Esta tese trata da contextualização da geometria descritiva como sistema de representação na arquitetura. Desenvolve-se a partir da desconstrução da Géométrie descriptive de Gaspard Monge, publicada em 1799, acompanhando a exposição de seu autor desde o conteúdo da capa até a sua última página, de onde são destacadas partes a serem estudadas entre os textos, desenhos e a própria apresentação da obra. Desconstruir a teoria mongeana apresenta-se relevante neste estudo por investigar sobre as lições dadas por Monge em 1799, que coexistem até o momento com outras representações, entretanto sem um questionamento e entendimento epistemológico. Considerando que existem distorções na exposição original das lições mongeanas em obras subseqüentes à Géométrie descriptive e que conceitos de representação determinam limites de compreensão do espaço que implicam na própria arquitetura, este estudo dá abertura de resignificação à teoria original de Monge no ensino de arquitetura. / This thesis is about the descriptive geometry contextualization as an Architecture representation system. It was developed after Gaspard Monge’s Géométrie descriptive deconstruction, published in 1799, accompanying its author exposition since its cover content until its last page, form where parts are detached to be studied among the texts, draws and the own handiwork presentation. To deconstruct Monge’s theory is relevantly presented in this study for investigate Monge’s lessons taught in 1799 that coexists until this moment with different representations, without any question and epistemology understanding. Considering that there are distortions in the original Monge lessons exposition in Géométrie descriptive following handiwork and that its representation concepts determinate limits to the space comprehension that imply the Architecture itself, this study gives opening to Monge’s original theory resignification in the Architecture teaching.

Esta tese trata da contextualização da geometria descritiva como sistema de representação na arquitetura. Desenvolve-se a partir da desconstrução da Géométrie descriptive de Gaspard Monge, publicada em 1799, acompanhando a exposição de seu autor desde o conteúdo da capa até a sua última página, de onde são destacadas partes a serem estudadas entre os textos, desenhos e a própria apresentação da obra. Desconstruir a teoria mongeana apresenta-se relevante neste estudo por investigar sobre as lições dadas por Monge em 1799, que coexistem até o momento com outras representações, entretanto sem um questionamento e entendimento epistemológico. Considerando que existem distorções na exposição original das lições mongeanas em obras subseqüentes à Géométrie descriptive e que conceitos de representação determinam limites de compreensão do espaço que implicam na própria arquitetura, este estudo dá abertura de resignificação à teoria original de Monge no ensino de arquitetura. / This thesis is about the descriptive geometry contextualization as an Architecture representation system. It was developed after Gaspard Monge’s Géométrie descriptive deconstruction, published in 1799, accompanying its author exposition since its cover content until its last page, form where parts are detached to be studied among the texts, draws and the own handiwork presentation. To deconstruct Monge’s theory is relevantly presented in this study for investigate Monge’s lessons taught in 1799 that coexists until this moment with different representations, without any question and epistemology understanding. Considering that there are distortions in the original Monge lessons exposition in Géométrie descriptive following handiwork and that its representation concepts determinate limits to the space comprehension that imply the Architecture itself, this study gives opening to Monge’s original theory resignification in the Architecture teaching.

Esta tese trata da contextualização da geometria descritiva como sistema de representação na arquitetura. Desenvolve-se a partir da desconstrução da Géométrie descriptive de Gaspard Monge, publicada em 1799, acompanhando a exposição de seu autor desde o conteúdo da capa até a sua última página, de onde são destacadas partes a serem estudadas entre os textos, desenhos e a própria apresentação da obra. Desconstruir a teoria mongeana apresenta-se relevante neste estudo por investigar sobre as lições dadas por Monge em 1799, que coexistem até o momento com outras representações, entretanto sem um questionamento e entendimento epistemológico. Considerando que existem distorções na exposição original das lições mongeanas em obras subseqüentes à Géométrie descriptive e que conceitos de representação determinam limites de compreensão do espaço que implicam na própria arquitetura, este estudo dá abertura de resignificação à teoria original de Monge no ensino de arquitetura. / This thesis is about the descriptive geometry contextualization as an Architecture representation system. It was developed after Gaspard Monge’s Géométrie descriptive deconstruction, published in 1799, accompanying its author exposition since its cover content until its last page, form where parts are detached to be studied among the texts, draws and the own handiwork presentation. To deconstruct Monge’s theory is relevantly presented in this study for investigate Monge’s lessons taught in 1799 that coexists until this moment with different representations, without any question and epistemology understanding. Considering that there are distortions in the original Monge lessons exposition in Géométrie descriptive following handiwork and that its representation concepts determinate limits to the space comprehension that imply the Architecture itself, this study gives opening to Monge’s original theory resignification in the Architecture teaching.

