In this talk, I will present a resolution method based on cobordant blow-ups with weighted centers, a natural and powerful tool for improving singularities in algebraic varieties. This approach refines the classical resolution process by using simple geometric invariants derived from the weighted normal cone, which effectively approximates singularities and guides the resolution through successive improvements.

The method applies to the resolution of varieties in characteristic zero and, in some cases, to varieties in positive characteristic. It also extends to the setting of foliated varieties, leading to the resolution of certain kinds of foliations.

The results come from a series of joint projects with D. Abramovich and M. Temkin, as well as with D. Abramovich, A. Belotto, and M. Temkin. The method relies on torus actions and is inspired by the idea of birational cobordisms.