Finite Generatedness of Algebras and Flips

该奖项支持肖库罗夫教授的一个项目。投射集中于代数的有限生成性--代数的一个基本问题。所提出的工作是在一类重要的代数(称为(FGA))中解决这一问题，目的是当对数最小模型规划(LMMP)，特别是对数终止在低维成立时，完成在任何维上存在对数翻转的证明。为了将这一归纳步骤应用于维度大于4的LMMP，主要调查者打算完成4倍的对数终止。这给出了5倍的对数反转，并建立了4维的LMP。他的学生朴继勋将在他的Del Pezzo纤维空间的双胞几何研究中完成这一技术的具体应用，以期了解它们的存在，它们某些模型的唯一性，以及它们的双胞分类。这是在代数领域中的研究方法和在代数几何中的应用。有限生成性对应于几何上的完备性，以及从计算角度看的有效性。代数和代数几何是非常古老的、传统的现代数学领域，但它们在过去的一个世纪里取得了革命性的繁荣。在它的起源中，代数几何处理的图形可以在平面上由简单的方程定义，即多项式，或在3-空间中通过最简单的几何结构给出，例如二次曲线。代数是关于这些方程的。这两个领域与大多数数学分支相互作用，例如分析、拓扑学和数学物理，在这些领域以及在数论、物理学、理论计算机科学和机器人学中的应用。

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.