim to be free from all constraints and work only « for the honor of the human mind » to use Jacobi's famous formula.