In the late 1980's, Jürgen Moser initiated a PDE version of the work of Aubry and of Mather for a class of dynamical systems. Moser was interested in solutions possessing certain minimality and 'monotonicity' properties. Over the past few years, there have been several extensions of this work.

In the first lecture, a survey of this research will be presented. In particular the application of minimization methods to find certain basic solutions of the equations and the use of the basic solutions to construct more complex solutions which undergo multiple transitions will be described.