These are notes related to a 4-lecture minicourse given during June 10-11, 2012, at a workshop just preceeding the SIAM conference on Nonlinear Waves and Coherent Structures in Seattle, WA, USA. The title of the workshop was “The stability of coherent structures and patterns,” and these four lectures concern stability theory for linear PDEs. The two other parts of the workshop are “Using AUTO f...