Neste trabalho estudamos a resolubilidade suave de campos vetoriais complexos suaves da forma L = L1 + iL2, em uma variedade M, com as seguintes propriedades: em cada ponto de M, os campos L1 e L2 são linearmente independentes , e seu colchete [L1, L2](x) é uma combinação linear de L1(x) e L2(x). Para tratar da resolubilidade local, nos utilizamos da teoria dos espaços Bp,k e operadores de força constante. Seguindo para a resolubilidade semiglobal, estudamos a folheação gerada por L1 e L2: mostramos que neste caso as folhas possuem estrutura de variedade complexa, o que nos permite obter um panorama bastante completo sobre o problema. Para encerrar, provamos que L é globalmente resolúvel se e somente se for semiglobalmente resolúvel e M for L-convexa; exibimos condições suficientes para que isto ocorra. / In this work we shall study the smooth solvability of smooth complex vector fields L = L1 + iL2 on a smooth manifold M, assuming the following properties: for any point of M, L1 and L2 are linearly independent and [L1,L2] is a linear combination of L1 and L2. Discussing local solvability, we shall employ the theory of Bp,k Spaces and Operators of Constant Strength. Moving on to Semi-Global Solvability, we shall study the foliation that is generated by L1 and L2: we prove that in this case the leaves are actually complex manifolds, which allow us to obtain an wide comprehension of the problem. Finally, we show that L is globally solvable if and only if it is semi-globally solvable and M is L-convex; we then exhibit sufficient conditions in order to it occur.

In this work, two particular orthogonal and conformal decompositions of the 3+1 dimensional Einstein equation and Arnowitt-Deser-Misner (ADM) formalism for general relativity are obtained. In order to do these, the 3+1 foliation of the four-dimensional spacetime, the fundamental conformal transformations and the Hamiltonian form of general relativity that leads to the ADM formalism, de

Στο πρώτο κεφάλαιο της εργασίας, παρουσιάζεται το πιο απλό παράδειγμα εμφύλλωσης και στη συνέχεια δίνεται ο ορισμός μιας εμφυλλωμένης πολλαπλότητας, υπό δύο διαφορετικές σκοπιές. Ο ορισμός συμπληρώνεται με τον σχολιασμό της τοπολογίας των φύλλων της εμφύλλωσης, δίνοντας το τοπολογικό πλαίσιο της πολλαπλότητας για τον ορισμό του κανονικού εμφυλλωμένου άτλαντα. Η εισαγωγή στη Θεωρία Εμφυλλώσεων ολοκληρώνεται με μία σειρά παραδειγμάτων εμφυλλώσεων, με επικεντρωμένο το ενδιαφέρον στην εμφύλλωση του Reeb και στην προσανατολισμένη εμφύλλωση του Seifert. Στο δεύτερο κεφάλαιο, συνδέεται η έννοια της γεωμετρικής ολοκληρωσιμότητας με την Θεωρία των Εμφυλλώσεων, μέσω του κλασικού θεωρήματος του Frobenius. Τα φύλλα της εμφύλλωσης του χώρου των φάσεων αποτελούν το γεωμετρικό πρότυπο επίλυσης δυναμικών συστημάτων, ως πρώτα ολοκληρώματα. Το κλασικό θεώρημα του Frobenius έδωσε τις αναγκαίες και ικανές συνθήκες ώστε η θεωρούμενη κατανομή να αποτελεί τον εφαπτόμενο χώρο της εμφύλλωσης. Το θεώρημα Frobenius δίνεται και αποδεικνύεται με πέντε ισοδύναμες εκδοχές. Μία από αυτές είναι η αλγεβρική εκδοχή, όπου τα πρώτα ολοκληρώματα καθορίζονται από τους γεννήτορες του ιδεώδους της εξωτερικής άλγεβρας, επιλύoντας τις εξισώσεις Pfaff. Οι παραγόμενες μορφές μέσω της εξωτερικής διαφόρισης των γεννητόρων του ιδεώδους, στην περίπτωση που ικανοποιούν τη συνθήκη ολοκληρωσιμότητας, συγκροτούν στο module των διαφορικών μορφών το διαφορικό ιδεώδες. Ακόμα, γίνεται αναφορά στο Λήμμα του Poincaré, που δίνει τις προϋποθέσεις για την ύπαρξη πρώτων ολοκληρωμάτων, στην περίπτωση απλά συνεκτικών πολλαπλοτήτων, και στην εύρεση ολοκληρωτικού παράγοντα. Στο τρίτο και τελευταίο κεφάλαιο, ως εφαρμογή στη Θεωρία Εμφυλλώσεων, αποδεικνύεται η ύπαρξη φύλλων μέσα στο σύνολο προσβασιμότητας, που καθορίζεται από το εκάστοτε σύστημα ελέγχου. Πρόκειται για το θεώρημα που δόθηκε τη δεκαετία του 70 από τον Sussmann. Ορίζοντας τη Lie άλγεβρα των κατανομών η οποία δημιουργείται από τις επαναλαμβανόμενες αγκύλες Lie. Στα πλαίσια αυτής ελέγχεται η συμπεριφορά των κατανομών, οι οποίες διαχωρίζονται σε ολοκληρώσιμες και bracket generating. Οι τελευταίες παράγουν τον εφαπτόμενο χώρο της πολλαπλότητας και αποτελούν βασική προϋπόθεση για να εφοδιαστεί η πολλαπλότητα με μια υπο-Riemannian δομή. Με αυτή τη δομή ορίζεται η υπο-Riemannian απόσταση από την οποία φτιάχνεται η βάση μιας τοπολογίας που συμπίπτει με τη φυσική τοπολογία της πολλαπλότητας. Σε αυτήν την τοπολογία ορίζονται τα φύλλα του συνόλου προσβασιμότητας. Επιπλέον, δίνεται μια απάντηση και στο πρόβλημα της ελεγξιμότητας, που διαπραγματεύεται η Θεωρία Ελέγχου. Τέλος, γίνεται αναφορά στις γεωδαισιακές εξισώσεις, όπως αυτές ορίζονται στο συνεφαπτόμενο ινώδες των τετραγωνικών μορφών, με χαρακτηριστικό παράδειγμα τις γεωδαισιακές που προκύπτουν από την ομάδα του Heisenberg. / --

