The Brauer-Picard groupoids of the ADE fusion categories

Cain will talk about work in progress on understanding the Brauer-Picard groupoids of the ADE fusion categories. The Brauer-Picard groupoid of a fusion category $\mathcal C$ has as objects all the fusion categories Morita equivalent to $\mathcal C$, and morphisms the Morita equivalences. Understanding it is essential to solving extension problems involving the category $\mathcal C$. The ADE fusion categories are amongst the simplest fusion categories (arising as "quantum subgroups" of $U_q \mathfrak{sl}_2$), and understanding their Brauer-Picard groupoids should help classify fusion categories with an object of dimension less than 2.