5. Chromatic Homotopy Theory

For a finite group $G$, Lurie constructs equivariant elliptic cohomology in commutative naive $G$spectra and in genuine $G$spectra compatibly [MR2597740].
Problem 5.1.
[C. Rezk] Can this construction be extended to commutative genuine $G$spectra or $G$commutative genuine $G$spectra? 
Problem 5.2.
[K. Ormsby] What would “rings of integers” in $tmf$ with level structure be? Also, is there a TAF analogue of such objects with connections to TMF? 
Problem 5.3.
[M. A. Hill] What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)? 
We may examine how viewing the Tate spectrum of a given $G$spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$action, we have \[ (KO_{(2)})^{tC_2} \simeq \bigvee_{k\in\Z} \Sigma^{4k} H\Q_{(2)}\]
Problem 5.4.
[M. Behrens] Why should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we’d like to look at "chromatic primes" and "equivariant primes" in the same place and understand how they interact.
[Needs editing  I’m sure Mark has a more eloquent statement] 
Problem 5.6.
[M. A. Hill] When $G = C_{2^n}$ what is the $G$equivariant Landweber exact functor theory for $MU^{(G)}$? 
Long Term
Problem 5.7.
[S. Glasman] Can we describe the moduli stack of formal groups in the derived setting?
Remark. [org.aimpl.user:carolyn.yarnall@uky.edu] Was this question really about the derived moduli stack of formal groups or the moduli stack for derived formal groups or should they be one and the same?


Problem 5.8.
[V. Stojanoska] Can we find explicit ways to write $S \in RAlg$ as an $R$module? What are objects that “untwist" when smashed together? (e.g. $KU \simeq KO \wedge C(\eta)$, $H\F_2 \simeq ko \wedge \A(1)$)
A slightly easier question: Is $tmf^{(n)}_0 \rightarrow tmf^{(n)}_1$ relatively Gorenstein?
Cite this as: AimPL: Equivariant derived algebraic geometry, available at http://aimpl.org/equideralggeom.