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: We introduce some standard techniques in algebraic number theory to investigate solutions of polynomials. If we consider the integers mod p, and our polynomial has no solution there (local), then it has no integer solution (global). But if there are solutions for every p, can we find a global solution? More generally, we can ask for rational solutions, and consider completions of the rationals as localizations. This is called the Hasse local-global principle. We will introduce and use Henselian lifting, the class number, and the Minkowski bound to give an example of a polynomial which does not satisfy the Hasse principle.