The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety Xres by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y → X in the sense that Yres → Y is the pullback of Xres → X.
In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems.
In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.