This thesis approaches the journeys and meetings I have had in the teaching. Part of subjectivity understanding produced from multiple heterogeneous components, social experience and their singular routes. Today, more than ever, subjectivity is convened to singularize itself. It is in this perspective of singling that this research enplanes. The analysis was done by lines of affection, which were treated as potency to act, not as feelings themselves, but based on speeches and writings fragments, when I approached my perception of what produces me in teaching, rubbing with what I experience in the daily of Farroupilha Federal Institute - Panambi campus, and the readings I have done. The research is sustained in the theoretical assumptions of Gilles Deleuze, Félix Guattari and Baruch Espinosa, in particular in the concepts of arrangement, subjectivity, potency to act and meetings. The des/reterritorialization, lines of escape and the subjectivity out of the subject insert as folds in the research. In this cartographic confection, among authors and fragments, it was necessary to find a method that could sustain the intense anxieties that follow these moments of creation. Thus, the cartography allowed me to trace the intensive lines that gave rise to the concerns presented here. The care I had was to not fall into the representation of the subject who produces and who is produced on campus Panambi, but rather think of the arrangements that this space allows, because the individual only constitutes himself when broker. I highlight as one of the contributions of this study, the teaching when takes on new proportions, having the possibility to consider producing up teaching as a micropolitics found every step of the way, in a move that pulses, that intends and finds resonance in the creation, in the collective production, in the authorship, in the transversality of saying and listening, in the constitution of new senses to see in its vents, pipes that leaves the air circular with the outside, even at the risk of not always find good winds: this was my trip s script, this was my proposal. / Esta tese aborda os percursos e encontros que tenho tido na docência. Parte da compreensão da subjetividade produzida a partir de múltiplos componentes heterogêneos, da experiência social e seus percursos singulares. Hoje, mais do que nunca, a subjetividade é convocada a se singularizar. É nessa perspectiva de singularização que esta pesquisa se insere. A análise se deu pelas linhas de afeto, que foram tratadas como potência de agir, e não como sentimentos em si, com base em fragmentos de falas e escritas, quando abordei minha percepção em relação ao que me produz na docência, friccionando com o que vivencio no cotidiano do Instituto Federal Farroupilha campus Panambi, e as leituras que tenho realizado. A investigação encontra-se nos pressupostos teóricos de Gilles Deleuze, Félix Guattari e Baruch Espinosa, em especial nos conceitos de agenciamento, subjetividade, potência de agir e encontros. A des/reterritorialização, as linhas de fuga e a subjetividade fora do sujeito inserem-se como dobras na pesquisa. Nesta confecção cartográfica, entre os autores e os fragmentos, foi necessário buscar um método que pudesse sustentar as intensas ansiedades que sucedem esses momentos de criação. Assim, a cartografia me permitiu traçar as linhas intensivas que deram origem às inquietações aqui apresentadas. O cuidado que tive foi o de não cair na representação do sujeito que produz e que se produz no campus Panambi, mas, sim, pensar nos agenciamentos que esse espaço possibilita, pois o indivíduo só se constitui ao se agenciar. Destaco como uma das contribuições deste estudo a docência quando assume novos contornos, tendo a possibilidade de considerar o produzir-se docente como uma micropolítica encontrada a cada passo do percurso, em um movimento que pulsa, que tenciona e que encontra ressonância na criação, na produção coletiva, na autoria, na transversalidade do dizer e do escutar, na constituição de novos sentidos para nela enxergar respiradouros, tubulações que deixa o ar circular com o exterior, mesmo com o risco de nem sempre encontrar bons ventos: este foi meu roteiro de viagem, esta foi minha proposta.

