Finite Generatedness of Algebras and Flips

该奖项支持Shokurov教授的项目。本文主要研究代数的基本问题--代数的有限生成性问题。所提出的工作是要解决的问题，在某一重要类代数（命名为（FGA））的目的是完成证明的存在性，在任何维的日志翻转时，日志最小模型程序（LMMP），特别是日志终止举行在较低的维度。为了将这一归纳步骤应用于大于4维的LMMP，主要研究者打算完成4倍的对数终止。这给出了5倍对数翻转，并在4维中建立了LMMP。他的学生Jihun Park在研究Del Pezzo纤维空间的双有理几何时，将具体应用这种技术，以证明它们的存在性、某些模型的唯一性以及双有理分类。这是研究在代数领域的方法和应用在代数几何。有限生成性对应于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，但在过去的世纪中却有了革命性的发展。代数几何最初是处理那些在平面上可以用最简单的方程即多项式来定义的图形，或者在三维空间中可以用最简单的几何构造即多项式来给出的图形。例如，在一个实施例中，圆锥曲线。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，拓扑学和数学物理，在这些领域的应用，以及在数论，物理学，理论计算机科学和机器人。

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.