Non-smooth dynamical systems arise in many applications, in particular to model mechanical systems subjected to impact or electrical systems with switches. The main goal of the subject is to develop a qualitative theory similar to the one existing for smooth systems. A natural approach to study such systems is the regularization, which produces a family of smooth dynamical systems converging to the initial system in some suitable topology. One of the delicate issues is to establish under which conditions the dynamics of this regularizing family allows to deduce information about the dynamics in the limit non-smooth case. In 2006, Buzzi, Teixeira and da Silva proposed a geometric framework to study such regularization through blowings-up. This framework naturally relates the regularizing families mentioned above to singular perturbation systems defined on manifolds with corners; thus establishing a strong bridge between these two subjects. There is now a large literature on this subject, but mainly limited to the so-called systems of regular type, where the non-smoothness locus is a smooth hypersurface and a single blowing-up suffices. However, there are several cases of interest (such as systems with multiple switches) where such regularity condition fails. In a joint paper from 2018, Panazzolo and da Silva proposed a general theory of geometric regularization for systems of non-regular type. Many basic problems are still open, such as a complete qualitative study of germs of planar vector fields possessing a discontinuity locus of cross type. The main goal of this project is to further explore these subjects, by enhancing the collaboration between the Brazilian team (da Silva and Buzzi) and the French team (Panazzolo, Fruchard). One of our priorities will be to recruit and form a PhD student to work in this subject. We expect to supervise this student in a collaborative way, allowing him/her to make long stays on both universities during the project.