BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor connection which yields an invariant prolongation of the first operator. The explicit calculation is presented in the case of conformal Killing forms.

We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.

In this work, we consider variational problems defined by $G$-invariant Lagrangians on the $r$-jet prolongation of a principal bundle $P$, where $G$ is the structure group of $P$. These problems can be also considered as defined on the associated bundle of the $r$-th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.

On démontre qu’il ne peut exister de système complet d’invariants analytiques pour l’action du groupe des germes des difféomorphismes sur les champs de vecteurs pour un champ dont la forme normale a un champ réduit associé nul.

On donne une construction géométrique d’invariants généralisant la classe de Maslov-Arnold d’une immersion lagrangienne dans un fibré cotangent et l’indice de Maslov-Arnold-Leray d’une immersion lagrangienne $2q$-orientée dans ${\mathbf{R}}^n\oplus {\mathbf{R}}^{n*}$: la classe de Maslov-Arnold universelle d’un fibré symplectique et l’indice de Maslov-Arnold-Leray d’un fibré $q$-symplectique, c’est-à-dire dont le groupe structural est le revêtement à $q$ feuillets de $Sp\left(n\right)$. Tout ceci relève d’une situation géométrique générale dans laquelle s’introduisent...

To a given complex-analytic equidimensional corank-1 germ f, one can associate a set of integer 𝓐-invariants such that f is 𝓐-finite if and only if all these invariants are finite. An analogous result holds for corank-1 germs for which the source dimension is smaller than the target dimension.

We study the inverse scattering problem for a waveguide $(M,\mathbf{g})$ with cylindrical ends, $M=M^c\cup \left({\cup }_{\alpha =1}^N({\Omega }^{\alpha }\times (0,\infty ))\right)$, where each ${\Omega }^{\alpha }\times (0,\infty )$ has a product type metric. We prove, that the physical scattering matrix, measured on just one of these ends, determines $(M,\mathbf{g})$ up to an isometry.

Given a Hermitian line bundle $L$ over a flat torus $M$, a connection $\nabla$ on $L$, and a function $Q$ on $M$, one associates a Schrödinger operator acting on sections of $L$; its spectrum is denoted $Spec(Q;L,\nabla )$. Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections $\nabla$, and we address the extent to which the spectrum $Spec(Q;L,\nabla )$ determines the potential $Q$. With a genericity condition, we show that if the connection is invariant under...

We show that the generating function for the higher Weil–Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten’s free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the moduli space of invertible Cohomological Field Theories, this suggests the existence of a “very large phase space”, correlation functions on which include Hodge integrals studied by C. Faber and R. Pandharipande....

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

We study a microfluidic flow model where the movement of several charged species is coupled with electric field and the motion of ambient fluid. The main numerical difficulty in this model is the net charge neutrality assumption which makes the system essentially overdetermined. Hence we propose to use the involutive and the associated augmented form of the system in numerical computations. Numerical experiments on electrophoresis and stacking show that the completed system significantly improves...

We study a microfluidic flow model where the movement of several charged species is coupled with electric field and the motion of ambient fluid. The main numerical difficulty in this model is the net charge neutrality assumption which makes the system essentially overdetermined. Hence we propose to use the involutive and the associated augmented form of the system in numerical computations. Numerical experiments on electrophoresis and stacking show that the completed system significantly improves...