pseudo-differential operators , Weyl quantization,PDE

This work dates from January 2016 as a result of my remarks related to physics made during my high studies. I try in this work to explain the cause behind the inability of Newtonian mechanics to describe correctly many phenomena where the studied object rotates at a very high linear speed. I proved that, in this case, the velocity field is not equiprojective and that the famous formula for changing the reference frame is not correct. I made an application to the case of the GPS system satellites, then I presented a new method for studying a rotating system velocity without needing the conventional steps of changing reference frames. I finished my work by demonstrating the formulas of the main differential operators and I presented them with all the related steps and calculations by using the elementary surfaces. I am eager to discuss the results of this work further with physics and mathematics specialists, and I hope that my formulas will help to simplify the study of many difficult physics phenomena.

Geometry and physics have a long history of mutual interactions. Among those coming from quantum mechanics, the early “Weyl quantization” of the phase space observables paved the way for the understanding of some deep relations between Poisson geometry and quantum algebras. As two outgrowths, of prime importance in the last decades were the Moyal and Weyl products, respectively in formal and non-formal deformation quantization. The notion of a midpoint of two given points appears naturally in the Weyl quantization and product, suggesting to consider symplectic symmetric spaces with midpoint map.

A conjecture of Weinstein in this context relates the associativity of non-formal products of functions with the symplectic area of so-called “double triangles”. The aim of this thesis is to improve the understanding of the structure of solvable symplectic symmetric spaces and to provide the necessary tools for an inductive construction of quantizations of these spaces satisfying the above conjecture.

For a general symmetric space, we relate the existence of a midpoint map to the existence and uniqueness of geodesics joining two points, and characterize the symmetric spaces for which each triangle has a unique double triangle. We introduce symplectic reduction and double extension for symplectic symmetric spaces, and uncover a “primitive structure” canonically attached to any such space, allowing the construction of T-star extensions à la Bordemann. We then study the behaviour of the main ingredient in our proposed quantization procedure, the Severa projection, under the above-mentioned operations. We end the thesis by providing explicit computations for a new class of examples yielding in particular new universal deformation formulas for the actions of some symplectic Lie groups on Fréchet algebras.

We consider two principal bundles of embeddings with total space Emb(M, N), with structure groups Dif f (M) and Dif f + (M), where Dif f + (M) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B(M, N) = Emb(M, N)/Dif f (M) and B + (M, N) = Emb(M, N)/Dif f + (M). From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in [38]. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B(M, N). A case of particular interest is M = S 1 , where connected components of B + (S 1 , N) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space T B + (S 1 , N) is a subgroup of some group GLres, following the classical notations of [45]. These constructions suggest some approaches in the spirit of [27] that could lead to knot invariants through a theory of Chern-Weil forms. MSC (2010) : 47G30, 47N50, 22E67

We state a Chern-Weil type theorem which is a generalization of a Chern-Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps M ap b (M, N) following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1], [19], [20], [14]. We specialize to the case N = G to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group H