Dans cette thèse, nous étudions les théories étroitement liées du contrôle, de la stabilisation et de la propagation des singularités, pour des équations aux dérivées partielles dispersives linéaires et non-linéaires. Les résultats principaux proviennent des travaux de l’auteur:[1] Zhu, H., 2016. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM : Control, Optimisation and Calculus of Variations (ESAIM: COCV), à paraître.[2] Zhu, H., 2017. Control of three dimensional water waves. arXiv preprint arXiv:1712.06130.[3] Zhu, H., 2018. Propagation of singularities for gravity-capillary water waves. arXiv preprint arXiv:1810.09339.Dans [1], nous avons étudié la stabilisation des ondes amorties sur les surfaces de révolution de Zoll. Nous avons donné un exemple où la région d’amortissement est à la limite de la condition du contrôle géométrique, alors que les ondes amorties présentent une décroissance exponentielle uniforme de l’énergie. Cet exemple généralise un résultat de Lebeau. Dans [2], nous avons étudié la contrôlabilité du système des ondes de surface avec tension superficielle. Nous avons démontré, en dimensions arbitraires, la contrôlabilité exacte pour des petites données spatialement périodiques à condition du contrôle géométriques. Ce résultat généralise le travail de Alazard, Baldi et Han-Kwan en dimension deux. Dans [3], nous avons étudié la propagation des singularités pour des ondes de surface avec tension superficielle. Nous avons défini le front d’onde quasi-homogène, généralisant le front d’onde de Hörmander et le front d’onde homogène de Nakamura et démontré des résultats de propagation des fronts d’onde quasi-homogènes par le système des ondes de surface avec tension superficielle. Comme corollaires, nous avons obtenu des effets régularisants locaux et micro-locaux pour les données initiales présentant une décroissance spatiale suffisante. / In this thesis, we study the closely related theories of control, stabilization and propagation of singularities for some linear and nonlinear dispersive partial differential equations. Main results come from the author’s works:[1] Zhu, H., 2016. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), to appear.[2] Zhu, H., 2017. Control of three dimensional water waves. arXiv preprint arXiv:1712.06130.[3] Zhu, H., 2018. Propagation of singularities for gravity-capillary water waves. arXiv preprint arXiv:1810.09339.In [1] we studied the stabilization of the damped wave equation on Zoll surfaces of revolution. We gave an example where the region of damping is at the borderline of the geometric control condition, yet the damped waves exhibit a uniform exponential decay of energy, generalizing an example of Lebeau.In [2] we studied the controllability of the gravity-capillary water wave equation. Under the geometric control condition, we proved in arbitrary spatial dimension the exact controllability for spatially periodic small data. This generalizes a result of Alazard, Baldi and Han-Kwan for the 2D gravity-capillary water wave equation.In [3] we studied the propagation of singularities for the gravity-capillary water wave equation. We defined the quasi-homogeneous wavefront set, generalizing the wavefront set of H¨ ormander and the homogeneous wavefront set of Nakamura, and proved propagation results for quasi-homogeneous wavefront sets by the gravity-capillary water wave equation. As corollaries, we obtained local and microlocal smoothing effects for gravity-capillary water waves with sufficient spatial decay.

