Recent developments in Moonshine
A 1979 paper by John Conway and Simon Norton laid out the basic framework of what they called Monstrous Moonshine, a relation between the representation theory of the Monster, the largest sporadic group,
and the modular j function. Later developments connected Monstrous moonshine to certain special solutions to string theory, also known as Vertex Operator Algebras, and these enabled the proof of the genus zero conjecture of Conway and Norton. More recently new kinds of connections between modular objects and sporadic groups have appeared which also have connections to string theory. Mathieu moonshine relates the representation theory of the Mathieu group to the elliptic genus of K3 surfaces and to a special mock modular form. Umbral and penumbral moonshine provide further generalizations. I will survey these new developments with an emphasis on general structures and important open problems.
Zoom Meeting: Link TBA via email