講座內容：In a sequence of papers, Donaldson initiated a program to study theextremal metrics on toric manifolds and solved the problem for cscK metrics ontoric surfaces. For toric manifolds, the equation of extremal metrics can bereduced to a real 4th-order partial differential equation on the Delzantpolytope, called the Abreu equation, which is similar to the maximal equationin affine geometry. The affine techniques play important role. We will discuss some recent developments.