Level structures and cyclic power operations on the homology of ${\mathbb{E}}_{\mathrm{\infty }}$ spaces

In this document, we discuss how cyclic power operations for periodic complex bordism can be understood through algebraic geometry, following from the theory of power operations for Morava $E$-theory developed by Ando, Hopkins, and Strickland. Using this perspective, we describe how one computes power operations on the $E$-homology of $BU$ and $BU\times \mathbb{Z}$, where $E$ is a 2-periodic even ${\mathbb{E}}_{\mathrm{\infty }}$ ring. Then we provide some explicit example computations at $p=2$; in particular, we compute the action of the additive operations $\mathrm{\Delta }$ on the indecomposables $\widehat{Q}(E_0(BU))$ for $E=KU,E_2$.