Este trabalho é o resultado de uma pesquisa de vários anos sobre as características espaciais da modernização no Município de Villavieja, região do Alto Magdalena, norte do Departamento do Huila, na Colômbia. Neste município, caracterizado pelo predomínio de grandes áreas de paisagem semiárida, tradicionalmente identificadas como o Deserto de La Tatacoa, a modernização foi identificada como um processo pelo qual a sociabilidade tende para a abstração, representada na monetarização como condição para a reprodução da vida de seus moradores. Identificamos um percurso da modernização associado à acumulação originária global que, para a área, surge na época colonial, a partir do século XVII, depois da aniquilação da população indígena originária, por parte dos conquistadores espanhóis; da redução dos sobreviventes em resguardos e da concentração da terra em mãos dos capitães desta empreitada. Conforma-se, assim, uma elite de fazendeiros criadores de gado para abastecer os assentamentos urbanos como Santa Fé e Popayán e uma massa de trabalhadores sem terra, que se tornarão agregados das grandes fazendas. Conforme as determinações do capital modificam o espaço a escala global, a região vai passando por diversos surtos econômicos, como a produção de fumo e anil, chegando, a partir da metade do século XX, à produção de bens agrícolas para o mercado interno, dentre os quais se destacam o arroz e o algodão. A produção se fez nas áreas de várzea de rios e córregos e ocupou os antigos agregados das fazendas, que, depois de uma intensa luta do movimento camponês foram se tornando pequenos proprietários em extensões de terras compradas pelo governo para fins de Reforma Agrária. A condição para a efetivação desta forma de reprodução da vida, entretanto, implicou a regularização fundiária, a aceitação do crédito, dos insumos agroindustriais, da organização cooperativa e das técnicas produzidas pela grande indústria. De modo geral, visava-se a subordinação dos pequenos proprietários camponeses ao capital industrial e financeiro. Os resultados deste processo dizem respeito à crise deste modelo pela concorrência com produtos importados, a partir da década de 1990 e da dificuldade dos camponeses em submeter seu tempo aos ritmos da produção industrial. Procuramos, a partir do uso do método dialético, as possibilidades de existência de outra forma de consciência, pautada na busca das diferenças acháveis na escala do povoado e no nível inferior de sociabilidade. Eles serão identificados como espaços diferenciais, seguindo a obra de Henri Lefebvre. / This paper is the result of a research of many years about the spatial characteristics of modernization at Villavieja Municipality, located in Upper Magdalena Region, at the north of Huila Department, in Colombia. In this area, characterized for the domain of wide semi-arid areas, traditionally referred as La Tatacoa Desert, modernization was identified as a process by which sociability tend to abstraction, represented by generalization of money as condition to reproduce life of their inhabitants. It is traced a path associated with the global primitive accumulation, that, at the area appears during colonial times, specifically since the beginnings of the XVII century, after the Spanish conquerors annihilation of indigenous people and the settlement of survivors in resguardos which generated land concentration by war leaders. Thus, it is conformed an elite of big cattle growers landowners who produced for the consumption of little villas like Santa Fé and Popayán, and a huge mass of landless workers which turn out to be aggregate peasants from the big farms. As far as capital determinations modify space at the global scale, this region passes trough different economic booms, like the production of tobacco and indigo, getting to, since half of 20th century, the production of agricultural goods for the inner market, specially rice and cotton. The production of this goods was developed in the dale of Magdalena river and its tributaries and was made by aggregate peasants of big farms who, after an intense fight for land from the peasant´s movement, turned out to be little land owners in areas of the big farms bought by the government for agrarian reform purposes. The condition for the effectiveness of this form of life reproduction, however, implied land regularization and the acceptance of credit, industrial inputs, cooperative organization and big industry techniques. By all means, it was aimed to subordinate little land owners to industrial and finance capital. The results of this process show the crisis of this model due to competition with imported products, since the 1990´s and peasants difficulty to undergo within industrial production rhythms. This paper seeks to, according to dialectical method; find the possibility of another conscience form, based upon the differences which can be found at the level of little villages and at the inferior level of sociability. They will be identified as differential spaces, according to Henri Lefebvre´s work.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the “intersection” of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the “intersection” of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action.