En turbulence pleinement développée et incompressible, on constate que l’énergie cinétique d’unécoulement est dissipée à un taux indépendant du nombre de Reynolds. C’est la loi zéro de la turbulence.Cette loi, qui fut découverte en 1935 par Taylor, a eu de nombreuses confirmations expérimentaleset numériques, et est au coeur de notre compréhension de la physique des régimes turbulents. Dansles années qui suivirent, Taylor proposa un mécanisme pour rendre compte de la loi zéro, basé sur laviscosité et sur l’idée d’une cascade d’énergie à travers les échelles. En 1949, Onsager se rend comptequ’une dissipation d’énergie peut aussi se produire sans l’assistance des forces visqueuses à petite échellesi le champ de vitesse devient suffisamment irrégulier, et propose une conjecture sur la régularité minimaleque devrait satisfaire le champ de vitesse pour assurer la conversation de l’énergie en l’absencede viscosité. En 2000, deux mathématiciens français, Jean Duchon et Raoul Robert, formalise pour lapremière fois les idées d’Onsager dans un cadre mathématique rigoureux. Ils établissent la forme exactede la dissipation d’énergie émanant de l’existence possible de singularités, et I’expriment en fonctiondes incréments de vitesse. Cependant, la pertinence de ces concepts en turbulence expérimentale resteà établir, et n’a jamais été étudiée.Dans cette thèse, nous proposons les premiers tests des idées d’Onsager à partir de données expérimentales,en se basant sur le travail de Duchon et Robert. Pour cela, nous nous plaçons dans le cadredes écoulements de von Kármán où la régularité des équations de Navier-Stokes n’est pas connue. Nousutilisons des mesures de vélocimétrie par image de particules pour obtenir les trois composantes duchamp de vitesse dans un plan méridien, et ainsi calculer ses incréments à l’échelle de résolution de notresystème de mesure. Le résultat principal de ce travail est la mise en évidence du caractère non-trivialdes écoulements turbulents à l’échelle de Kolmogorov, où l’on observe des topologies très irrégulièresdu champ de vitesse coïncidant avec des évènements extrêmes de transferts inertiels d’énergie. / The zeroth law of turbulence states that fully developed turbulent incompressible flows dissipatetheir kinetic energy independently of the Reynolds number. Since its discovery by Taylor in 1935, thislaw has had many experimental and numerical confirmations, and is at the heart of our understandingof turbulence. In the following years, Taylor proposed a mechanism for the zeroth law, based onviscosity and the idea of a cascade of energy through scales. In 1949, Onsager realized that energydissipation could occur without the final assistance by viscosity at small scales if the velocity fieldbecomes sufficiently irregular, and conjectured the minimum regularity condition above which energyconservation is ensured in the absence of viscosity. In 2000, two french mathematicians, Jean Duchonand Raoul Robert, were able to derive the analytical expression for the inertial dissipation in termsof velocity increments, along with the corresponding energy balance. However, the relevance of theseideas for real turbulence has never been studied.In this thesis, we present the first tests of Onsager’s idea from experimental data, based on thework of Duchon and Robert. We enter the framework of von Kármán flows for which the regularity ofNavier-Stokes equations is unknown. We use particle image velocimetry measurements which provideus with the three components of the velocity field on a meridional plane, and allows for the computationof velocity increments at the resolution scale of our measurement set-up. In this work, we point out thenon-trivial character of turbulent flows at the Kolmogorov scale, where we observe irregular

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

En 1979, Trotman a démontré que les stratifications réelles lisses qui satisfont la condition de (a)-régularité sont précisément celles pour lesquelles la transversalité aux strates des applications est une condition stable dans la topologie forte. C'était un résultat surprenant puisque la (t)-régularité semblait être plus appropriée pour la stabilité de la transversalité, une erreur qui a été faite dans plusieurs articles avant que ce résultat soit montré par Trotman. Notre premier résultat est un analogue au résultat de Trotman pour la topologie faible.Il y a une dizaine d'années Trotman a demandé si le même résultat est valable pour les stratifications analytiques complexes. Dans ce travail on démontre un analogue du résultat de Trotman dans le cas complexe, en utilisant la notion de variété de Oka introduite par Forstneric et on montre que la conjecture n'est pas vraie en général en donnant des contre-exemples.Dans sa thèse, Trotman a formulé une conjecture pour généraliser son résultat pour les stratifications (a_f)-régulières de Thom. Dans une tentative de résolution de cette conjecture on a observé que la transversalité par rapport à un feuilletage est une condition stable, cependant ce n'est pas une condition générique. Donc, en voulant imiter la preuve de Trotman on ne pourra pas obtenir cette généralisation. Néanmoins, on donne ici une preuve de cette conjecture. Ce résultat peut être résumé en disant que les (a_f)-défauts dans une stratification peuvent être détectés en perturbant les applications transverses au feuilletage induit par f. Certaines techniques pour détecter (a_f)-défauts sont aussi données vers la fin. / Trotman in 1979 proved that real smooth stratifications which satisfy the condition of $(a)$-regularity are precisely those stratifications for which transversality to the strata of smooth mappings is a stable condition in the strong topology. This was a surprising result since $(t)$-regularity seemed to be more appropriate for stability of transversality, a mistake that was made in several articles before this result of Trotman. Our first result is an analogue of this result of Trotman for the weak topology.Trotman asked more than ten years ago whether a similar result holds for complex analytic stratifications. We will give an analogue of Trotman's result in the complex setting using Forstneriv c's notion of Oka manifolds and show that the result is not true in general by giving counterexamples.In his Ph.D. thesis Trotman conjectured a generalization of his result for Thom $(a_f)$-regular stratifications. In an attempt to prove this conjecture we noticed that while transversality to a foliation is a stable condition, it is not generic in general. Thus, mimicking the proof of the result of Trotman would not suffice to obtain this generalization. Nevertheless, we will present a proof of this conjecture in this work. This result can be summarized by saying that Thom $(a_f)$-faults in a stratification can be detected by perturbation of maps transverse to the foliation induced by $f$. Some other techniques of detecting $(a_f)$-faults are also given towards the end.

