Boolean function analysis: beyond the Boolean cube.

Boolean function analysis traditionally studies Boolean functions on the Boolean cube, using Fourier analysis on the group Z_2^n. Other domains of interest include the biased Boolean cube, other abelian groups, and Gaussian space. In all cases, the focus is on results which are independent of the dimension.

Recently, Boolean function analysis has expanded in scope to include other domains such as the symmetric group, the "slice", and the Grassmann scheme. The latter has figured in the recent proof of the 2-to-1 conjecture (a step toward the unique games conjecture) by Dinur, Khot, Kindler, Minzer, and Safra.

In this talk, we first survey classical Boolean function analysis, to set the scene. We then describe in some detail the recent work on the 2-to-1 conjecture, focusing on the connection to Boolean function analysis.