Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

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Le problème de l'indice est de calculer l'indice d'un opérateur elliptique en termes topologiques. Ce problème fut résolu par M. Atiyah et I. Singer en 1963 dans "The index of elliptic operators on compact manifolds". Quelques années plus tard, ces auteurs ont fourni une nouvelle preuve dans "The index of elliptic operators I" permettant plusieurs généralisations et applications. La première est la prise en compte de l'action d'un groupe compact G, dans ce cadre on obtient une égalité dans l'anneau des représentations de G. Par la suite ils ont généralisé ce résultat au cadre des familles d'opérateurs elliptiques paramétrées par un espace compact dans "The index of elliptic operators IV", ici l'égalité vit dans la K-théorie de l'espace paramétrant la famille.Une autre généralisation importante est celle des opérateurs transversalement elliptiques par rapport à l'action d'un groupe G, c'est-à-dire elliptiques dans le sens transverse aux orbites de l'action d'un groupe sur une variété. Cette classe d'opérateurs a été étudié pour la première fois dans le cadre d'un opérateur P agissant sur une variété M par M. Atiyah (et I. Singer) dans "Elliptic operators and compact groups", en 1974. Dans cet article l'auteur définit une classe indice et montre qu'elle ne dépend que de la classe du symbole en K-théorie. Il montre ensuite qu'elle vérifie différents axiomes : action libre, multiplicativité et excision. Ces différents axiomes permettent alors de ramener le calcul de l'indice à un espace euclidien muni de l'action d'un tore. Par la suite, cette classe d'opérateurs a été étudier du point de vue de la K-théorie bivariante par P. Julg [1982] et plus récemment dans le cadre des actions propres sur une variété non compacte par G. Kasparov [2016].Dans cette thèse, nous nous intéressons aux familles d'opérateurs G-transversalement elliptiques. Nous définissons une classe indice en K-théorie bivariante de Kasparov. Nous vérifions qu'elle ne dépend que de la classe du symbole de la famille en K-théorie. Nous montrons que notre classe indice vérifie les propriétés d'action libre, de multiplicativité et d'excision espérées en K-théorie bivariante. Nous montrons ensuite un théorème d'induction et de compatibilité avec les applications de Gysin. Ces derniers théorèmes permettent de ramener le calcul de l'indice au cas d'une famille triviale pour l'action d'un tore comme dans le cadre d'un seul opérateur sur une variété. Nous démontrons ensuite qu'on peut associer à cette classe indice un caractère de Chern à coefficients distributionnels sur G à valeurs dans la cohomologie de de Rham de l'espace paramétrant lorsque c'est une variété. Pour ce faire, nous utilisons l'homologie locale de M. Puschnigg [2003] et une technique de M. Hilsum et G. Skandalis [1987]. Par la suite, nous nous intéressons aux formules de Berline et Vergne dans ce cadre. Avant de passer aux formules générales pour une famille d'opérateurs G-transversalment elliptiques, on commence par regarder si on obtient les mêmes formules dans le cadre elliptique. On montre alors des égalités similaires à celles obtenues par N. Berline et M. Vergne [1985] dans le cadre d'un opérateur elliptique G-invariant. Dans un dernier chapitre, on montre la formule de Berline-Vergne dans le cadre des familles d'opérateurs G-transversalement elliptiques. On utilise ici la formule de Berline-Vergne pour un opérateur G-transversalement elliptique et les différentes techniques mises en place dans les chapitres précédents. / The index problem is to calculate the index of an elliptic operator in topological terms. This problem was solved by M. Atiyah and I. Singer in 1963 in "The index of elliptic operators on compact manifolds". Few years later, these authors have given a new proof in "The index of elliptic operators I" allowing several generalizations and applications. The first is taking into account of the action of a compact group G, in this frame they obtain an equality in the ring of the representations of G. Later they generalized this result to the framework of the families of elliptic operators parameterized by a compact space in "The index of elliptic operators IV", here equality lives in the K-theory of the space of parameter.Another important generalization is the transversely elliptic operators with respect to a group action, that is to say, elliptic in the transverse direction to the orbits of a group action on a manifold. This class of operators has been studied for the first time by M. Atiyah (and I. Singer) in "Elliptic operators and compact groups", in 1974. In this article the author defines an index class and shows that it depends only on the symbol class in K-theory. Then he shows that it verifies different axioms: free action, multiplicativity and excision. These different axioms allows to reduce the calculation of the index to an Euclidean space equipped with an action of a torus. Next, this class of operators has been studied from the point of view of bivariant K-theory by P. Julg [1982] and more recently in the context of proper action on a non-compact manifolds by G. Kasparov [2016].In this thesis, we are interested in families of G-transversely elliptic operators. We define an index class in Kasparov bivariant K-theory. We verify that it depends only on the class of the symbol of the family in K-theory. We show that our index class satisfies the expected free action, multiplicativity and excision properties in bivariant K-theory. We then show a theorem of induction and compatibility with Gysin maps. These last theorems allows to reduce the calculation of the index to the case of a trivial family for the action of a torus as in the framework of a single operator on a manifold. We then prove that we can associate to this index class a Chern character with distributional coefficients on G with values ​​in the de Rham cohomology of the parameter space when it is a manifold. To do this, we use the bivariant local cyclic homology of M. Puschnigg [2003] and a technique of M. Hilsum and G. Skandalis [1987].Before treating the general framework of families of G-transversely elliptic operators, we look at the elliptic case. We show that the expected formulas are true in this context. In the last chapter, we show the Berline-Vergne formula in the context of families of G-transversely elliptic operators. We use here the Berline-Vergne formula for a G-transversely elliptic operator and the different methods used in the previous chapters.

The thesis comprises two parts. In the first part, we investigate various cohomological aspects of hypercomplex manifolds and analyse the existence of special metrics. In the second part, we define Seiberg-Witten equations on the leaf space of manifolds which admit a Riemannian foliation of codimension four. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The Synchronous Reluctance Motor (SynRM) has been studied. A suitable machine vector modelhas been derived. The influence of the major parameters on the motor performance has beentheoretically determined.Due to the complex rotor geometry in the SynRM, a suitable and simple combined theoretical(analytical) and finite element method has been developed to overcome the high number ofinvolved parameters by identifying some classified, meaningful, macroscopic parameters.Reducing the number of parameters effectively was one of the main goals. For this purpose,attempt has been made to find and classify different parameters and variables, based on availableliteratures and studies. Thus a literature study has been conducted to find all useful ideas andconcepts regarding the SynRM. The findings have been used to develop a simple, general, finiteelement aided and fast rotor design procedure. By this method rotor design can be suitablyachieved by related and simplified finite element sensitivity analysis.The procedure have been tested and confirmed. Then it is used to optimize a special rotor for aparticular induction machine (IM) stator. This optimization is mainly focused on the torquemaximization for a certain current. Torque ripple is also minimized to a practically acceptablevalue. The procedure can also be used to optimize the rotor geometry by considering the othermachine performance parameters as constrains.Finally full geometrical parameter sensitivity analysis is also done to investigate the influence ofthe main involved design parameters on the machine performance.Some main characteristics like magnetization inductances, power factor, efficiency, overloadcapacity, iron losses, torque and torque ripple are calculated for the final designs and in differentmachine load conditions.Effects of ribs, air gap length and number of barriers have been investigated by means of suitableFEM based method sensitivity analysis.