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.
