Resolution of singularities of complex algebraic varieties and their families

TL;DR

This paper reviews Hironaka's theorem on resolving singularities in complex algebraic varieties, discusses recent advancements in simplifying proofs and extending results to families of varieties, including work on toroidal orbifolds.

Contribution

It presents recent progress in simplifying Hironaka's proof and extends resolution techniques to families of varieties, including toroidal orbifolds.

Findings

Improved methods for resolution of singularities

Extensions to families of varieties

Joint work on toroidal orbifolds

Abstract

We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jaros{\l}aw W{\l}odarczyk.