We define a transverse Dolbeault cohomology associated to any almost complex structure $j$ on a smooth manifold $M$. This we do by extending the notion of transverse complex structure and by introducing a natural j-stable involutive limit distribution with such a transverse complex structure. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of (M,j) introduced by Cirici and Wilson, showing that the (p,0) cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure.

We study the L∞-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras. We show that for these examples formality in Kontsevich's sense does NOT hold, but we compute the L∞ structure on the cohomology given by homotopy transfer in certain cases.

On 4-symmetric symplectic spaces, invariant almost complex structures -up to sign- arise in pairs. We exhibit some 4-symmetric symplectic spaces, with a pair of “natural” compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non-integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non-integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.

We look at methods to select triples $(M,\omega,J)$ consisting of a symplectic manifold $(M,\omega)$ endowed with a compatible positive almost complex structure $J$, in terms of the Nijenhuis tensor $N^J$ associated to $J$. We study in particular the image distribution $\Image N^J$.

In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ defined on the twistor space $J(M,g)$ of an even-dimensional manifold $M$ with additional structures $g$ and $\nabla$ a $g$-connection. We also look at the question of the compatibility of $J^{\pm}_\nabla$ with a natural closed $2$-form $\omega^{J(M,g,\nabla)}$ defined on $J(M,g)$. For $(M,g)$ we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection $\nabla$. In all cases $J(M,g)$ is a bundle of complex structures on the tangent spaces of $M$ compatible with $g$ and we denote by $\pi \colon J(M,g) \longrightarrow M$ the bundle projection. In the case $M$ is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.

We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their duals are studied in detail and compared to other approaches. As application we discuss the convergence of the complete growth function introduced by Grigorchuk and Nagnibeda.

This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -- with intersection -- the introduction to formal deformation quantization and group actions, corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are implemented by a moment map and we give indications to build a corresponding quantum moment map. Reduction is a construction in classical mechanics with symmetries which allows to reduce the dimension of the manifold; we describe one of the various quantum analogues which have been considered in the framework of formal deformation quantization. We end up by some considerations about convergence of star products.

This is a report on some ongoing work with Michel Cahen and Thibaut Grouy: the aim of our project is to define Radon-type transforms in symplectic geometry. The chosen framework is that of symplectic symmetric spaces whose canonical connection is of Ric

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in Kählerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type transforms, associating to a smooth compactly supported function on a homogeneous manifold [Formula: see text], a function on a homogeneous space [Formula: see text] of totally geodesic submanifolds of [Formula: see text], and vice versa. We describe some spaces [Formula: see text] and [Formula: see text] in such Radon-type duality with [Formula: see text] a model of symplectic symmetric space with Ricci-type canonical connection and [Formula: see text] an orbit of totally geodesic symplectic submanifolds.

The aim of the project is to define and study Radon-type transforms in a symplectic framework. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as a symplectic analogue of the Riemannian symmetric spaces with constant sectional curvature on which many Radon transforms have been widely investigated. A broad range of examples can be found in S. Helgason’s book [1]. These transforms associate to a compactly supported continuous function on such a space, another function on a class of subspaces by mean of the integration with respect to invariant measures. In our cases, the subspaces for integration are the symplectic totally geodesic submanifolds. They can be endowed with an invariant measure. The same holds for the orbits of those submanifolds under the action of a natural group. First of all, we describe all the spaces involved. On the one hand, we pursue the study of symplectic symmetric spaces with Ricci-type curvature, initiated by M. Cahen, S. Gutt and J. Rawnsley [2]. On the other hand, we prove that the orbits of the selected submanifolds are also symmetric spaces. Then, we define the associated Radon-type transforms thanks to the existence of invariant measures. The next step is to study these Radon transforms through the exponential mapping and to understand the obstructions of getting an inversion formula with the same method as used in the Riemannian cases (cf. S. Helgason [1]). For some of our chosen spaces, we know that the exponential mapping is a diffeomorphism from an open subset of their tangent space at a base-point onto a dense open subset of themselves.

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group Mpc and classify G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space.

We advertise the use of the group Mpc (a circle extension of the symplectic group) instead of the metaplectic group (a double cover of the symplectic group). The essential reason is that Mpc-structures exist on any symplectic manifold. They first appeared in the framework of geometric quantization [4, 10]. In a joint work with John Rawnsley [1], we used them to extend the definition of symplectic spinors and symplectic Dirac operators which were first introduced by Kostant [9] and K. Habermann [6] in the presence of a metaplectic structure. We recall here this construction, stressing the analogies with the group Spinc in Riemannian geometry. Dirac operators are defined as a contraction of Clifford multiplication and covariant derivatives acting on spinor fields; in Riemannian geometry, the contraction is defined using the Riemannian structure. In symplectic geometry one contracts using the symplectic structure or using a Riemannian structure defined by the choice of a positive compatible almost complex structure. We suggest here more general contractions yielding new Dirac operators.

