The series aims to introduce resolution of singularities for non-experts, with foliation specialists in mind. The work discussed is joint with Andé Belotto da Silva, Michael Temkin and Jaroslaw Wlodarczyk.

Talk 1: Resolution of singularities in characteristic 0 - why does it work?

I continue a long struggle to explain to non-experts why resolution of singularities in characteristic zero works. I explain a criterion, one paragraph in an article by Wlodarczyk, which tells you what you need in order to resolve singularities.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

The Minimal Model Program is a (partially conjectural) framework of the classification of algebraic varieties. In the early 2000s Brunella, Mendes and McQuillan observed that this framework could be adapted to the study of foliations on projective surfaces. In recent years this program of study has been developed for foliations on higher dimensional projective varieties. Our first lecture will review the Minimal Model Program for surface foliations. We will then survey the recent developments in the topic, focusing especially on three cases where the theory of minimal models of foliations is most developed, namely for rank one foliations, co-rank one foliations and algebraically integrable foliations. Time permitting we will explain a very recent development: adjoint foliated structures. These structures arise naturally as a way to address some of the unique challenges which arise when studying minimal model techniques in the setting of foliations.[-]

The Minimal Model Program is a (partially conjectural) framework of the classification of algebraic varieties. In the early 2000s Brunella, Mendes and McQuillan observed that this framework could be adapted to the study of foliations on projective surfaces. In recent years this program of study has been developed for foliations on higher dimensional projective varieties. Our first lecture will review the Minimal Model Program for surface foliations. We will then survey the recent developments in the topic, focusing especially on three cases where the theory of minimal models of foliations is most developed, namely for rank one foliations, co-rank one foliations and algebraically integrable foliations. Time permitting we will explain a very recent development: adjoint foliated structures. These structures arise naturally as a way to address some of the unique challenges which arise when studying minimal model techniques in the setting of foliations.[-]

The Minimal Model Program is a (partially conjectural) framework of the classification of algebraic varieties. In the early 2000s Brunella, Mendes and McQuillan observed that this framework could be adapted to the study of foliations on projective surfaces. In recent years this program of study has been developed for foliations on higher dimensional projective varieties. Our first lecture will review the Minimal Model Program for surface foliations. We will then survey the recent developments in the topic, focusing especially on three cases where the theory of minimal models of foliations is most developed, namely for rank one foliations, co-rank one foliations and algebraically integrable foliations. Time permitting we will explain a very recent development: adjoint foliated structures. These structures arise naturally as a way to address some of the unique challenges which arise when studying minimal model techniques in the setting of foliations.[-]