We implement a singularity theory approach, the path formulation, to classify ${𝔻}_3$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a ${𝔻}_3$-miniversal unfolding $F_0$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of $F_0$ onto its unfolding parameter space. We apply our results to degenerate...

We use the theory of partially hyperbolic systems [HPS] in order to find singularities of index $1$ for vector fields with isolated zeroes in a $3$-ball. Indeed, we prove that such zeroes exists provided the maximal invariant set in the ball is partially hyperbolic, with volume expanding central subbundle, and the strong stable manifolds of the singularities are unknotted in the ball.

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi :P\to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N\left(\xi \right)$ of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from $N\left(\xi \right)$ onto a certain representation space ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$, in fact a diffeomorphism, with reference to suitable smooth structures $C^{\infty }\left(N\right(\xi \left)\right)$ and $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$, where $\Gamma$ denotes the universal central extension of...

Using the notion of the maximal polar quotient we characterize the critical values at infinity of polynomials in two complex variables. As an application we give a necessary and sufficient condition for a family of affine plane curves to be equisingular at infinity.

We generalize the Malgrange preparation theorem to matrix valued functions $F(t,x)\in C^{\infty }(\mathbf{R}\times {\mathbf{R}}^n)$ satisfying the condition that $t\mapsto \det F(t,0)$ vanishes to finite order at $t=0$. Then we can factor $F(t,x)=C(t,x)P(t,x)$ near (0,0), where $C(t,x)\in C^{\infty }$ is inversible and $P(t,x)$ is polynomial function of $t$ depending $C^{\infty }$ on $x$. The preparation is (essentially) unique, up to functions vanishing to infinite order at $x=0$, if we impose some additional conditions on $P(t,x)$. We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...

Singular projections of generic 2-dim surfaces in ℝ³ with singular boundary to 2-space are studied. The case of projections of surfaces with nonsingular boundary has been treated by Bruce and Giblin. The aim of this paper is to generalise these results to the simplest singular case where the boundary of the surface consists of two transversally intersecting lines. Local models for germs of generic singular projections of corank ≤ 1 and codimension ≤ 3 are given. We also present geometrical realisations...