Distinguished Lecture: Filling a few holes in the classical resolution of singularities

"...in this field, almost everything is already discovered, and all that remains is to fill a few unimportant holes."  - Philipp von Jolly in his recommendation to Max Planck not to go into physics.

Since 2015 I am taking part in a long project (more precisely, a series of projects) with Dan Abramovich and Jarek Wlodarczyk on resolution of singularities in characteristic zero -- a field which was (and sometimes still is) considered as accomplished up to a few unimportant holes. To our surprise it turned out that there were (and still are) quite a few fundamental things to discover in this classical and thoroughly explored field, and the new discoveries even provide a more conceptual view on what was known before we started our project. It is impossible to compress all results of this journey in one talk, but I will try to outline a unified view on most of our discoveries in these projects. If time permits in the end I will also say a couple of words about our new project in progress with Andre Belotto -- still in characteristic zero...

(This colloquium-style talk is intended for a broad mathematical audience.)