Cette these est consacree a l'etude de deux sujets de geometrie symplectique inspiresde la physique mathematique. Les themes que nous developperons mettent en evidence certaines connexions avec la topologie symplectique d'une part, la geometrie Riemannienne d'autre part.Dans la partie 1, nous etudions la quantification par deformation formelle d'une variete symplectique, a l'aide de produits star. Nous definissons le groupe des automorphimeshamiltoniens d'un produit star formel. En nous inspirant d'idees de Banyaga, nous identifions ce groupe comme etant le noyau d'un morphisme remarquable sur le groupedes automorphismes du produit star. Nous relions certaines proprietes geometriques de ce groupe d'automorphismes hamiltoniens a la topologie du groupe des diffeomorphismeshamiltoniens.Dans la partie 2, nous etudions les operateurs de Dirac symplectiques. Les ingredientsnecessaires a leur construction (algebre de Weyl, structures $Mp^c$, champs de spineurs symplectiques, connexions symplectiques,...) sont egalement utilises en quantification geometrique et enquantification par deformation formelle. Les operateurs de Dirac symplectiques sont construitsde maniere analogue a l'operateur de Dirac de la geometrie Riemannienne. Une formule de Weitzenbocklie les operateurs de Dirac symplectiques a un operateur elliptique $mathcal{P}$ d'ordre 2. Nous etudionsles noyaux de ces operateurs de Dirac symplectiques et leur lien avec le noyau de P.Sur l'espace hermitien symetrique $CP^n$, nous calculerons le spectre de $mathcal{P}$ et nous prouverons un theoreme de Hodge pour les operateurs de Dirac-Dolbeault symplectiques./In this thesis we study two topics of symplectic geometry inspired from mathematical physics.Part 1 is devoted to the study of deformation quantization of symplectic manifolds. More precisely, we consider formal star products on a symplectic manifold. We define the group of Hamiltonian automorphisms of a formal star product. Following ideas of Banyaga, we describe this group as the kernelof a morphism on the group of automorphisms of the star product. We relate geometric properties of the group of Hamiltonian automorphisms to the topology of the group of Hamiltonian diffeomorphisms. Part 2 is devoted to the study of symplectic Dirac operators. The construction of those operators relies on many concepts used in geometric quantization and formal deformation quantization such as Weyl algebra, $Mp^c$ structures, symplectic spinors, symplectic connections,... The construction of symplectic Dirac operators is analogous to the one of Dirac operators in Riemannian geometry. A Weitzenbock formula relates the symplectic Dirac operators to an elliptic operator $mathcal{P}$ of order 2. We study the kernels of the symplectic Dirac operators and relate them to the kernel of $mathcal{P}$. On the hermitian symmetric space $CP^n$, we compute the spectrum of $mathcal{P}$ and we prove a Hodge theorem for the symplectic Dirac-Dolbeault operator.

In this article we relate the construction of Ricci type symplectic connections by reduction to the construction of star product by reduction yielding rather explicit descriptions for the star product on the reduced space.

Symmetric symplectic spaces of Ricci type are a class of symmetric symplectic spaces which can be entirely described by reduction of certain quadratic Hamiltonian systems in a symplectic vector space. We determine, in a large number of cases, if such a space admits a subgroup of its transvection group acting simply transitively. We observe that the simply transitive subgroups obtained are one dimensional extensions of the Heisenberg group.

Given a symplectic manifold $(M,\omega)$ admitting a metaplectic structure, and choosing a positive $\omega$-compatible almost complex structure $J$ and a linear connection $\nabla$ preserving $\omega$ and $J$, Katharina and Lutz Habermann have constructed two Dirac operators $D$ and ${\wt{D}}$ acting on sections of a bundle of symplectic spinors. They have shown that the commutator $[ D, {\wt{D}}]$ is an elliptic operator preserving an infinite number of finite dimensional subbundles. We extend the construction of symplectic Dirac operators to any symplectic manifold, through the use of $\Mpc$ structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in $H^2(M,\Z)$. For any $\Mpc$ structure, choosing $J$ and a linear connection $\nabla$ as before, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator $\mathcal{P}$ is elliptic. Using the Fock description of the spinor space allows the definition of a notion of degree and the construction of a dense family of finite dimensional subbundles; the operator $\mathcal{P}$ stabilizes the sections of each of those.