This thesis is divided in two parts. The first one is devoted to the dynamics of polariton condensates, with particular attention to their pattern-forming capabilities. In many configurations of physical interest, the dynamics of polariton condensates can be modelled by means of a non-linear PDE which is strictly related to the Gross-Pitaevskii and the complex Ginzburg-Landau equations. Numerical simulations of this equation are used to investigate the robustness of the rotating vortex lattice which is predicted to spontaneously form in a non-equilibrium trapped condensate. An idea for a polariton-based gyroscope is then presented. The device relies on peculiar properties of non-equilibrium condensates - the possibility of controlling the vortex emission mechanism and the use of pumping strength as a control parameter - and improves on existing proposals for superfluid-based gyroscopes. Finally, the important rôle played by quantum pressure in the recently observed transition from a phase-locked but freely flowing condensate to a spatially trapped one is discussed. The second part of this thesis presents work done in the context of the geometrical models of matter framework, which aims to describe particles in terms of 4-dimensional manifolds. Conserved quantum numbers of particles are encoded in the topology of the manifold, while dynamical quantities are to be described in terms of its geometry. Two infinite families of manifolds, namely ALF gravitational instantons of types A_k and D_k, are investigated as possible models for multi-particle systems. On the basis of their topological and geometrical properties it is concluded that A_k can model a system of k+1 electrons, and D_k a system of a proton and k-1 electrons. Energy functionals which successfully reproduce the Coulomb interaction energy, and in one case also the rest masses, of these particle systems are then constructed in terms of the area and Gaussian curvature of preferred representatives of middle dimension homology. Finally, an idea for constructing multi-particle models by gluing single-particle ones is discussed.

<p>Let $p>5$ be a prime and $X_0$ a simply-connected $4$-manifold with boundary the Poincar\'e homology sphere $\Sigma(2,3,5)$ and even negative-definite intersection form $Q_=\text_8$ . We obtain restrictions on extending a free $\bZ/p$-action on $\Sigma(2,3,5)$ to a smooth, homologically-trivial action on $X_0$ with isolated fixed points. It is shown that for $p=7$ there is no such smooth extension. As a corollary, we obtain that there does not exist a smooth, homologically-trivial $\bZ/7$-equivariant splitting of $\#^8 S^2 \times S^2=E_8 \cup_ \overline$ with isolated fixed points. The approach is to study the equivariant version of Donaldson-Floer instanton-one moduli spaces for $4$-manifolds with cylindrical ends. These are $L^2$-finite anti-self dual connections which asymptotically limit to the trivial product connection.</p> / Doctor of Philosophy (PhD)

Soit (M, ω) une variété symplectique. Nous construisons une version de l’éclatement et de la contraction symplectique, que nous définissons relative à une sous-variété lagrangienne L ⊂ M. En outre, si M admet une involution anti-symplectique ϕ, et que nous éclatons une configuration suffisament symmetrique des plongements de boules, nous démontrons qu’il existe aussi une involution anti-symplectique sur l’éclatement ~M. Nous dérivons ensuite une condition homologique pour les surfaces lagrangiennes réeles L = Fix(ϕ), qui détermine quand la topologie de L change losqu’on contracte une courbe exceptionnelle C dans M. Finalement, on utilise ces constructions afin d’étudier le packing relatif dans (ℂP²,ℝP²). / Given a symplectic manifold (M,ω) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, if M admits an anti-symplectic involution ϕ, i.e. a diffeomorphism such that ϕ2 = Id and ϕ*ω = —ω , and we blow-up an appropriately symmetric configuration of symplectic balls, then we show that there exists an antisymplectic involution on the blow-up ~M as well. We derive a homological condition for real Lagrangian surfaces L = Fix(ϕ) which determines when the topology of L changes after a blow down, and we then use these constructions to study the real packing numbers for real Lagrangian submanifolds in (ℂP²,ℝP²).

Soit (M, ω) une variété symplectique. Nous construisons une version de l’éclatement et de la contraction symplectique, que nous définissons relative à une sous-variété lagrangienne L ⊂ M. En outre, si M admet une involution anti-symplectique ϕ, et que nous éclatons une configuration suffisament symmetrique des plongements de boules, nous démontrons qu’il existe aussi une involution anti-symplectique sur l’éclatement ~M. Nous dérivons ensuite une condition homologique pour les surfaces lagrangiennes réeles L = Fix(ϕ), qui détermine quand la topologie de L change losqu’on contracte une courbe exceptionnelle C dans M. Finalement, on utilise ces constructions afin d’étudier le packing relatif dans (ℂP²,ℝP²). / Given a symplectic manifold (M,ω) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, if M admits an anti-symplectic involution ϕ, i.e. a diffeomorphism such that ϕ2 = Id and ϕ*ω = —ω , and we blow-up an appropriately symmetric configuration of symplectic balls, then we show that there exists an antisymplectic involution on the blow-up ~M as well. We derive a homological condition for real Lagrangian surfaces L = Fix(ϕ) which determines when the topology of L changes after a blow down, and we then use these constructions to study the real packing numbers for real Lagrangian submanifolds in (ℂP²,ℝP²).

L’objectif de ce travail est d’établir des modèles de comportement mécaniques pour les matériaux en grandes déformations. Au lieu des approches classiques en 3D dans lesquelles la notion d'objectivité est ambigüe et pour lesquelles différentes dérivées objectives sont utilisées arbitrairement, le formalisme quadridimensionnel dérivé des théories de la Relativité est appliqué. En 4D, les deux aspects de la notion d’objectivité, l’indépendance du référentiel (ou covariance) et l’invariance à la superposition de mouvement de corps rigide, peuvent désormais être distinguées. En outre, l’utilisation du formalisme 4D assure la covariance des modèles. Pour les modèles incrémentaux, la dérivée de Lie est choisie permettant une variation totale par rapport au temps, tout en étant à la fois covariante et invariante à la superposition des mouvements de corps rigide. Dans ce formalisme 4D, nous proposons également un cadre thermodynamique en 4D pour développer des modèles de comportement en 4D tels que l’hyperélasticité, l’élasticité anisotrope, l’hypoélasticité et l’élastoplasticité. Ensuite, les projections en 3D sont obtenus à partir des modèles en 4D et étudiés en les testant sur des simulations numériques par éléments finis avec le logiciel Zset / The objective of this work is to establish mechanical constitutive models for materials undergoing large deformations. Instead of the classical 3D approaches in which the notion of objectivity is ambiguous and different objective transports may be arbitrarily used, the four-dimensional formalism derived from the theories of Relativity is applied. Within a 4D formalism, the two aspects of notion of objectivity: frame-indifference (or covariance) and invariance to the superposition of rigid body motions can now be distinguished. Besides, the use of this 4D formalism ensures the covariance of the models. For rate-form models, the Lie derivative is chosen as a total time derivative, which is also covariant and invariant to the superposition of rigid body motions. Within the 4D formalism, we also propose a framework using the 4D thermodynamic to develop 4D constitutive models for hyperelasticity, anisotropic elasticity, hypoelasticity and elastoplasticity. Then, 3D models are derived from 4D models and studied by applying them in numerical simulations with finite element methods using the software Zset