Genericity, Bifurcation and Symmetry

In these lectures I would like to discuss how the existence of symmetries alters the type of bifurcation behavior that one expects to observe. In the first lecture I will concentrate on the structure and dynamics of steady-state bifurcation from equilibria. It is here that the influence of symmetries on linearized equations will be discussed and some facts from elementary representation theory introduced. The second lecture will be devoted to effects of symmetry on period-doubling in maps with a short description of an application to large arrays of Josephson junctions. In the final lecture I will describe how certain standard choices of boundary conditions (particularly Neumann) can be thought of as symmetry constraints and how this fact alters notions of genericity. It accord with the style that has developed in the lectures at this workshop, the lectures are of different length.