| The minimal model program has two conjectural outputs, a minimal model or a Mori fibre space and in neither case is the output unique in general. However Kawamata has recently shown that any two minimal models of the same variety are connected by a sequence of flops. It is then natural to consider how two Mori fibre spaces are connected. The Sarkisov program attempts to factorise birational map between two Mori fibre spaces as a sequence of elementary links. In the case of surfaces these links would be elementary transformations between two P1-bundles, and the Sarkisov program provides a natural framework to prove the classical result that the Birational automorphism group of P2 is generated by a Cremona transformation and PGL(3). In this talk, I will describe recent work with Christopher Hacon which show that the Sarkisov program works in all dimensions, using some ideas of Shokurov on finiteness of minimal models. |