Boundedness and termination

PI将致力于代数变种的两国分类。最小模型计划是一个雄心勃勃的计划，分类品种，直到两国等价。要完成程序并建立最小模型的存在性，只需证明翻转的终止性即可。PI将使用有关ACC的最新结果作为日志规范阈值以终止攻击。一个相关的课题是研究Fano品种的双族有界性，特别是Borisov-Alexeev-Borisov猜想。双几何还提供了一种研究射影空间上的向量束的潜在方法，作为攻击Hartshorne猜想的一种手段。最后，PI还计划研究双几何、叶积和丰度猜想之间的联系。代数几何是数学中最古老和最具挑战性的研究领域之一，它结合了一些非常经典的几何，例如二次曲线和更现代的代数技术。最近在高维几何方面有很多令人兴奋的工作。PI将为《皇家学会学报》写一篇关于这项工作的综述文章，这篇文章是为一般有科学素养的读者写的，PI也将在他的教学中向本科生和研究生传授一些令人兴奋的代数几何研究。

The PI will work on the birational classification of algebraic varieties. The minimal model program is an ambitious program to classify varieties up to birational equivalence. To finish the program and to establish existence of minimal models, it suffices to show termination of flips. The PI will use recent results concerning ACC for the log canonical threshold to attack termination. A related project is to study birational boundedness of Fano varieties, especially the conjecture of Borisov-Alexeev-Borisov. Birational geometry also offers a potential way to study vector bundles on projective space, to be used as a means to attack Hartshorne's conjectures. Finally, the PI also plans to study the connection between birational geometry, foliations and the abundance conjecture.Algebraic Geometry is one of the oldest and most challenging of areas of research in mathematics, which combines some very classical geometry, for example that of conic sections and the more modern techniques of algebra. There has been a lot exciting recent work in higher dimensional geometry. The PI will write a survey article for the Proceedings of the Royal Society on this work which is intended for a general scientific literate audience and the PI will also try to impart some of the exciting research in algebraic geometry to undergraduate and graduate students in his teaching.