Minimal Model Program in Birational Geometry

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Algebraic Geometry is the study of algebraic varieties, i.e. the setof points satisfying a system of polynomial equations. In the last two decades, algebraic geometry has played a central role in Mathematics, due to its many applications in different fields. Thus, it is of crucial importance to find a classification of thesevarieties. The Minimal Model Program, started by Mori around the 1970's, aims to generalize the classification of projective surfaces,i.e. two dimensional projective varieties, to higher dimensional varieties. This Program was successfully carried out in the 1980's for projectivethree-folds, i.e. projective varieties of dimension three. In higher dimension, many problems concerning the geometry of projective varieties are still open. The proposed research aims to solve some of these outstanding problems, making progress in the Minimal Model Program.