Introduction to vertex algebras

Vertex algebras are a class of noncommutative, nonassociative algebras that arose out of conformal field theory in the 1980s, and were axiomatized by R. Borcherds in his proof of the Moonshine Conjecture. They have found applications in many areas of mathematics including representation theory, combinatorics, finite group theory, number theory, and geometry. I will give an introduction to the subject starting from scratch, with emphasis on the relationship between vertex algebras and the geometry of arc spaces.