[pt] Seja (Es)t o espaço de germes na origem de funções suaves entre os espaços euclidianos de dimensões e t. Nesta dissertação, apresentamos a parte da Teoria de Mather que descreve hipóteses suficientes para k-determinação em (Es)t sob duas ações diferentes, induzindo as chamadas R- e K-equivalências. Um germe é k-determinado se é equivalente a qualquer perturbação que deixa invariante seu k-jato, os termos de ordem até k de sua expansão de Taylor na origem. A R-equivalência consiste em compor germes com germes de difeomorfismos µa direita. A K- equivalência é mais difícil de descrever. / [en] Let (Es)t be the space of smooth map-germs at the origin between Euclidian spaces of dimensions s and t. In this dissertation, we present a section of Mather theory describing su±cient conditions for k- determinacy of this map-germs under two different actions, inducing the so called R- e K- equivalences. A map-germ is k-determined if it is equivalent to any perturbation that leaves invariant its k-jet, i.e., the terms up to order k of its Taylor expansion at the origin. The R-equivalence consists of compositions with germs of diffeomorphisms to the right. The K- equivalence is harder to describe.

A proposta deste trabalho é apresentar resultados para o estudo sistemático de sistemas dinâmicos reversíveis-equivariantes, ou seja, em presença simultânea de simetrias e antisimetrias. Este é o caso em que o domínio e as equações que regem o sistema são invariantes pela ação de um grupo de Lie compacto Γ formado pelas simetrias e anti-simetrias do problema. Apresentamos métodos de teoria de Singularidades e teoria de invariantes para classificar bifurcações a um parâmetro de pontos de equilíbrio destes sistemas. Para isso, separamos o estudo de aplicações Γ-reversíveis-equivariantes em dois casos: auto-dual e não auto-dual. No primeiro caso, a existência de um isomorfismo linear Γ-reversível-equivariante estabelece uma correspondência entre a classificação de problemas Γ-reversíveis-equivariantes e a classificação de problemas Γ-equivariantes associados, para os quais todos os elementos de Γ agem como simetria. Os resultados obtidos para o caso não auto-dual se baseiam em teoria de invariantes e envolvem técnicas algébricas que reduzem a análise ao caso polinomial invariante. Dois algoritmos simbólicos são estabelecidos para o cálculo de geradores para o módulo das funções anti-invariantes e para o módulo das aplicações reversíveis-equivariantes. / The purpose of this work is to present results for the sistematic study of reversible-equivariant dynamical systems, namely in simultaneous presence of symmetries and reversing simmetries. This is the case when the domain and the equations modeling the system are invariant under the action of a compact Lie group Γ formed by the symmetries and reversing symmetries of the problem. We present methods in Singularities and Invariant theory to classify oneparameter steady-state bifurcations of these systems. For that, we split the study of the ¡¡reversible-equivariant mapping into two cases: self-dual and non self-dual. In the first case, the existence of a Γ-reversible-equivariant linear isomorphism establishes a one-toone correspondence between the classification of Γ-reversible-equivariant problems and the classification of the associated Γ-equivariant problems, for which all elements in Γ act as symmetries. The results obtained for the non self-dual case are based on Invariant theory and involve algebraic techniques that reduce the analysis to the invariant polynomial case. Two symbolic algorithms are established for the computation of generators for the module of anti-invariant functions and for the module of reversible-equivariant mappings.

