What is moonshine?

We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.

This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.

This article is based on a talk given at a one-day meeting in NUI, Maynooth on the Fourth of April, 2008, held to honour David Walsh and Richard Watson. It surveys joint work with the author.

In 1975 Ogg offered a bottle of Jack Daniels for an explanation of the fact that the prime divisors of the order of the monster are the primes p for which the characteristic p supersingular j-invariants are all defined over the field with p elements. This coincidence is often suggested to be the first hint of monstrous moonshine, the deep unexpected interplay between the monster and modular functions. We revisit Ogg's problem, and we point out (using existing tools) that the ...

Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zer...

In this article, we discuss the relation between Kac-Moody algebras, the monstrous moonshine, Jacobi forms and infinite products. We also review Borcherds' solution of the Moonshine Conjecture and his work of constructing automorphic forms on the orthogonal group which can be written as infinite products.

In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to...

This paper has been withdrawn by the author, due to a mistake on page 29.

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Talk at the International Conference ``G\'eom\'etrie au vingti\`eme ci\`ecle: 1930--2000'', Paris, Institut Henri Poincar\'e, Sept. 2001. The title is a homage to Hans Rademacher and Otto Toeplitz whose book fascinated the author many years ago.