0 Se p 20 01 Postcards from the Edge , or Snapshots of the Theory of Generalised Moonshine †

I dedicate this paper to a man who throughout his career has exemplified the power of conceptual thought in math: Bob Moody. In 1978, John McKay made an intriguing observation: 196 884 ≈ 196 883. Monstrous Moonshine is the collection of questions (and a few answers) inspired by this observation. In this paper we provide a few snapshots of what we call the underlying theory. But first we digress with a quick and elementary review. By a lattice in C we mean a discrete subgroup of C under addition. We can always express this (nonuniquely) as the set of points Zw + Zz def = Λ{w, z}. We dismiss as too degenerate the lattice Λ = {0}. Call two lattices Λ, Λ ′ similar if they fall into each other once the plane C is rescaled and rotated about the origin — i.e. Λ ′ = αΛ for some nonzero α ∈ C. In Figure 1 we draw (parts of) two similar lattices. For another example, consider the degenerate case where w and z are linearly dependent over R: then in fact w and z are linearly dependent over Z (otherwise discreteness would be lost) and any such lattice is similar to Z ⊂ C. We're interested in the set of all equivalence classes [Λ] of similar lattices. There is a natural topology on this set, and in fact a differentiable structure. Now, it's a lesson of modern geometry that one probes a topological set by considering the functions which † This is the text of my talk at the Banff conference in honour of R.V. Moody's 60th birthday. A streamlined version of this paper (with the pedagogy removed) is my contribution to a volume in his honour.