Minimal Model Program in Birational Geometry

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The principal investigator proposes to continue the study of the classification of Projective Varieties in Birational Geometry. The Minimal Model Program, started by Mori around the 1970's, aims to generalize the classification of projective surfaces to higher dimensional varieties. This Program was successfully carried out in the 1980's for projective three-folds. The principal investigator plans to carry out the Minimal Model Program in higher dimension, aiming to complete the classification of complex projective varieties.Moroever, the principal investigator plans to extend the Minimal Model Program to a broader class of varieties defined in positive characteristic. Although the techniques involved in this program are very different, he expect to obtain results that are as strong as in the classicalMinimal Model Program. Quite apart from its own interest, it is hoped that this study will be very useful in completing the classification of complex projective varieties. Finally, the principal investigator intends to continue his study of the Kahler-Ricci flow on a wide range of projective varieties, by translating Mori's work into an analytic language.Mathematical tools and concepts have been extensively applied in a wide range of sciences such as physics, engineering and economics. In particular, birational geometry has proven to be a very useful tool in theoretical physics, especially in string theory. Cascini's recent work has already inspired several important conferences. In particular, the Mathematical Sciences Research Institute organized a one-week workshop entitled "Hot Topics:Minimal and Canonical Models in Algebraic Geometry" to discuss the aforementioned results obtained by Cascini and his collaborators. At the same time, seminars on the same topics were organized in many departments of Mathematics in this country.