Este trabalho é o resultado de uma pesquisa de vários anos sobre as características espaciais da modernização no Município de Villavieja, região do Alto Magdalena, norte do Departamento do Huila, na Colômbia. Neste município, caracterizado pelo predomínio de grandes áreas de paisagem semiárida, tradicionalmente identificadas como o Deserto de La Tatacoa, a modernização foi identificada como um processo pelo qual a sociabilidade tende para a abstração, representada na monetarização como condição para a reprodução da vida de seus moradores. Identificamos um percurso da modernização associado à acumulação originária global que, para a área, surge na época colonial, a partir do século XVII, depois da aniquilação da população indígena originária, por parte dos conquistadores espanhóis; da redução dos sobreviventes em resguardos e da concentração da terra em mãos dos capitães desta empreitada. Conforma-se, assim, uma elite de fazendeiros criadores de gado para abastecer os assentamentos urbanos como Santa Fé e Popayán e uma massa de trabalhadores sem terra, que se tornarão agregados das grandes fazendas. Conforme as determinações do capital modificam o espaço a escala global, a região vai passando por diversos surtos econômicos, como a produção de fumo e anil, chegando, a partir da metade do século XX, à produção de bens agrícolas para o mercado interno, dentre os quais se destacam o arroz e o algodão. A produção se fez nas áreas de várzea de rios e córregos e ocupou os antigos agregados das fazendas, que, depois de uma intensa luta do movimento camponês foram se tornando pequenos proprietários em extensões de terras compradas pelo governo para fins de Reforma Agrária. A condição para a efetivação desta forma de reprodução da vida, entretanto, implicou a regularização fundiária, a aceitação do crédito, dos insumos agroindustriais, da organização cooperativa e das técnicas produzidas pela grande indústria. De modo geral, visava-se a subordinação dos pequenos proprietários camponeses ao capital industrial e financeiro. Os resultados deste processo dizem respeito à crise deste modelo pela concorrência com produtos importados, a partir da década de 1990 e da dificuldade dos camponeses em submeter seu tempo aos ritmos da produção industrial. Procuramos, a partir do uso do método dialético, as possibilidades de existência de outra forma de consciência, pautada na busca das diferenças acháveis na escala do povoado e no nível inferior de sociabilidade. Eles serão identificados como espaços diferenciais, seguindo a obra de Henri Lefebvre. / This paper is the result of a research of many years about the spatial characteristics of modernization at Villavieja Municipality, located in Upper Magdalena Region, at the north of Huila Department, in Colombia. In this area, characterized for the domain of wide semi-arid areas, traditionally referred as La Tatacoa Desert, modernization was identified as a process by which sociability tend to abstraction, represented by generalization of money as condition to reproduce life of their inhabitants. It is traced a path associated with the global primitive accumulation, that, at the area appears during colonial times, specifically since the beginnings of the XVII century, after the Spanish conquerors annihilation of indigenous people and the settlement of survivors in resguardos which generated land concentration by war leaders. Thus, it is conformed an elite of big cattle growers landowners who produced for the consumption of little villas like Santa Fé and Popayán, and a huge mass of landless workers which turn out to be aggregate peasants from the big farms. As far as capital determinations modify space at the global scale, this region passes trough different economic booms, like the production of tobacco and indigo, getting to, since half of 20th century, the production of agricultural goods for the inner market, specially rice and cotton. The production of this goods was developed in the dale of Magdalena river and its tributaries and was made by aggregate peasants of big farms who, after an intense fight for land from the peasant´s movement, turned out to be little land owners in areas of the big farms bought by the government for agrarian reform purposes. The condition for the effectiveness of this form of life reproduction, however, implied land regularization and the acceptance of credit, industrial inputs, cooperative organization and big industry techniques. By all means, it was aimed to subordinate little land owners to industrial and finance capital. The results of this process show the crisis of this model due to competition with imported products, since the 1990´s and peasants difficulty to undergo within industrial production rhythms. This paper seeks to, according to dialectical method; find the possibility of another conscience form, based upon the differences which can be found at the level of little villages and at the inferior level of sociability. They will be identified as differential spaces, according to Henri Lefebvre´s work.

Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.

Les systèmes intégrables toriques sont des systèmes intégrables dont toutes les composantes de l'application moment sont périodiques de même période. Il s'agit donc de variétés symplectiques munies d'actions Hamiltoniennes de tores. Au début des années 80, Atiyah-Guillemin-Sternberg ont démontré que l'image de l'application moment était un polytope convexe à face rationnelles. Peu de temps après, Delzant a démontré que dans le cas intégrable qui nous intéresse, ce polytope caractérisait entièrement le système : la variété symplectique comme l'action du tore. Le champs d'étude s'est ensuite élargi aux systèmes dits semi-toriques. Ce sont des systèmes intégrables dont toutes les composantes de l'application moment sauf une sont périodiques de même période. En outre, pour simplifier l'étude de ces systèmes, on demande que tous les points critiques du systèmes soient non-dégénérés, et sans composante hyperbolique pour la hessienne. En revanche les points critiques des systèmes semi-toriques peuvent comporter des composantes dites "foyer-foyer". Celles-ci ont une dynamique plus riche que les singularités elliptiques, mais conservent certaines propriétés qui rendent leur analyse plus aisée que les singularités hyperboliques. San Vu-Ngoc et Alvaro Pelayo ont réussi à étendre pour ces systèmes semi-toriques les résultats d'Atiyah-Guillemin-Sternberg et Delzant en dimension 2. L'objectif de cette thèse est de proposer une extension de ces résultats en dimension quelconque, à commencer par la dimension 3. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. / Semi-toric integrable systems are integrable systems whose every component of the moment map are periodic of the same period. They are symplectic manifolds endowed with a Hamiltonian torus actions. At the beginning of the 80's, Atiyah-Guillemin-Sternberg proved that the image of the moment map was a polytope with rational faces. A bit after that, Delzant showed that in the integrable case that matters to us, this polytope characterized entirely the system, that is, the symplectic manifold as well as the torus action. Next, field of study widened to semi-toric systems. They are integrable systems whose all components except one are periodic with the same period. Moreover, to simplify their study, we ask that these systems have only non-degenerate critical points without hyperbolic components. On the other hand, critical points of semi-toric systems can have so-called ''focus-focus'' components. They have a richer dynamic than elliptic singularities, but it retains some properties that makes them easier to study than hyperbolic singularities. San Vu-Ngoc and Alvaro Pelayo have managed to extend to these semi-toric systems the results of Atiyah-Guillemin-Sternberg and Delzant in dimension 2. The objective of this thesis is to propose an extension of these results to any dimension, starting with dimension 3. Techniques involved are analysis as well as symplectic geometry, and Morse theory in stratified differential spaces.