Dans cette thèse nous nous intéressons aux flots de Monge-Ampère complexes, à leurs généralisations et à leurs applications géométriques sur les variétés hermitiennes compactes. Dans les deux premiers chapitres, nous prouvons qu'un flot de Monge-Ampère complexe sur une variété hermitienne compacte peut être exécuté à partir d'une condition initiale arbitraire avec un nombre Lelong nul en tous points. En utilisant cette propriété, nous con- firmons une conjecture de Tosatti-Weinkove: le flot de Chern-Ricci effectue une contraction chirurgicale canonique. Enfin, nous étudions une généralisation du flot de Chern-Ricci sur des variétés hermitiennes compactes, le flot de Chern-Ricci tordu. Cette partie a donné lieu à deux publications indépendantes. Dans le troisième chapitre, une notion de C -sous-solution parabolique est introduite pour les équations paraboliques, étendant la théorie des C -sous-solutions développée récem- ment par B. Guan et plus spécifiquement G. Székelyhidi pour les équations elliptiques. La théorie parabolique qui en résulte fournit une approche unifiée et pratique pour l'étude de nombreux flots géométriques. Il s'agit ici d'une collaboration avec Duong H. Phong (Université Columbia ) Dans le quatrième chapitre, une approche de viscosité est introduite pour le problème de Dirichlet associé aux équations complexes de type hessienne sur les domaines de Cn. Les arguments sont modélisés sur la théorie des solutions de viscosité pour les équations réelles de type hessienne développées par Trudinger. En conséquence, nous résolvons le problème de Dirichlet pour les équations de quotient de hessiennes et lagrangiennes spéciales. Nous établissons également des résultats de régularité de base pour les solutions. Il s'agit ici d'une collaboration avec Sl-awomir Dinew (Université Jagellonne) et Hoang-Son Do (Institut de Mathématiques de Hanoi). / In this thesis we study the complex Monge-Ampère flows, and their generalizations and geometric applications on compact Hermitian manifods. In the first two chapters, we prove that a general complex Monge-Ampère flow on a compact Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti- Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow. This part gave rise to two independent publications. In the third chapter, a notion of parabolic C -subsolution is introduced for parabolic non-linear equations, extending the theory of C -subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows. This part is a joint work with Duong H. Phong (Columbia University) In the fourth chapter, a viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in Cn. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions. This part is a joint work with Sl-awomir Dinew (Jagiellonian University) and Hoang-Son Do (Hanoi Institute of Mathematics).

L'objet de cette thèse est d'étudier la théorie du transport optimal sur un espace de Wiener abstrait. Les résultats qui se trouvent dans quatre principales parties, portent :Sur la convexité de l'entropie relative. On prolongera des résultats connus en dimension finie, sur l'espace de Wiener muni d'une norme uniforme, à savoir que l'entropie relative est (au moins faiblement) 1-convexe le long des géodésiques induites par un transport optimal sur l'espace de Wiener.Sur les mesures à densité logarithmiquement concaves. Le premier des résultats importants consiste à montrer qu'une inégalité de type Harnack est vraie pour le semi-groupe induit par une telle mesure sur l'espace de Wiener. Le second des résultats obtenus nous fournit une inégalité en dimension finie (mais indépendante de la dimension), contrôlant la différence de deux applications de transport optimal.Sur le problème de Monge. On s'intéressera au problème de Monge sur l'espace de Wiener, muni de plusieurs normes : des normes à valeurs finies, ou encore la pseudo-norme de Cameron-Martin.Sur l'équation de Monge-Ampère. Grâce aux inégalités obtenues précédemment, nous serons en mesure de construire des solutions fortes de l'équation de Monge-Ampère (induite par le coût quadratique) sur l'espace de Wiener, sous de faibles hypothèses sur les densités des mesures considérées

