Pure spinor description of maximally supersymmetric gauge theories

Using non-minimal pure spinor superspace, Cederwall has constructed BRST-invariant actions for D=10 super-Born-Infeld and D=11 supergravity which are quartic in the superfields. But since the superfields have explicit dependence on the non-minimal pure spinor variables, it is non-trivial to show these actions correctly describe super-Born-Infeld and supergravity. In this talk, I will expand solutions to the equations of motion from the pure spinor action for D=10 abelian super Born-Infeld to leading order around the linearized solutions and show that they correctly describe the interactions expected. If I have time, I will explain how to generalize these ideas to D=11 supergravity.