Foliation Theory in Birational Geometry

最小模型程序的目标是将Castelnuovo-Enrique-Severi代数曲面分类的主要结果推广到高维投影变体。该项目由森稔于上世纪80年代启动，并因此获得了久负盛名的菲尔兹奖。Kawamata, Kollár, Miyaoka, Mori, Reid， Shokurov和其他许多人成功地进行了三倍投影。最近，其中一些结果也被推广到任何维度。这项研究的目的是开发新的技术，利用叶理理论，解决目前在最小模型程序中的一些障碍。这将通过结合来自不同数学领域的工具来实现，例如数论、Kähler几何和分析。特别是，我计划使用最小模型程序中的新结果来研究投影品种上的叶的分类。

The goal of the Minimal Model Program is to generalise the main results of the Castelnuovo-Enrique-Severi classification of algebraic surfaces, to higher dimensional projective varieties. The programme was started by Mori in the 1980s, for which he won the prestigious Fields Medal. It was successfully carried out for projective threefolds by Kawamata, Kollár, Miyaoka, Mori, Reid, Shokurov and many others. More recently, some of these results were also generalised to any dimension.The purpose of the proposed research is to develop new techniques, usingFoliation Theory, to solve some of the current obstacles in the MinimalModel Program. This will be achieved by combining together tools from different areas of mathematics, such as Number Theory, Kähler Geometry and Analysis. In particular, I plan to use the new results in the Minimal Model Program to study the classification of foliations over a projective variety.