Finite Generatedness of Algebras and Flips

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This award supports a project of Professor Shokurov. The projectis focused on the finite generatedness of algebras--a fundamental problem of algebra. The work proposed is to resolve the problem in a certain important class ofalgebras (named (FGA)) with the aim to finish the proof of the existence of log flips in any dimension whenever the Log Minimal Model Program (LMMP) and, in particular, the log terminations hold in the lower dimensions. To apply this inductive step to the LMMP in dimensions higher than 4the principal investigator intends to finish the log termination for 4-folds. This gives the 5-fold log flips, and establishes the LMMP in dimension 4. A concrete application of this techniquewill be done by his student, Jihun Park, in his study of birational geometry of Del Pezzo fibre spaces, with a view to their existence, uniqueness of their certain models, and to the birational classification. This is research in the field of algebra with methods and applications in algebraic geometry. The finite generatedness corresponds tocompleteness in geometry, and effectiveness from the computational point of view. Algebra and algebraic geometry are very old, traditional areas of modern mathematics, but which have had a revolutionary flowering in the past century. In its origin, algebraic geometrytreated figures that could be defined in the plane by the simplestequations, namely polynomials, or be given in the 3-space by the simplestgeometric constructions, e. g., conic sections.Algebra is about these equations. Both fields interacts with most of branches of mathematics, e.g., analysis, topology andmathematical physics, with applications in thosefields as well as in number theory, physics, theoretical computer science,and robotics.