** **

** **

** **

**Abstract:**

Monstrous moonshine initially arose about 30
years ago from apparent numerical coincidences between modular functions on the
complex upper half plane and linear representations of the monster simple
group. Later computations
suggested that certain subgroups of the monster also yielded modular functions
when combinations of their irreducible representations were assembled into
graded vector spaces and their characters taken. This was codified by Norton in
his generalized moonshine conjecture, which asserts the existence of a
generalized character that associates a modular function to any commuting pair
of elements in the monster. I will describe some recent progress on this
conjecture, using a construction of generalized Kac-Moody algebras via orbifold
conformal blocks to reduce it to a conjecture that about 50 unknown rational
numbers are in fact equal to one.