Existence of minimal models for varieties of log general type II

We prove the existence of pl-flips

On the Minimization of Maximum Transient Energy Growth.

The problem of minimizing the maximum transient energy growth is considered. This problem has importance in some fluid flow control problems and other classes of nonlinear systems. Conditions for the existence of static controllers that ensure strict dissipativity of the transient energy are established and an explicit parametrization of all such controllers is provided. It also is shown that by means of a Q-parametrization, the problem of minimizing the maximum transient energy growth can be posed as a convex optimization problem that can be solved by means of a Ritz approximation of the free parameter. By considering the transient energy growth at an appropriate sequence of discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite program. The theoretical developments are demonstrated on a numerical example

A geometric characterisation of toric varieties

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.Comment: 40 page

Existence of minimal models for varieties of log general type

We prove that the canonical ring of a smooth projective variety is finitely generated