The principal investigator will conduct research on the birational classification of algebraic varieties, in which spectacular breakthroughs have been made recently. The principal investigator proposes to prove the remaining major conjectures of the minimal model program,namely the existence of minimal models, the abundance conjecture, termination of flips, and the conjecture of Alexeev-Borisov concerning boundedness of Fano varieties.
Algebraic Geometry is one of the oldest and most challenging of areas of research in mathematics, which combines some very classicial geometry, for example that of conic sections and the more modern techniques of algebra, which have had some recent spectacular successes, for example the work of Wiles on Fermat's Last Theorem. The principal investigator is preparing a chapter of a book on some recent exciting work in higher dimensional geometry, whose aim is to disseminate the seminal work of Shokurov in a form which will be accessible to a wide audience. The investigator will also try to impart some of the interesting research in algebraic geometry to undergraduate and graduate students in his teaching.
Algebraic Geometry is the study of the solutions to a collection of polynomial equations. Even though it has a very long history, starting with the Greeks who were interested in conic sections, lines, circles, ellipses, parabolas and hyperbolas, there are still many fundamental questions which don't have satisfactory answers. For example, given a collection of polynomial equations, it takes a long time, even with the aid of the fastest computer, to find the number of solutions. A method to quickly determine the number of solutions would be very useful in applications to engineering and science. Together with Birkar, Cascini and Hacon, the PI found a method to represent the solutions to a collection of polynomial equations in a simple way, which is known as the canonical model. Together with Hacon he studied solutions with a particularly simple geometry and how to connect together different solutions. Together with Hacon and Xu, the PI studied the symmetries of a collection of polynomial equations; they showed that the number of symmetries is in general quite small. Building on this work they investigated how the solutions to a system of polynomial equations varies, as the coefficients, or moduli, of the polynomial equations varies and they were able to prove some basic results about moduli. The PI helped to organise a number of workshops, conferences, conference proceedings, and gave some lecture series to train early career mathematicians.