Aquest treball es refereix a foliacions i camps de Killing. Les foliacions, com a part individualitzada dins la Geometria Diferencial, pot considerarse que neixen a partir de la teoria dels sistemes dinàmics en varietats i de la teoria de connexions en fibrats desenvolupada per Ch. Ehresmann i G. Reeb entre 1940 i 1960. Resultats d'aquesta disciplina s'utilitzen en camps com ara sistemes d'equacions diferencials, termodinàmica, teoria del control…

Depuis 1980, il est un problème ouvert de donner des formules cohomologiques pour l'indice basique d'un opérateur différentiel basique transversalement elliptique sur un fibré vectoriel au dessus d'une variété feuilletée. Dans les années 1990, El Kacimi-Alaoui a proposé d'utiliser la théorie de Molino pour étudier cette indice. Molino a montré qu'à tout feuilletage Riemannien transversalement orienté, nous pouvons associer une variété, appelée variété basique, qui est munie d'une action du groupe orthogonal, El Kacimi-Alaoui a montré comment associer à l'opérateur basique transversalement elliptique un opérateur sur un fibré vectoriel, appelé fibré utile, au dessus de la variété basique.L'idée est d'obtenir la formule cohomologique espérée à partir des résultats sur l'opérateur sur le fibré utile. Cette thèse est une première étape dans cette direction. Lorsque le feuilletage Riemannien est de Killing, Goertsches et Töben ont remarqué qu'il existe un isomorphisme cohomologique naturel entre la cohomologie basique équivariante du feuilletage de Killing et la cohomologie équivariante de la variété basique.Le résultat principal de cette thèse est de donner une réalisation géométrique de l'isomorphisme cohomologique ci-dessus à travers les caractères de Chern sous certaine Hypothèse. / From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversally oriented Riemannien foliation, we can associate a manifold, called basique manifold, which is équiped with an action of orthogonal group, El Kacimi-Alaoui has shown how to associate a transversally elliptic basic differential operator an operator on a vector bundle, called useful bundle, over the basique manifold.The idea is to obtain the desired cohomologic formula from résultats about the operator on the useful bundle. This thesis is a first step in this direction. While the Riemannien foliation is Killing, Goertsches et Töben have remarked that there exists a naturel cohomologic isomorphism between the equivariant basique cohomology of the Killing foliation and the equivariant cohomology of the basique manifold.The principal result of this thesis is the geometric realisation of the cohomologic isomorphism by Chern characters under some hypothèses.

