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.[-]

The goal of the Minimal Model Program (MMP) is to provide a framework in which the classification of varieties or foliations can take place. The basic strategy is to use surgery operations to decompose a variety or foliation into "building block” type objects (Fano, Calabi-Yau, or canonically polarized objects).

We first review the basic notions of the MMP in the case of varieties. We then explain work on realizing the MMP for foliations on threefolds (both in the case of codimension =1 and dimension =1 foliations). We explain and pay special attention to results such as the Cone and Contraction theorem, the Flip theorem and a version of the Basepoint free theorem.[-]

The goal of the Minimal Model Program (MMP) is to provide a framework in which the classification of varieties or foliations can take place. The basic strategy is to use surgery operations to decompose a variety or foliation into "building block” type objects (Fano, Calabi-Yau, or canonically polarized objects).

We first review the basic notions of the MMP in the case of varieties. We then explain work on realizing the MMP for foliations on threefolds (both in the case of codimension =1 and dimension =1 foliations). We explain and pay special attention to results such as the Cone and Contraction theorem, the Flip theorem and a version of the Basepoint free theorem.[-]