donaldson and chen introduced the j-functional in '99, and explained its importance in the existence problem for constant scalar curvature metrics on compact kahler manifolds. an important open problem is to find algebro-geometric conditions under which the j-functional has a critical point. the critical points of the j-functional are described by a fully-nonlinear pde called the j-equation. i will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the j-equation.