Neste trabalho, estabelecemos uma equação que relaciona a curvatura de Ricci de uma variedade riemanniana M e as segundas formas fundamentais de duas folheações ortogonais de dimensões complementares, F e F, definidas em M. Usando essa equação, encontramos uma estimativa da curvatura média da folheação F e uma condição necessária e suficiente para que tal folheação seja totalmente geodésica. Mostramos também uma condição suficiente para que M seja localmente um produto riemanniano das folhas de F e F, se uma das folheações for totalmente umbílica. Por fim, provamos ainda uma fórmula integral válida para tais folheações. / In this work, we and an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F, defined on M. Using this equation, we and an estimate of the mean curvature of the foliation F and a necessary and suficient condition for the foliation F to be totally geodesic. We also show a suficient condition for the manifold M to be locally a riemannian product of the leaves of F and F, if one of the foliations is totally umbilical. Finally, we also prove an integral formula for such foliations.

<p> The Peter Strides Pond study area, approximately 150 km^2, is located in the Southern Long Range Mountains in southwestern Newfoundland. It lies at the southern margin of the Paleozoic Central Mobile Belt in the Dunnage tectonostratigraphic zone. A felsic intrusion, granoblastic gneiss, heterolithic conglomerate and basalt form foliated, linear bodies across the study area parallel to the regional, northeast striking, structural grain; gabbro, diabase and rhyolite have non-linear exposure and no internal fabric. Two parallel mylonite zones traverse the study area. The Victoria River Shear Zone (VRSZ) to the north and the Peter Strides Pond Shear Zone (PSPSZ) to the south are concordant with the regional fabric and separate domains which increase in metamorphic grade from lower greenschist in the northwest to upper amphibolite facies to the southeast. Veins and lenses of variable form and composition are observed in several lithologies but the focussed occurrence of milky white, foliation parallel quartz veins in both the VRSZ and the felsic pluton is significant to these units.</p> <p> A tentative stratigraphic succession for the study area utilizes regional correlations, radiometric ages and fossil evidence because limited exposure does not reveal contacts between adjacent lithologies. Basalt of the Victoria Lake Group and gneiss of the Bay du Nord Group are the oldest units in the area. Conglomerate unconformably overlies the Victoria Lake Group and contains volcanic and sedimentary clasts derived from the underlying group. Mafic to felsic plutonic igneous rocks intrude volcanic rocks of the Victoria Lake Group. Undated gabbro, diorite and diabase may be coeval with Devonian adamellite-granodiorite. Parallel shear zones are concordant with the regional structural grain and follow lithologic contacts closely. Megacrystic granite cross-cuts PSP mylonite, adamellite and Bay du Nord gneiss. It is the youngest unit in the study area. The study area's tectonic framework represents a compressional environment which dominated during, and continued after Iapetus closure. Mafic volcanic rocks adjacent to ophiolite complexes represent back arc basins preserved via obduction. Continued compression was accomodated by crustal thickening through reverse, northwest directed, thrust faults. The inhomogeneously thickened crust provided a host of potential magma compositions to be intruded locally.</p> / Thesis / Master of Science (MSc)

