The Tate-Oort group scheme $\mathbb{TO}_p$

Godeaux surfaces over $\mathbb{C}$ are constructed in terms of $\mu_5$-equivariant quintic hypersurfaces in $\mathbb{P}^3$, giving a quotient surface $X$ having $\mathbb{Z}/5$ in $\mathrm{Pic} X$. The Tate-Oort group scheme $\mathbb{TO}_p$ answers the same kind of questions in characteristic $p$ and in mixed characteristic at $\mathrm{p}$. (I lectured on this at HSE in November 2017.) My lecture will give an overview of theoretical points in the construction of $\mathbb{TO}_p$, its Cartier dual $(\mathbb{TO}_p)^{\vee}$, its representation theory and many types of applications.
For more details and the current draft of a preprint, see my website $+\mathrm{TO_p}$.