An introduction to Borcherds-Kac-Moody Lie algebras, vertex algebras, and related automorphic forms

In this talk, I will explain what these Lie algebras, which generalize the semi-simple finite dimensional ones, are and why they were originally studied.  The more interesting ones can be constructed from lattice vertex algebras and hence I will give an idea about this construction.  As we will see, the essential information about the structure of these Lie algebras is contained in a formula known as the denominator formula.  In the cases of interest today, this gives an infinite product expansion of a function on a hyperbolic space transforming nicely under the action of its automorphism group -- i.e. an automorphic form on a Grassmannian -- with the property that the exponents of the product factors are coefficients of a vector valued modular form.  I will end by mentioning some of the main open questions in this area.