The Hippo signaling pathway was first discovered in Drosophila melanogaster and is involved in organ size control by regulating cell proliferation and apoptosis. This well conserved pathway is activated by various signal inputs, including cell-cell contact, mechanotransduction, and G-protein coupled receptors, with signals converging on the downstream effector protein Yap and its homologue Taz, which are transcriptional co-activators. When the Hippo pathway is activated, Yap/Taz are phosphorylated, leading to cytoplasmic retention and degradation, and diminishing their transcriptional activity. Yap has also been recently implicated as a potential oncogene, as it is upregulated and transcriptionally active in several tumor types. Furthermore, inhibiting Yap activity in various cancer models has been shown to revert cancer cells to a normal phenotype. Although the role of Yap has been described in several organ systems, there is a paucity of information about the function of Yap in the central nervous system. I investigated the function of Yap/Taz in the murine cerebellum to determine its significance during normal development and a potential role for Yap/Taz in medulloblastoma, a tumor that arises in the cerebellum. In Chapter 2, I describe the expression pattern of Yap from embryonic through adult stages in mice, and demonstrate the functional significance of Yap/Taz in different cell populations using conditional knockout mouse models. I show that Yap plays a significant role in cell fate determination as well as in cerebellar foliation: Yap is highly expressed in the ventricular zone and is required for the proper formation of ependymal cells, and is also strongly expressed in Bergmann glia (BG) during early developmental stages, where Yap, together with Taz, plays a significant role in cerebellar foliation. Furthermore, Yap/Taz-deficient BG exhibit migrational defects, as their cell bodies can be found mislocalized to the molecular layer (ML), rather than remaining tightly associated with Purkinje Cells (PCs) in the PC layer. BG support the health of PCs, and severely defective BG positioning eventually leads to a loss of PCs. However, although Yap is highly expressed in granule neuron progenitors (GNPs) during the rapid postnatal expansion stage, it does not appear to play a major role in proliferation of these cells as conditionally knocking-out Yap/Taz in GNPs does not alter their proliferative capacity. Our observations demonstrate that in the cerebellum, Yap has a novel function in glia that is required for the development of normal foliation and organization, but plays a minimal role in GNP proliferation. Importantly, I also show that the reduction of sphingosine-1-phosphate G-protein-coupled receptor (S1P1) signal transduction activates the upstream kinase Lats with concomitant increases of phosphorylated Yap as well as a reduction of the known Yap target connective tissue growth factor (CTGF). This study identifies a novel function of Yap/Taz in cerebellar glia that is required for the development of normal foliation and laminar organization with sphingosine-1-phosphate (S1P) signaling as a potential extracellular cue regulating Yap activity during cerebellar development. In Chapter 3, I present further support for the finding that Yap/Taz are not required for GNP proliferation in vivo by discussing the failure of Yap/Taz loss to rescue the Sonic-hedgehog (Shh) mediated medulloblastoma phenotype, in which GNPs are considered to be the tumor cell of origin. Furthermore, I provide evidence suggestive of a tumor suppressive function of Yap/Taz in the cerebellum. Together, previously unknown functions of Yap in the developing and malignant cerebellum are described, providing a foundation for future studies of Yap in the central nervous system (CNS). / Biomedical Sciences

Neste trabalho nós estudamos os expoentes de Lyapunov de aplicações f : Td &rarr; Td homotópicas a uma aplicação Anosov linear e a continuidade absoluta de folheações. Nós mostramos para algumas classes de homotopia de aplicações que a soma dos expoentes de Lyapunov está limitado pela soma dos expoentes de Lyapunov da aplicação Anosov linear. Além disso, admitindo uma propriedade conhecida como densidade uniformemente limitada (UBD) nas folheações, mostramos uma igualdade entre a soma dos expoentes de Lyapunov de f e do Anosov linear. Também construímos um conjunto C1 aberto de difeomorfismos parcialmente hiperbólicos do toro T4, preservando volume, com folheação central bidimensional não compacta e não absolutamente contínua. Ainda construímos um exemplo parcialmente hiperbólico com folhas centrais bidimensionais, não compactas onde a desintegração do volume ao longo da folheação central não é nem Lebesgue nem atômica. / In this work we study the Lyapunov exponents of maps f : Td &rarr; Td homotopic to a linear Anosov map. We proof for some homotopic classes of maps which the sum of Lyapunov exponents is bounded by the sum of the Lyapunov exponents of the linear Anosov map. Moreover, by assuming a property known as uniformly bounded density (UBD) in the foliations, we show an equality between the sum of the Lyapunov exponents of f and the linear Anosov. We also construct an C1 open set of volume preserving partially hyperbolic diffeomorphisms with non compact two dimensional center foliation and non absolutely continuous. We still build an example of partially hyperbolic diffeomorphism with non compact bidimensional center leaves where the disintegration of volume along the center foliation is neither Lebesgue nor atomic.

