Problems in moonshine
نویسندگان
چکیده
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], [DM96], [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. We first quickly review the original moonshine conjectures of McKay, Thompson, Conway and Norton [C-N]. The classification of finite simple groups shows that every finite simple group either fits into one of about 20 infinite families, or is one of 26 exceptions, called sporadic simple groups. The monster simple group is the largest of the sporadic finite simple groups, and was discovered by Fischer and Griess [G]. Its order is
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