In the late 90's, Bierstone and Milman proved the first result of resolution of singularities of ideals which is valid in quasianalytic classes. Since then, variations of the result have been used to treat several algebraic-geometric problems in quasianalytic geometry. For example, in the celebrated work of Rolin, Speissegger and Wilkie, the authors use a local version of resolution of singularities to show that the structure generated by restricted quasianalytic functions is o-minimal.

This course section has three objectives. First, we will provide statements of resolution of singularities (local and global) of ideals in quasianalytic classes, and we will provide the idea behind the proof of the local version. Second, we will discuss some applications to algebraic-geometric problems in quasianalytic geometry, such as division. In particular, we will discuss some of the current limitations of the technique. Finally, we will introduce a new result on resolution of singularities of morphisms, called monomialization of morphisms. We will provide the idea for the proof, and will finish by discussing a few applications.