Stable mappings of 3-manifolds into the plane
Stokes' phenomenon; smoothing a victorian discontinuity
Stratifications of polynomial spaces
In the paper we construct some stratifications of the space of monic polynomials in real and complex cases. These stratifications depend on properties of roots of the polynomials on some given semialgebraic subset of R or C. We prove differential triviality of these stratifications. In the real case the proof is based on properties of the action of the group of interval exchange transformations on the set of all monic polynomials of some given degree. Finally we compare stratifications corresponding...
Sufficiency of Jets via Stratification Theory.
Sufficient decrease principle on Riemannian manifolds.
Sur le théorème des fonctions composées différentiables
Soit un morphisme propre relativement algébrique entre espaces semi-analytiques. On montre que si désigne l’anneau des fonctions de classe sur , l’image par de est fermée dans muni de sa topologie naturelle d’espace de Frechet ; ceci généralise un résultat précédent de J.-C. Tougeron (lui-même généralisant un résultat de Glaeser) qui traite du cas semi-algébrique. La méthode est tout à fait analogue et utilise des propriétés algébriques de l’anneau des fonctions Nash-analytiques introduit...
Sur les singularités des formes différentielles
On étudie, sur le modèle de la théorie des singularités d’applications différentiables, les singularités des formes différentielles extérieures sur une variété différentiable. Les invariants fondamentaux utilisés sont le rang et la classe (au sens de E. Cartan) d’une forme différentielle. On étudie leur comportement générique à l’aide des théorèmes de transversalité. Par exemple, l’ensemble des points d’une variété de dimension où la classe d’une forme de Pfaff est égale à est génériquement...
Survey of recent results on singularities of equivariant maps
Symmetric caustics and Curie's principle
Physical systems producing caustics may possess symmetries. In that case the relation between the symmetry of the system, considered as a whole, and the symmetry of the caustic follow a very general symmetry principle, the Curie principle. We give various examples of application of the Curie principle to caustics produced by the deflection of light in liquid crystals: the so called squint effect, the visualization of a new type of roll structure, etc. We show also that the Curie principle applies...
Symmetry Properties of Singularities of C°°-Functions.
Systems of Clairaut type
A characterization of systems of first order differential equations with (classical) complete solutions is given. Systems with (classical) complete solutions that consist of hyperplanes are also characterized.
Tangencies of generic real projective hypersurfaces.
The duals of generic hypersurfaces.
The Geometry of Critical and Near-Critical Values of Differentiable Mappings.
The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Given a real analytic vector field tangent to a hypersurface with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra associated with the singularity of the vector field on . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...
The linking number of singular maps.
The Rate of Convergence of a Harmonic Map at a Singular Point.
The second Yamabe invariant with singularities
Let be a compact Riemannian manifold of dimension .We suppose that is a metric in the Sobolev space with and there exist a point and such that is smooth in the ball . We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to and of volume . We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with...
The singularities of Yang-Mills connections for bundles on a surface.
The singularities of Yang-Mills connections for bundles on a surface.