This course will be organized around two main themes. One will be the analysis of symmetries, in particular the symmetries of tiling patterns and crystals. The other will be classical polynomial algebra, in particular the analysis of the extent to which polynomial equations may be solved explicitly (and what that means). The relevant mathematical topics are what are known as group theory and Galois theory. Our treatment of group theory will be fairly abstract, while the treatment of polynomial algebra and Galois theory will be very concrete, classical, and historically motivated.