Problems in moonshine.  
Richard E. Borcherds, 
Mathematics department, Evans Hall #3840, University of California 
at Berkeley, CA 94720-3840 U. S. A. 
www home page www.math.berkeley.edu/~reb 

The talk at the I.C.C.M. was an introduction to moonshine. As there are 
already several survey articles on what is known about moonshine ([B94], 
[B99], [D-M94], [D-M96], [F-H98], [J]), this paper differs from the talk 
and will concentrate mainly on what we do not know about it. 

Moonshine is not a well defined term, but everyone in the area recognizes 
it when they see it. Roughly speaking, it means weird connections between 
modular forms and sporadic simple groups. It can also be extended to 
include related areas such as infinite dimensional Lie algebras or complex 
hyperbolic reflection groups. Also, it should only be applied to things 
that are weird and special: if there are an infinite number of examples of 
something, then it is not moonshine.