UNAM

Let $F:R^n \rightarrow R^k$ be a homogeneous quadratic mapping, and consider its zero set $V=F^{-1}(0)$ and
the intersection of $V$ with the unit sphere $Z=V\cap S^{n-1}$. The topology of these varieties has been studied for decades,
but in the last five years many old and new topological problems about them have been solved: the topological description
of the generic case for $k=2$, of many infinite families for $k>2$ (where complete results look out of reach)
and of some of their variants (open-book structures on $Z$, semi-algebraic versions,
smoothings of $V,\dots$), under some restrictions that should be removed soon by the same methods.

After reviewing briefly those results I will turn to some work in progress on questions that seem to
require new methods: the topological description of their projective versions and of some interesting
families that admit contact structures. The construction of another interesting family with dihedral symmetry
depends on certain conjectures about the Vandermonde matrix on roots of unity.

(Joint work with Yadira Barreto, Samuel Gitler, Vinicio Gómez Gutiérrez and Alberto Verjovsky).