Preface

Summary

In 1979 John Conway and Simon Norton published their famous paper entitled “Monstrous Moonshine.” This paper greatly expanded on earlier observations and ideas of John McKay and John Thompson and on an observation of Andrew Ogg stimulated by a lecture of Jacques Tits on the conjectured Fischer-Griess Monster sporadic finite simple group. The paper presented a number of conjectures relating the conjugacy classes of the Monster to certain meromorphic modular invariant functions, called Hauptmoduln (= principal moduli), for a particular set of genus zero modular groups. The search for an explanation of this remarkable connection between finite group theory and number theory involved the development and application of many diverse areas of mathematics including vertex (operator) algebras, Borcherds algebras, or generalized Kac-Moody algebras, automorphic forms and elliptic cohomology, together with string theory and conformal field theory in theoretical physics. Robert Griess constructed the Monster; Igor Frenkel, James Lepowsky and Arne Meurman constructed a “Moonshine Module” for the Monster by means of vertex operator theory, proving the McKay-Thompson conjecture; and Richard Borcherds proved the remaining Conway-Norton conjectures for the Moonshine Module, which carries the structure of a vertex operator algebra. Many new problems remain — problems that could not even have been formulated in 1979.

To mark the 25th anniversary of the publication of the Monstrous Moonshine paper, a workshop entitled “Moonshine – the First Quarter Century and Beyond, a Workshop on the Moonshine Conjectures and Vertex Algebras” was hosted by the International Centre for Mathematical Sciences at Heriot-Watt University, Edinburgh from 5th July to 13th July in 2004 (www.icms.org.uk/archive/meetings/2004/moonshine).