Boundedness and termination

PI将致力于代数变种的混生分类。最小模式计划是一个雄心勃勃的计划，旨在将品种分类到种族等值。为了完成程序并建立最小模型的存在性，只要显示翻转的终止就足够了。PI将使用关于ACC的最新结果作为对数规范阈值来攻击终止。一个相关的项目是研究Fano变种的双生有界性，特别是Borisv-Alexeev-Borisv猜想。旋转几何也为研究射影空间上的向量丛提供了一种潜在的方法，可以用来作为一种方法来攻击哈特肖恩的猜想。最后，PI还计划研究二次几何、叶层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，例如二次曲线几何和更现代的代数技巧。最近在高维几何方面有很多令人兴奋的工作。该协会将为《皇家学会会报》撰写一篇有关这项工作的调查文章，面向有科学素养的普通读者，该协会还将试图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.