Birational geometry in positive characteristic

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Algebraic geometry is the study of algebraic varieties, i.e. the sets of solutions of a collection of polynomial equations. Even though algebraic geometry started more than two thousand years ago, with the study of the geometry of conic sections, such as circles, we still cannot completely answer many simple and fundamental questions. For example, given a collection of polynomials, with finitely many solutions, we would like to know the number of these solutions. We know this number in many special cases, and these cases have interesting applications in many different fields, such as engineering and biology. The goal of the Minimal Model Programme, started by Fields medalist S. Mori in the 1980s, is to generalise the main results of the classification of algebraic surfaces, due to Castelnuovo, Enrique and Severi, to higher dimensional projective varieties. In particular, it predicts the existence of a birational model for any complex projective variety which is as simple as possible. The last decade has seen exceptional activity towards Mori's programme, and it is very reasonable to expect that much more progress will be made in the near future. The aim of this proposal is to study new methods, by combining tools in commutative algebra and birational geometry, which would improve these results and solve some of the main open problems in the Minimal Model Programme over any algebraically closed field.