Title: Artin approximation


Abstract:

A significant part of Mathematics boils down to "resolving systems of equations", e.g. equations of implicit function type, F(x,y)=0. In many cases one cannot get any solvability, and has only some ``order-by-order" solutions. The obtained power series, y(x), do not need to be analytic.

Artin approximation (A.P.) ensures: every formal solution is approximated by analytic solutions. This goes in contrast to various other (functional or differential) equations, for which the formal and analytic words are very different. 

I will give a brief introduction to this topic, and then explain several recent results. 
The classical Artin approximation does not immediately hold for morphisms of germs of schemes. (The involved functional equations are not of implicit function type.) Yet, A.P. holds for analytic morphisms that are 'of weakly finite singularity type'. In the algebraic case the situation is much better, A.P. holds for quivers of algebraic morphisms of scheme-germs.


Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=f64fe104-9bf4-45ee-906c-b38300dc543f