Finite Generatedness of Algebras and Flips

这个奖支持Shokurov教授的一个项目。这些项目的重点是代数的有限生成性——代数的一个基本问题。提出的工作是解决某一类重要的代数（FGA）中的问题，目的是在对数极小模型程序（lmpp），特别是对数终止在低维上成立的情况下，完成在任何维上存在对数翻转的证明。为了将此归纳步骤应用于高于4维的LMMP，主要研究者打算完成4倍的日志终止。这给出了5次log翻转，并建立了维度4中的lmpp。该技术的具体应用将由他的学生Jihun Park完成，在他对Del Pezzo纤维空间的双几何的研究中，着眼于它们的存在，它们的某些模型的唯一性，以及双几何分类。这是在代数领域的研究与代数几何的方法和应用。有限生成性对应几何上的完备性和计算上的有效性。代数和代数几何是现代数学中非常古老的传统领域，但在过去的一个世纪里，它们出现了革命性的发展。在它的起源中，代数几何处理的图形可以在平面上由简单方程（即多项式）定义，或者在三维空间中由最简单的几何结构（如二次曲线）给出。代数就是关于这些方程的。这两个领域都与数学的大多数分支相互作用，例如分析、拓扑和数学物理，并在这些领域以及数论、物理学、理论计算机科学和机器人技术中应用。

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.