Manipuladores robóticos com cinemática paralela apresentam alta rigidez, alta relação carga/peso próprio e boa precisão quando comparados a manipuladores de cinemática serial. No entanto, a região de trabalho dos manipuladores paralelos é limitada devido à presença de singularidade. Com o objetivo de aumentar a região de trabalho, redundâncias cinemáticas podem ser introduzidas nas cadeias cinemáticas. Devido à sua arquitetura paralela, a incerteza nos parâmetros geométricos pode ter grande influência no comportamento cinemático e no desempenho dinâmico. O estudo do impacto dessas incertezas quando redundâncias são introduzidas em uma manipulador robótico planar de cinemática paralela é o objetivo desse trabalho. Distribuições normais foram adotadas para a avaliação do comprimento dos elos. O impacto dessas variações foi avaliado numericamente através da comparação de resultados da simulação de trajetórias para os diferentes manipuladores robóticos. Além disso, verificou-se o impacto dessas variações nas regiões de singularidades dos sistemas robóticos. Essas avaliações numéricas foram realizadas para o manipulador robótico 3(P)RRR. Este manipulador consiste de 3 cadeias cinemáticas em paralelo. Cada cadeia apresenta uma junta prismática ativa (P), uma junta de revolução ativa (R) e duas juntas de revolução passivas (RR). Através desse trabalho, uma metodologia de avaliação do impacto de incerteza geométricas em manipuladores robóticos paralelos com redundância de atuação foi proposta e investigada. / Parallel kinematic manipulators present higher rigidity, better load capacity and improved accuracy when compared to serial kinematic manipulators. However, the workspace of parallel kinematic manipulator is usually limited due to the presence of singularity regions. In order to enlarge the workspace, kinematic redundancy can be introduced in the kinematic chains. Due to its parallel architecture, the uncertainty and variability of some geometric parameters may have great influence on its kinematic behavior and dynamic performance. The impact of these variabilities when redundancies are considered should also be verified. The aim of this study is to evaluate some geometric uncertainties in the links\' dimensions of a planar parallel robot manipulator with kinematic redundancy. Normal distributions are adopted for evaluating the variability of length of the links. The impact of these changes was evaluated numerically by comparing the results obtained by simulating trajectories for different robotic manipulators. In addition, the impact of these variabilities in the singularity regions is also assessed. These numerical evaluations have been performed for the redundant manipulator 3(P)RRR. This manipulator consists of three kinematic chains in parallel. Each chain has an active prismatic joint (P), an active revolute joint (R) and two passive revolute joints (RR). Through this work, a methodology for assessing the impact of geometric uncertainty in parallel robotic manipulators with kinematic redundancy has been proposed and investigated.

"Optical vortices are singularities in phase fronts of laser beams. They are characterized by a dark core whose size may dramatically affect their behavior upon propagation. Previously, only large-core vortices have been extensively studied. The object of the research presented in this dissertation was to explore ways of generating small-core optical vortices (also called optical vortex filaments), and to examine their propagation using analytical, numerical and experimental methods. Computer-generated holography enabled us to create arbitrary distributions of optical vortex filaments for experimental exploration. We used hydrodynamic paradigms to develop an heuristic model which described the dependence of vortex motion on other vortices and the background beam, both qualitatively and quantitatively. We predicted that pair of optical vortex filaments will rotate with angular rates inversely proportional to their separation distance (just like vortices in a fluid). We also reported the first experimental observation of this novel fluid-like effect. It was found, however, that upon propagation in linear media, the fluid-like rotation was not sustained owing to the overlap of diffracting vortex cores. Further numerical studies and experiments showed that rotation angle may be enhanced in nonlinear self-defocusing media. The results presented in this thesis offer us a better understanding of dynamics of propagating vortices which may result in applications in optical switching, manipulation of micro-particles and optical limiting."