Les strates de l'espace de modules des di__erentielles ab_eliennes sont naturellementmunies d'un feuilletage holomorphe, appel_e feuilletage isop_eriodique (ou feuilletagesdes p_eriodes aboslues, ou encore feuilletage du noyau). Celui-ci a _et_e introduit il y a25 ans, d'abord par A. Eskin et M. Kontsevitch, puis par K. Calta et C. McMullenavant de devenir un objet important en dynamique de Teichmuller. La questiong_en_erale abord_ee dans ce texte est la suivante :Comment les feuilles du feuilletage isop_eriodique se r_epartissent-ellesdans l'espace de module ?McMullen a d_emontr_e l'ergodicit_e du feuilletage dans les strates principales (o_u toutesles singularit_es sont simples) en genre 2 et 3 en utilisant des techniques issue dela dynamique homog_ene. Calsamiglia, Deroin & Francaviglia ont ensuite _etenduce resulat et obtenu une classi_cation _a la Ratner des ensembles ferm_es satur_espar le feuilletage. Simultan_ement, Hamenstadt a fourni une preuve alternative del'ergodicit_e, toujours dans la strate principale. De fa_con _etonnante, le seul r_esulatconnu pour les autres strates est d^u _a P. Hooper et B. Weiss : les feuilles des surfacesde Arnoux-Yoccoz sont denses dans les strates qui les contiennent.La question de la dynamique du feuilletage isop_eriodique peut ^etre formul_ee dansle contexte plus g_en_eral des sous vari_et_es a_nes. Avila, Eskin et Moller ont prouv_eque la codimension des feuilles est alors paire. Le cas de la codimension 2, ou rang1, est d_ej_a riche. Nous _etablissons un cri_ete de densit_e des feuilles et l'appliquons_a di__erentes familles de vari_et_es a_nes de rang 1. Parmi celles-la, les lieux Prymoccupent une place importante. Nous d_emontrons dans ce cadre que les feuilles sontsoit ferm_ees, soit denses, en fonction de l'artithm_eticit_e du lieu. Dans le cas nonarithm_etique, nous prouvons que le feuilletage est ergodique pour la mesure a_neassoci_ee. Cela aboutit _a la d_ecouverte de nouvelles feuilles denses dans des strates _asingularit_es multiples. Ces r_esultats sugg_erent une connection entre la g_eometrie desvari_et_es a_nes et la dynamique isop_eriodique. L'exploitation de cette connection engenre 3 aboutit _a la classi_cation des vari_et_es a_nes non arithm_etiques ne provenantpas d'orbites ferm_ees dans les strates _a deux singularit_es. / The strata of the moduli space of abelian di_erentials are endowed with a naturalholomorphic foliation, known as the isoperiodic foliation (or absolute period foliationor kernel foliation). It has been introduced 25 years ago by A. Eskin and M. Kontsevichand later by K. Calta and C. McMullen before it became a central object inTeichmuller dynamics. The general question addressed in this text is the following:How do the leaves of the isoperiodic foliation wander around in themoduli space ?McMullen proved the ergodicity of the foliation in the principal stratum (where thesingularities of the abelian di_erentials are all simple) in genus 2 and 3 using resultsfrom group actions on homogeneous space. Calsamiglia, Deroin & Francavigliageneralized this result in higher genera and obtained a Ratner-like classi_cation ofthe closed saturated subsets. Simultaneously, Hamenstadt gave an alternative proofof the ergodicity. Surprisingly enough, for the strata where at least one zero isnot simple, the only result available was due to Hooper and Weiss: the leaf of theArnoux-Yoccoz surface is dense in the stratum in which it belongs.The question of the dynamics of the isoperiodic foliation can be rephrased in the moregeneral context of a_ne manifolds. Avila, Eskin, M^oller proved that the codimensionof the leaves is even. The codimension 2 case, also known as rank 1, already displaysa rich and contrasted picture. We give a criterion for density of the leaves, and applyit to di_erent families of rank one a_ne manifolds. Among those, special attention isdedicated to the Prym eigenform loci. We prove that the leaves are either compactor dense, depending on the arithmeticity of the locus. In the non arithmetic case, weprove that the foliation is ergodic with respect to the a_ne measure. In turn, thisgives new examples of dense leaves in strata where at least one of the singularity isnot simple. The aforementioned results suggest a connection between the dynamicsof the isoperiodic foliation and the geometry of a_ne manifolds. This connection isanalyzed in genus 3 and results in a classi_cation of the proper non arithmetic a_nemanifolds in strata with 2 singularities.