Abstract: This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

Abstract: Galois referred to his theory as the ``Theory of Ambiguities". In his last letter he writes: “Mes principales m'editations depuis quelque temps etaient dirigees sur l’application a l’analyse transcendante de la theorie de l’ambiguite. Il s’agissait de voir a priori dans une relation entre quantites ou fonctions transcendantes quels echanges on pouvait faire, quelles quantites on pouvait substituer aux quantites donnees sans que la relation put cesser d’avoir lieu. Cela fait reconnaitre tout de suite l’impossibilite de beaucoup d’expressions que l’on pourrait chercher. Mais je n’ai pas le temps et mes id´ees ne sont pas encore bien d´eveloppees sur ce terrain qui est immense...”

This idea of Galois is indeed so rich and the territory is so vast that even after 200 years of mathematics we are still not sure it has fully delivered all its potential. I shall introduce the idea of analytic continuation and use it to define the monodromy of an algebraic function, as an instance of the application of ambiguities in analysis and algebra. I shall then indicate how this quickly leads to a proof of impossibility of solving general quintics by radicals, the Abel-Ruffini theorem.