Sections of corank 1 germs
Normal forms of corank 1 maps $f(x,y)=(x,p(x,y),q(x,y))$ suggest, at least in some cases, that they could be seen as $1$-parameter unfoldings of the plane curves $\gamma_0(y)=(p(0,y),q(0,y))$. If a certain genericity condition is satisfied then the transverse slice curve $\gamma_0$ contains information on the geometry of $f$. We introduce invariants $C,J,T$ related to the Reidemeister moves (codimension 1 transitions) that appear in a stable perturbation of $\gamma_0$ and relate them to Mond's invariants of $f.$ Some results on the geometry of the map germ $f$ include: $f$ is finitely determined if and only if $C,J,T<\infty.$
Juan Jose Nuno-Ballesteros
Equisingularity of families of Isolated Determinantal Singularities
Abstract: We study necessary and sufficient conditions for a family of isolated determinantal singularities to be Whitney or topologically equisingular. The topological triviality of the family is related to the constancy of the vanishing Euler characteristic and Whitney equisingularity is characterized in terms of the constancy of the polar multiplicities. We generalize the results of Lê-Ramanujam and Teissier for hypersurfaces and the results of Gaffney for complete intersections.
(Based on joint work with B. Oréfice-Okamoto and J. N. Tomazella)
Multiple points of map-germs
In 1986 D. Mond introduced two ideals which define double-points of map-germs and showed equality between them in the corank 1 case. In this work we prove their equality in some situations, show a counterexample in another, and extend one of them to define multiple-points (of any multiplicity) of map-germs of any corank, possibly with singular domain.