Soit Γ un sous-groupe fini de SU2(ℂ). Alors le quotient ℂ2/Γ peut être plongé dans ℂ3 sous la forme d'une surface munie d'une singularité isolée. Le quotient ℂ2/Γ est appelé singularité de Klein, d'après F. Klein qui fut le premier à les décrire en 1884. A travers leurs résolutions minimales, ces singularités ont un lien étroit avec les diagrammes de Dynkin simplement lacés de types Ar, Dr et Er. Dans les années 1970, E. Brieskorn et P. Slodowy ont tiré profit de cette connection pour décrire les résolutions et les déformations de ces singularités à l'aide de la théorie de Lie. En 1998 P. Slodowy et H. Cassens ont construit les déformations semiuniverselles des ℂ2/Γ à l'aide de la théorie des carquois ainsi que des travaux de P.B. Kronheimer en géométrie symplectique datant de 1989. En théorie de Lie, la classification des algèbres de Lie simples divisent ces dernières en deux classes: les algèbres de Lie de types Ar, Dr et Er qui sont simplement lacées, et celles de types Br, Cr, F4 et G2 appelées non-homogènes. A l'aide d'un second sous-groupe fini Γ' de SU2(ℂ) tel que Γ ⊲ Γ', P. Slodowy a étendu en 1978 la notion de singularité de Klein aux algèbres de Lie non-homogènes en ajoutant à ℂ2/Γ le groupe d'automorphismes Ω= Γ'/Γ du diagramme de Dynkin associé à la singularité. L'objectif de cette thèse est de généraliser la construction de H. Cassens et P. Slodowy à ces singularités de types Br, Cr, F4 et G2. Il en résultera des constructions explicites des déformations semiuniverselles de types inhomogènes sur les fibres desquelles le groupe Ω agit. Le passage au quotient d'une telle application révèle alors une déformation d'une singularité de type ℂ2/Γ' / Let Γ be a finite subgroup of SU2(ℂ). Then the quotient ℂ2/Γ can be embedded in ℂ3 as a surface with an isolated singularity. The quotient ℂ2/Γ is called a Kleinian singularity, after F. Klein who studied them first in 1884. Through their minimal resolutions, these singularities have a deep connection with simply-laced Dynkin diagrams of types Ar, Dr and Er. In the 1970's E. Brieskorn and P. Slodowy took advantage of this connection to describe the resolutions and deformations of these singularities in terms of Lie theory. In 1998 P. Slodowy and H. Cassens constructed the semiuniversal deformations of the Kleinian singularities using quiver theory and work from 1989 by P.B. Kronheimer on symplectic geometry. In Lie theory, the classification of simple Lie algebras allows for a separation in two classes: those simply-laced of types Ar, Dr and Er, and those of types Br, Cr, F4 and G2 called inhomogeneous. With the use of a second finite subgroup Γ’ of SU2(ℂ) such that Γ ⊲ Γ’, P. Slodowy extended in 1978 the definition of a Kleinian singularity to the inhomogeneous types by adding to ℂ2/Γ the group of automorphisms Ω= Γ’/Γ of the Dynkin diagram associated to the singularity. The purpose of this thesis is to generalize H. Cassens' and P. Slodowy's construction to the singularities of types Br, Cr, F4 and G2. It will lead to explicit semiuniversal deformations of inhomogeneous types on the fibers of which the group Ω acts. By quotienting such a map we obtain a deformation of a singularity ℂ2/Γ’