Borcherds Lie algebras, modular forms, and moonshine

This minicourse is intended as an introduction to the “moonshine conjecture” and to the beautiful circle of ideas involved in its resolution: finite group theory, the theory of modular forms, and infinite dimensional Lie algebras.

Topics Covered.

In approximate chronological order.

• Modular forms and functions, theta functions, and Eisenstein series.
• The Weyl character formula for finite dimensional Lie algebras, and for Borcherds- Kac-Moody Lie algebras.
• Integral Lattices; structure and classification. The Leech lattice.
• The monster group, the moonshine module, and the moonshine conecture.
• The fake monster Lie algebra, the monster Lie algebra, and Borcherds’ proof of the moonshine conecture.
• Other moonshines.

Intended audience.
Masters and doctoral students, and summer program students.

The course is intended to be as self-contained as possible, so there are no formal pre-requisites, but some familiarity with Lie algebras and with the basic theory of complex functions would be helpful. For Lie algebras, Reimundo Heluani’s summer course would be more than enough.

Timeline and Duration.
Total lecture time of 6 hours, spread over 2 weeks (either as four lectures of 90 minutes or six lectures of 60 minutes).
2019, February 11 – 22.