This is a reading group on the new approach to resolution of singularities, due to Abramovich, Temkin, Wlodarcyzk.

Resolution of singularities used to be a fundamental result (due to Hironaka) that everyone used and had to spend years to fully understand but whose proof hardly anyone had read. The situation got much better in the early 2000s: efforts by various people made the proof more conceptual, and in the end a comprehensible summary could fit into a book. Well, things got even better this year, with a new approach that just takes 20 pages. The only disadvantage second advantage of the new approach is that along the way we can learn about weighted blowups (which are also more efficient in practice), and see stacks in action: everything becomes equivariant with respect to finite group actions.

This seminar will meet **Mondays at 1pm** in JCMB 5327.

Some references:

Igor Dolgachev Weighted projective varieties

Miles Reid Graded rings and varieties in weighted projective space

Timothy Hosgood An introduction to varieties in weighted projective space

Barbara Fantechi Stacks for everybody

Dan Abramovich, Michael Temkin, and Jaroslaw Wlodarczyk Functorial embedded resolutions for blowing up

Dan Abramovich, Michael Temkin, and Jaroslaw Wlodarczyk Principalization of ideals on toroidal orbifolds

Janosz Kollar Resolution of singularities – Seattle lecture

Slides of the corresponding talk (good brief introduction to the traditional setup): http://www.math.columbia.edu/~thaddeus/seattle/kollar.pdf

Janosz Kollar Resolution of singularities – Annals of Mathematics Studies.

MSRI talk by Dan Abramovich: https://www.msri.org/workshops/869/schedules/26587