AbstractsMihai Tibar: Real map germs and open book structuresAbstract: In his 1968 Princeton lecture notes on complex hypersurface singularities, Milnor shows that the link $S_\varepsilon^{2n-1}\cap f^{-1}(0)$ of a holomorphic function germ $f : (\mathbb C^n, 0) \to (\mathbb C, 0)$ is actually a fibered link. Equivalently, one says that the sphere $S_\varepsilon^{2n-1}$ is endowed with an ``open book decomposition''. The complex setting has been further investigated by Milnor and Brieskorn in the 1960's and by many other authors ever since. In a short section about real analytic map germs $\psi :(\mathbb R^m ,0) \to (\mathbb R^p ,0)$, where $m\ge p \ge 2$, Milnor points out the elusiveness of the real setting compared to complex one. We focus here on the existence of fibered links defined by real analytic map germs. We discuss old and new criteria in case of isolated critical points and we indicate some issues in case of non-isolated singularities.
David Mond: Vanishing homology, Lefschetz thimbles, and the Fourier-Laplace transform
Ricardo Uribe-Vargas: Differential geometry of surfaces in the neighbourhood of a godron Abstract: We show some generic (robust) properties of smooth surfaces immersed in the real $3$-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (called also {\em cusp of Gauss}): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the tangent plane, the self-intersection line, the cuspidal edge and the {\em flecnodal curve} at a generic swallowtail in $\R^3$. We present some global results, for instance: {\em In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection).
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