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Tuesday, April 27, 2021

**Abstract:** Quotients, localizations and completions of $BP$ play a central role in chromatic homotopy theory. For example, the Johnson-Wilson spectra $E(h)$ obtained by a quotient and localization of $BP$ are key players in the chromatic story at height $h$. However, working only with $E(h)$, the equivariance inherent to the chromatic story coming from the Morava stablizer group is obscured. A first step is to instead consider $E_{\mathbb{R}}(h)$, the Real Johnson-Wilson $C_2$-spectrum. However, for many heights $h$ there are bigger subgroups of the Morava stabilizer group lurking and $E_{\mathbb{R}}(h)$ does not capture their action. Indeed, restricting to finite cyclic 2-groups and for $h=2^{n-1}m$, the stabilizer group contains a subgroup isomorphic to $C_{2^n}$. In this talk, I will explain how one can instead consider quotients of norms of $N_{C_2}^{C_{2^n}}BP_{\mathbb{R}}$ to construct height $h$, $C_{2^n}$-equivariant analogues of $E(h)$.

For Zoom info, please contact vesna@illinois.edu