Daniel Bergh (Copenhagen): Weak factorization and the Grothendieck group of Deligne-Mumford stacks

The weak factorization theorem for varieties roughly says that any proper birational map of smooth varieties factors as a sequence of blow-ups and blow-downs in smooth centres. I will show that a similar theorem holds for Deligne-Mumford stacks, provided that we enlarge the class of birational modifications used to include so called root stacks (there also are independent proofs for this by Harper and by Rydh). Furthermore, I will show how to use this to get a presentation of the Grothendieck group of Deligne-Mumford stacks with generators given by smooth and proper Deligne-Mumford stacks. Time permitting I will also mention some joint work with Gorchinskiy, Larsen and Lunts, where we use the results above to prove a conjecture by Galkin-Shinder on the categorical zeta function.