Around 1985 three groups of jugglers independently created the same notational system for juggling patterns. This has been very useful for recording, transmitting, and categorizing old patterns, and especially for creating new ones. I'll explain this system and demonstrate many examples.
While this theory has obvious connections to permutations, only very recently have I and my coauthors noticed it to be the natural setting for studying a certain decomposition of the space of matrices. Quite surprisingly, it is easier to understand this finite-dimensional geometry in terms of a much more familiar decomposition in infinite dimensions (the "Bruhat decomposition of the affine flag manifold"), and I'll explain how "antimatter juggling" naturally suggested this.