||Mock modular forms are generalizations of modular forms
first introduced by Ramanujan. Their properties had been mysterious
for 80 years until various breakthroughs in the past 10 years. In the
last century, the Monstrous Moonshine Conjecture initiated the study
of the fascinating relation between modular forms and sporadic groups.
In this colloquium I will report a conjecture on a new type of "umbral
moonshine" relating a set of mock modular forms, including many of
Ramanujan's original examples, and the representation theory of a set
of finite groups. One instance of such a surprising umbral moonshine
phenomenon relates the largest Mathieu group to the geometry of K3
surfaces, as was first observed by Euguchi-Ooguri-Tachikawa in 2010.
However, the origin and the explanation of this umbral moonshine is
still unknown. This talk is based on the arXiv pre-print: 1201.4140,
1204.2779 with John Duncan and Jeff Harvey.