Resolution of singularities for the dynamical mathematician
Dan Abramovich
TL;DR
The work provides a coherent, characteristic-0 framework for resolution of singularities grounded in blowups realized through extended Rees algebras and deformation to the normal cone, with a unifying treatment of weighted and ordinary blowups. It introduces a robust invariant-and-center strategy, built recursively via maximal contact and coefficient ideals, to guarantee termination by progressively lowering the invariant. The exposition connects resolution to foliations, showing how principalization can be adapted to foliated settings and preserved under aligned blowups in several good foliation classes. The methods yield a canonical, functorial process with broad applicability to embedded resolution, coefficient-ideal machinery, and foliations, offering practical guidance through examples and exercises.
Abstract
I begin by explaining to non-specialists why resolution of singularities in characteristic 0 works. Then I go into some ideas telling how it actually works. I finish with a brief discussion of related results on foliations. I report on work with André Belotto da Silva, Michael Temkin, and Jarosław Włodarczyk; any claim to originality is joint with them and appears in the paper [AdSTW25].