James Lepowsky
Vertex-alg intertwining ops and q-series
The course will focus on selected aspects of vertex operator algebra theory, adapted to students' interests, background, and research needs. (Note: The courses "Topics in Vertex Operator Algebra Theory I" (Math 557) and "Topics in Vertex Operator Algebra Theory II" (Math 558) are generally independent of each other. These two course designations are for topics courses in this field, and vary from semester to semester.)
In Spring, 2026, I will review basic foundations of the theory, for the benefit of less-experienced and also more-experienced students, depending on the students' backgrounds. Vertex-algebraic intertwining operators relate modules among vertex operator algebras, and are foundational in the representation theory of vertex operator algebras. One of the many manifestations of the theory of intertwining operators lies in gaining deeper, representation-theoretic understanding of important classical and new q-series identities, the prototypical examples being the Rogers-Ramanujan identities, whose theory I will introduce. I will develop the basics of intertwining operator theory, and discuss new developments in both the theory of q-series identities and the theory of intertwining operators. Along the way, we will see a lot of the deeper aspects of vertex operator algebra theory itself.
Basic references: I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988; I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs AMS, Vol. 104, Number 494, 1993; J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2004.
References on q-series identities: G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1984; A. V. Sills, An Invitation to the Rogers-Ramanujan Identities, CRC Press, Taylor and Francis, 2018.
Other materials will be recommended and supplied.
Basic algebra