Integrable hierarchies arise by starting with a differential equation that models a real life process and constructing a system of infinitely many differential equations that can be solved simultaneously. Some classical examples of integrable hierarchies are the KP, KdV, and Toda hierarchies. Solutions to such hierarchies can be studied algebraically using tools such as Darboux transformations. I will begin by introducing the extended bigraded Toda hierarchy (EBTH). Darboux transformations for the EBTH were determined by Li and Song in 2016. I will conclude by showing that the action of the Darboux transformation on solutions to the EBTH is given by a vertex operator, and that this result leads to Fay identities for the EBTH. This talk will be accessible for undergraduates.

This is joint work with B. Bakalov.