Periodic phenomena in equivariant stable homotopy theory

ABSTRACT Building off of many recent advances in the subject by many different researchers, we describe a picture of $A$-equivariant chromatic homotopy theory, which mirrors the now classical non-equivariant picture of Morava, Miller–Ravenel–Wilson and Devinatz–Hopkins–Smith, where $A$ is a finite abelian $p$-group. Specifically, we review the structure of the Balmer spectrum of the category of $A$-spectra, and the work of Hausmann–Meier connecting this to $\mathrm{MU}_A$ and equivariant formal group laws. Generalizing work of Bhattacharya et al., we introduce equivariant analogues of $v_n$-self maps, and generalizing work of Carrick and Balderrama, we introduce equivariant analogues of the chromatic tower, and give equivariant analogues of the smash product and chromatic convergence theorems. The equivariant monochromatic theory is also discussed. We explore computational examples of this theory in the case of $A = C_2$, where we connect equivariant chromatic theory with redshift phenomena in Mahowald invariants.