For a germ (X,0) of normal complex space of dimension n + 1 with an isolated singularity at 0 and a germ f: (X,0) → (ℂ,0) of holomorphic function with df(x) ≤ 0 for x ≤ 0, the fibre-integrals     $s\mapsto {\int }_{f=s}\varrho {\omega }^{\text{'}}\bigwedge \overline{{\omega }^{\text{'}\text{'}}},\varrho \in C_c^{\infty }\left(X\right),{\omega }^{\text{'}},{\omega }^{\text{'}\text{'}}\in {\Omega }_X^n$, are $C^{\infty }$ on ℂ* and have an asymptotic expansion at 0. Even when f is singular, it may happen that all these fibre-integrals are $C^{\infty }$. We study such maps and build a family of examples where also fibre-integrals for ${\omega }^{\text{'}},{\omega }^{\text{'}\text{'}}\in {⍹}_X$, the Grothendieck sheaf, are $C^{\infty }$.

The singularities occurring in any sort of ordering are known in physics as defects. In an organized fluid defects may occur both at microscopic (molecular) and at macroscopic scales when hydrodynamic ordered structures are developed. Such a fluid system serves as a model for the study of the evolution towards a strong disorder (chaos) and it is found that the singularities play an important role in the nature of the chaos. Moreover both types of defects become coupled at the onset of turbulence....

We describe singularities of the convex hull of a generic compact smooth hypersurface in four-dimensional affine space up to diffeomorphism. It turns out that the boundary of the convex hull is the front of a Legendre variety. Its singularities are classified up to contact diffeomorphism.