Cet article considère des équations aux dérivées partielles non linéaires de la forme $F(x,X^{\alpha }u)=0$, $\left|\alpha \right|\le m$, où les $X_1,...,X_p$ sont des champs de vecteur vérifiant la condition de Hörmander. Soit $u$ une solution réelle de classe $C^{2m+1}$; on suppose que la localisation de l’opérateur linéarisé sur le groupe de Lie associé au système $\left\{X_j\right\}$ est hypoelliptique; nous démontrons sous ces hypothèses que $u$ est de classe $C^{\infty }$.

We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if ${}_{\Gamma }$ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space $k{\oplus }_{\Gamma }$ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra $U{(}_{\Gamma })$ is a bialgebra in the category of A-comodules.

This is a survey article based on the author’s Master thesis on affine representations of a gauge group. Most of the results presented here are well-known to differential geometers and physicists familiar with gauge theory. However, we hope this short systematic presentation offers a useful self-contained introduction to the subject.In the first part we present the construction of the group of motions of a Hilbert space and we explain the way in which it can be considered as a Lie group. The second...

It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podleś spheres are related by restriction. In this approach, the equatorial Podleś sphere is distinguished because only in this case the restricted spectral triple admits an equivariant grading operator together with a real structure (up to infinitesimals of arbitrary high order). The real structure is expressed by the Tomita operator on quantum SU(2) and it is...

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparametrizations. In particular we investigate the metric, for a constant $A>0$, $G_c^A(h,k):={\int }_{S^1}(1+A{\kappa }_c{\left(\theta \right)}^2)\langle h\left(\theta \right),k\left(\theta \right)\rangle \left|c^{\text{'}}\left(\theta \right)\right|d\theta$ where ${\kappa }_c$ is the curvature of the curve $c$ and $h$, $k$ are normal vector fields to $c$. The term $A{\kappa }^2$ is a sort of geometric Tikhonov regularization because, for $A=0$, the geodesic distance between any two distinct curves is 0, while for $A>0$ the...