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Explore groundbreaking developments in resolution of singularities through this Members' Colloquium lecture delivered at the Institute for Advanced Study. Discover how the field has evolved since Hironaka's foundational 1964 work in characteristic zero, which required four decades of intensive mathematical research to simplify and establish functoriality. Learn about the mysterious uniqueness of the basic resolution algorithm despite various descriptions, and understand how this situation transformed dramatically in the last decade with the discovery of analogues and generalizations across different mathematical settings including logarithmic, relative, stack theoretic, and foliated contexts. Focus primarily on the classical algorithm and the revolutionary "dream algorithm" independently discovered in 2019 by Abramovich-Temkin-Wlodarczyk and McQuillan, which employs weighted stack-theoretic blowings up to improve singularities at each step. Gain insights into the continuation of this mathematical story through recent work on resolutions of varieties with foliations by Abramovich-Belotto-Temkin-Wlodarczyk, representing the cutting edge of algebraic geometry research.
