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 classical Minimal 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.
"Algebraic geometry" is the study of geometric objects which can be described as the set of solutions of polynomial equations in many variables. This field of mathematics has important connections to topology (which describes the basic shapes of the solution sets) and to Differential Geometry (in which one measures the distances between pairs of points on the solution set). One of the key problems in algebraic geometry is to describe all of the "algebraic varieties" (the individual solutions sets studied in the field) which have the same collection of rational functions on them (i.e., functions which are ratios of polynomials in many variables). This problem has generated much work in the field in recent decades, including the award of a Fields Medal (the "Nobel Prize of Mathematics") to Shigefumi Mori for work in this area. Among the main accomplishments of this project was a new approach to studying this "minimal model problem", which has already simplified the proofs and strengthened the theorems in the area. The "minimal model problem" also has an important connection to Differential Geometry. In fact, the very technique of "Ricci flow" which was used by Perelman to solve the Poincaré conjectture can be appled to the Minimal Model Problem as well, and one finds that the shrinking and topological elimination of pieces of the space (from Ricci flow) replicates the processes which are found directly in Algebraic Geometry in addressing the Minimal Model Problem. In addition to important advances in the field, this project provided an important training opportunity for graduate students working in Algebra and Algebraic Geometry, Differential Geometry, and related fields. Five PhD theses have been written under the direction of this project, with another five still in progress.
