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.
The structure of the cut locus in dimension less than or equal to six
The topology of integrable differential forms near a singularity
The unfoldings of a germ of vector fields in the plane with a singularity of codimension 3
The versality discriminant and local topological equivalence of mappings
We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations of germs of mappings , , to non-finitely determined germs (these occur generically outside the “nice dimensions” for Mather, even among topologically stable mappings). The failure of finite determinacy is described geometrically by the “versality discriminant”, which is the set of points where is not stable (i.e. viewed as an unfolding it is not versal). The criterion asserts that...
Thom polynomials for open Whitney umbrellas of isotropic mappings
A smooth mapping of a smooth n-dimensional manifold L into a smooth 2n-dimensional symplectic manifold (M,ω) is called isotropic if f*ω vanishes. In the last ten years, the local theory of singularities of isotropic mappings has been rapidly developed by Arnol’d, Givental’ and several authors, while it seems that the global theory of their singularities has not been well studied except for the work of Givental’ [G1] in the case of dimension 2 (cf. [A], [Au], [I2], [I-O]). In the present paper,...
Topological Sufficiency of Smooth Map-Germs.
Topological Triviality of Deformations of Functions and Newton Filtrations.
Topological triviality of versal unfoldings of complete intersections
We obtain algebraic and geometric conditions for the topological triviality of versal unfoldings of weighted homogeneous complete intersections along subspaces corresponding to deformations of maximal weight. These results are applied: to infinite families of surface singularities in which begin with the exceptional unimodular singularities, to the intersection of pairs of generic quadrics, and to certain curve singularities.The algebraic conditions are related to the operation of adjoining powers,...
Topological Type in Families of Germs.
Transversalities for Lagrange singularities of isotropic mappings of corank one
Travaux de Thom sur les singularités