First Swig of Moonshine: An Introduction to Modular Forms, Representation Theory, and the Monster

This thesis is a synthesis of existing material in the mathematical field of “monstrous moonshine.” This theory pertains to the almost serendipitous relation between two ostensibly divorced areas of mathematics—modular function theory (proper to complex analysis, well within the realm of “continuous” mathematics) and the properties of the so-called monster group from group theory (firmly within the realm of combinatorial or “discrete” mathematics). The study of monstrous moonshine was initiated in 1978 by a chance observation by John McKay (namely, the first Fourier coefficient of the j-invariant corresponded to one plus the dimension of the smallest irreducible character of the monster), and the true explanation of monstrous moonshine was presented by Richard Borcherds in 1992, for which he won a Field’s medal in 1998. As this is an exceedingly higher-level area of modern math that is still under active research, it would be difficult to give a fully in-depth analysis of Borcherds’ proof; however, this paper tackles the prerequisite theory necessary to rigorously establish the context in which monstrous moonshine lies, as well as a general summary of the steps of Borcherds’s resolution of Norton and John McKay’s original conjecture. In my demonstrations I assume no more than familiarity with undergraduate-level mathematics.