Members' Colloquium
New Methods in Resolution of Singularities
Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However, one point remained quite mysterious: despite different descriptions of the basic resolution algorithm, it was essentially unique.
The situation changed in the last decade, when a series of analogues and generalizations were discovered in different settings: logarithmic, relative, stack theoretic and foliated. In my talk I'll mainly discuss the classical algorithm and the so-called dream algorithm, independently discovered in 2019 by Abramovich-Temkin-Wlodarczyk and McQuillan, which uses weighted stack-theoretic blowings up and improves the singularity at each step. If time permits I will also describe continuation of this story to resolutions of varieties with a foliation due to Abramovich-Belotto-Temkin-Wlodarczyk.