UCLA

We begin by investigating analogues of the Ravenel conjectures in chromatic homotopy in the setting of Real-oriented homotopy theory, where one carries the data of canonical group actions by the cyclic group of order 2 via complex conjugation. This analysis yields a formula for Bousfield localization of $C_2$-spectra at the Real Johnson-Wilson theories, $E_\R(n)$, from which follows a smash product theorem and a chromatic convergence theorem for cofree $C_2$-spectra.

We turn to a systematic study of cofreeness in Real-oriented homotopy theory and establish the cofreeness of the norms of Real bordism theory, $N_{C_2}^{C_{2^n}}MU_\R$, for all $n\ge1$, recovering a result of Hu and Kriz at $n=1$. The method of proof establishes a connection to the Segal conjecture for $C_2$ - also known as Lin's theorem - and yields a new, conceptual proof of this classical result.

We finish by bringing various equivariant spectra in Real-oriented homotopy theory into the world of stacks and chromatic homotopy by applying a method of Hopkins' to their fixed point spectra. We demonstrate this in detail for the Real Johnson-Wilson theories and give several modular descriptions of the stacks $\mc M_{ER(n)}$, recovering and generalizing Hopkins' description at $n=1$ of $\mc M_{KO}$ as the moduli stack of nonsingular quadratic equations.