The proof for the resolution of singularities in characteristic $0$ is built on a complicated induction frame, doing one main job: it allows one to define at each stage of the resolution process and at each singular point of the current variety a local invariant $\operatorname{inv}_a(X) $ — a string of natural integers $(n_1,n_2,...,n_k)$, considered lexicographically.

This invariant, in turn, performs two jobs:

1. It defines the center of the blowups to which the singularities are submitted as the locus of points where the invariant assumes its maximal value.
2. It drops after each blowup at the points which have been modified by the blowup.

As the lexicographic order is a well ordering, one arrives in finitely many steps at the minimal value of the invariant, corresponding to a smooth variety.

In the talk, which is for a general audience, we will explain the main ideas of how to construct the invariant. There are some basic principles to observe, and putting these together, everything then evolves quite systematically.