Le résultat principal de cette thèse est l'obtention d'une condition nécessaire et suffisante pour l'existence d'une métrique de Kähler-Einstein sur une compactification bi-équivariante lisse et Fano d'un groupe complexe réductif connexe. Ces variétés comprennent les variétés toriques et les compactifications magnifiques de groupes semisimples adjoints.Dans la première partie de ce travail sont développés les outils nécessaires à l'étude de l'existence de métriques de Kähler-Einstein sur ces variétés. Nous calculons en particulier la Hessienne complexe d'une fonction $Ktimes K$-invariante sur la complexification d'un groupe compact $K$. Nous associonségalement, à toute métrique invariante à courbure positive sur un fibré linéarisé ample sur une compactification de groupe, une fonction convexe dont le comportement asymptotique est prescrit. Ceci est utilisé une première fois pour obtenir une formule pour l'invariant alpha d'un fibré en droite ample sur une compactification de groupe Fano. Cette formule est obtenue par le calcul des seuils log canoniques des métriques hermitiennes invariantes à courbure positive, et induit, dans le cas particulier des variétés toriques, un résultat obtenu auparavant, figurant dans l'article par ailleurs inclus en appendice de la thèse.Nous prouvons ensuite le résultat principal en obtenant des estimées $C^0$ le long de la méthode de continuité, en se ramenant à une équation de Monge-Ampèreréelle sur un cône. La condition obtenue est que le barycentre du polytope associé à la compactification de groupe, par rapport à la mesure de Duistermaat-Heckman, doit être dans une zone particulière du polytope. Cette condition peut être vérifiée sur les exemples, donne de nouveaux exemples de variétés deKähler-Einstein Fano, et donne aussi un exemple qui n'admet aucun soliton de Kähler-Ricci. Nous calculons de plus la plus grande borne inférieure de Ricci lorsqu'il n'y a pas de métrique de Kähler-Einstein. / The main result of this work is a necessary and sufficient condition for the existence of a Kähler-Einstein metric on a smooth and Fano bi-equivariant compactification of a complex connected reductive group. Examples of such varieties include wonderful compactifications of adjoint semisimple groups.The tools needed to study the existence of Kähler-Einstein metrics on these varieties are developed in the first part of the work, including a computation of the complex Hessian of a $Ktimes K$-invariant function on the complexification of a compact group $K$. Another step is to associate to any non-negatively curved invariant hermitian metric on an ample linearized line bundle on a group compactification a convex function with prescribed asymptotic behavior. This is used a first time to derive a formula for the alpha invariantof an ample line bundle on a Fano group compactification. This formula is obtained through the computation of the log canonical thresholds of any non-negatively curved invariant hermitian metric, and gives the sameresult, for toric manifolds, as the one we obtained before, in an article that is included in this thesis as an appendix.Then we prove the main result by obtaining $C^0$ estimates along the continuity method, using the tools developed to reduce to a real Monge-Ampère equation on a cone. The condition obtained is that the barycenter of the polytope associated to the group compactification, with respect to the Duistermaat-Heckman measure, lies in a certain zone in the polytope. This condition can be checked on examples, gives new examples of Fano Kähler-Einstein manifolds, and also gives an example that admits no Kähler-Ricci solitons. We also compute the greatest Ricci lower bound when there are no Kähler-Einstein metrics.

Cette thèse porte sur la théorie du pluripotentiel et des problèmes d'équidistribution. Elle consiste en 4 chapitres. Le premier chapitre se consarce à l'étude de la régularité de la solution de l'équation de Monge-Ampère complexe sur une variété kahlérienne compacte X. Plus précisement, à l'aide des outils de la géométrie Cauchy-Riemann, on montre que la dernière équation possède une (unique) solution holdérienne pour une large classe géométrique de mesures de probabilités supportées par des sous-variétés réelles de X. Dans le chapitre 2, on étudie l'intersection des courants positifs fermés de grand bidegré. On y prouve que le produit extérieur de deux courants positifs fermés dont l'un possède un superpotentiel continu est positif fermé. Ceci généralise un résultat classique pour les courants de bidegré (1,1). Les deux chapitres suivants sont des applications de la théorie du pluripotentiel à des problèmes d'équidistribution. Dans le chapitre 3, on donne une vitesse explicite de convergence pour l'équidistribution des points de Fekete dans un compact K de l'espace euclidien à bord lisse par morceaux vers la mesure d'équilibre de K. Ici, les points de Fekete sont des bons points dans le problème d'interpolation d'une fonction continue sur K par des polynômes. Un tel contrôle de vitesse est crucial en pratique qu'on utilise les points de Fekete. La thèse se termine par le chapitre 4 où on prouve un analogue de la loi de Weyl pour les résonances d'un opérateur de Schodinger générique sur l'espace euclidien de dimension impair. Les résonances sont des objets centraux dans l'étude des opérateurs de Schrodinger. Elles jouent un rôle similaire à celui des valeurs propres dans le cadre compact. / This thesis concerns the pluripotential theory and equidistribution problems. It consists of 4 chapters. The first chapter is dedicated to the study of the regularity of the solution of the complexe Monge-Ampère equation on a compact Kahler manifold X. More precisely, using tools from the Cauchy-Riemann geometry, we prove that the last equation possesses a unique Holder continuous solution for a large geometric class of probability measures supported on real submanifolds of X. In the chapter 2, we study the intersecton of positive closed currents of higher bidegree. We prove there that the wedge product of two such currents one of which has a continuous superpotential est closed and positive. This property generalises a classical result for currents of bidegree (1,1). The next two chapters are applications of the pluripotential theory to equidistribution problems. In the chapter 3, we give an explicit speed of convergence for the equidistribution of Fekete's points in a compact subset K of the Euclidean space with piecewise smooth boundary toward the equilibrium measure of K. Here, the Fekete's points are good points for the interpolation problem of continuous functions by polynomials on K. A such control of speed is crucial in practice when ones use Fekete's points. The thesis is ended by the chapter 4 where we prove an analogue of Weyl's law for the resonances of a generic Schrodinger operator on an Euclidean space of odd dimension. The resonances are central objects in the research of Schrodinger operators. They play a similar role to that of eigenvalues in the compact setting.

