Blowing up points

A celebrated theorem in algebraic geometry due to Hironaka states that (over an algebraically closed field of characteristic zero) any algebraic variety admits a so-called resolution of singularities; and that, moreover, such resolution can be obtained by a sequence of finitely many “surgeries” called blow-ups. Blow-ups, however, are not only important to algebraic geometers, but they are relevant in the world of symplectic geometry and, remarkably (to me), they are quite useful in the realm of dynamical systems as well. Motivated by this, this introductory talk will be a tale of two stories: On the one hand, I will present an overview of some of my own research interests, explaining how blow-ups show up in the picture and discussing a few concrete examples. And, on the other hand, I will attempt to further describe how this type of geometric transformation relates to the theme of the symposium: "fast and slow”.