James Lepowsky
Vertex Operator Algebras and Introduction to Monstrous Moonshine
I will introduce the subject of vertex operator algebras for non-experts, mostly using selected parts of the monographs [FLM] and [LL] listed below.
Topics: formal calculus and introduction to the use of vertex operators in representation theory; the notion of vertex operator algebra; introduction to the theory of modules for a vertex operator algebra; introduction to applications to "Monstrous Moonshine." This is the connection between the Monster finite simple group and the theory of modular functions, based on a certain vertex operator algebra constructed in [FLM], leading to the proof of the Conway-Norton Monstrous Moonshine conjectures. The deeper theory is too much to cover in one semester but the ideas covered will enable interested people to continue with these subjects. In addition, I'll be introducing ideas designed to be of interest to people who already have experience in vertex operator algebra theory. Research problems will be discussed.
Administrative note: From now on, the course that has been called "Math 557 - Topics in Vertex Operator Algebra Theory" for many recent semesters will sometimes continue to be called "Math 557 - Topics in Vertex Operator Algebra Theory" but will instead sometimes be called "Math 558 - Theory of Algebras," which is the case for Spring 2025. As was the case with Math 557 in the past semesters, the topics covered each semester in either "Math 557 - Topics in Vertex Operator Algebra Theory" or "Math 558 - Theory of Algebras" will be independent of one another and will be adapted to the needs and research directions of the currently interested students. Each semester, the subtitle of the course will be unique. The reason for the new use of the course name "Math 558 - Theory of Algebras" is purely administrative, namely: Students who take these courses multiple times, with distinct course subtitles, will routinely be able to get credit on their transcript, with fewer repetitions of the course designation "Math 557 - Topics in Vertex Operator Algebra Theory" on their transcript.
The course descriptions of Math 557 for earlier semesters are archived under the course title 16:640:557 - Topics in Vertex Operator Algebra Theory in the Course Descriptions
Basic references:
[FLM] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988
[FHL] I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs AMS, Vol. 104, Number 494, 1993
[LL] J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2004
Many other materials will be recommended and supplied.
Basic algebra