Boundedness and termination

PI将致力于代数簇的双有理分类。 最小模型计划是一个雄心勃勃的计划，分类品种的双有理等价。 为了完成程序并建立最小模型的存在性，只需显示翻转的终止即可。 PI将使用关于ACC的最新结果作为攻击终止的对数规范阈值。 一个相关的项目是研究Fano簇的双有理有界性，特别是Borisov-Alexeev-Borisov猜想。 双有理几何也提供了一个潜在的方式来研究向量丛的射影空间，被用来作为一种手段，攻击哈茨霍恩的结构。 最后，PI还计划研究双有理几何，叶理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，例如圆锥曲线和更现代的代数技术。 最近在高维几何中有许多令人兴奋的工作。 PI将写一篇调查文章的程序的皇家学会对这项工作的目的是为一般科学素养的观众和PI也将试图传授一些令人兴奋的研究代数几何本科生和研究生在他的教学。

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.