Abstract: Any birational geometer would agree that the best algorithm
for resolution of singularities should run by defining a simple invariant of
the singularity and iteratively blowing up its maximality locus.
The only problem is that already the famous example of Whitney umbrella
shows that this is impossible, and all methods following Hironaka had
to use some history and resulted in more complicated algorithms.
Nevertheless, in a recent work with Abramovich and Wlodarczyk we did
construct such an algorithm, and an independent description of a similar
algorithm was done by McQuillan and Marzo. The novelty is in enlarging
the pool of admissible blow ups: we allow weighted blow ups of smooth
centers, and working with orbifolds we achieve that the ambient space
nevertheless remains smooth.
In this talk I will explain some ideas of the old and new algorithms and describe
simple examples, including the Whitney umbrella, in the old and new
algorithms.