An aim of this diploma thesis is to create interactive descriptive geometry coursebook. As a basis for coursebook creation was used "Present State of Descriptive Geometry Education on High Schools Survey" which was proceeded via questionnaire. Before the creation of coursebook itself were made analyses of accessible textbooks, geometric softwares in use and possible environments for coursebook content presentation. A result is integrated part of subject matter taught on high schools complemented by solved examples and interactive applets created in software GeoGebra. The coursebook content is presented in a form of HTML sites on a public website http://www.sadsam.cz/dg/ and also as an off- line version on DVD disc enclosed.

Os Trapistas no Brasil é o objeto do nosso trabalho. A primeira experiência trapista no Brasil ocorreu no início do século XX no Vale do Paraíba Paulista. Monges Trapistas franceses deram início ao Mosteiro Nossa Senhora de Maristela, na cidade de Tremembé, estado de São Paulo. Passou a ser também a primeira Trapa da América do Sul. Com esse mosteiro, houve o início da vida e da tradição monástica cisterciense no Brasil. No Vale do Paraíba, revolucionaram a agricultura com modernas técnicas agrícolas, principalmente na rizicultura, e prestaram também relevantes serviços no campo socioeclesial. Posteriormente, com o regresso destes para Europa, houve, em 1977, a fundação do Mosteiro Trapista Nossa Senhora do Mundo, no estado do Paraná. A relação entre o primeiro Mosteiro Trapista no Brasil, fundado no início do século XX, e o segundo Mosteiro, Nossa Senhora do Novo Mundo, iniciado em 1977, mostra que o passado e presente caminham nessa relação: a ponte entre história e memória. O passado e a sua relação com ele são elementos centrais da identidade da Ordem Cisterciense da Estrita Observância / The Trappists in Brazil is the aim o four assignment. The first Trappist experience in Brazil happened in the beginning of the XX century in the region of Paraíba Valley. French Trappist monks started the Monastery of Our Lady Maristela, in Tremembé city state of São Paulo. It also became the first Trapa of South America. With this monastery, we had the beginning of Cistercian monastic life and tradition in Brazil. In the region of Paraíba Valley, modern agricultural techniques were revolutionized, especially in rice growing; they also provided relevant relevant services in the church social field . Later, with the return of these to Europe in 1977, we had the foundation of the Trappist Monastery of Our Lady of the New World, in the state of Paraná. The relationship between the first Trappist Monastery in Brazil, founded in the early twentieth century, and the second Trappist Monastery of Our Lady of the New World, which started in 1977, shows that the past and present walk together, the bridge between history and memory. The past and its relationship with it are core elements to the identity of the Cistercian Order of Strict Observance

In the context of moment maps and diffeomorphisms of Kähler manifolds, Donaldson introduced a fully nonlinear Monge-Ampère type equation. Among the conjectures he made about this equation is that the existence of solutions is equivalent to a positivity condition on the initial data. Weinkove later affirmed Donaldson's conjecture using a gradient flow for the equation in the space of Kähler potentials of the initial data. The topic of this thesis is the case when the initial data is merely semipositive and the domain is a closed Kähler surface. Regularity techniques for degenerate Monge-Ampère equations, specifically those coming from pluripotential theory, are used to prove the existence of a bounded, unique, weak solution. With the aid of a Nakai criterion, due to Lamari and Buchdahl, it is shown that this solution is smooth away from some curves of negative self-intersection.