Traditionally the functions being considered are on the Boolean cube $\{0,1\}^n$, or more rarely on other product domains, usually finite. We explore functions on more exotic domains such as finite groups and association schemes, concentrating on two examples: the symmetric group and the Johnson association scheme (the "slice"), which consists of all vertices in the Boolean cube with a specified weight.

We will survey a few classical results in analysis of Boolean functions on the Boolean cube and their generalizations to more exotic domains. On the way, we will explore questions such as: Which functions on the symmetric group are "dictatorships" (depend on one "coordinate") or "juntas" (depend on a few "coordinates")? Is there a "Fourier expansion" for functions on the slice? Is the middle slice a "representative" section of the Boolean cube?

Joint work with David Ellis, Ehud Friedgut, Guy Kindler, Elchanan Mossel, and Karl Wimmer.