2. Equivariant Stable Homotopy Theory

When $G = C_p$, any genuine $G$spectrum can be described via gluing data of the form $(X,Y, X\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$spectra for any finite group $G$.
Problem 2.1.
[S. Glasman] What is the description of $Comm_G(Sp^G)$ using the above model? 
In [MR1410465], Hesselholt and Madsen show that for any commutative ring $R$, \[\pi_0 THH(HR)^{C_{p^n}} \cong W_{n+1}(R),\] the $p$typical Witt vectors of length $n+1$. Furthermore, in [1401.5001v2], Angeltveit, et. al. describe a construction of the cyclotomic structure on THH using $N^{S^1}_e$, an extension of the HillHopkinsRavenel multiplicative norm.
Problem 2.2.
[L. Hesselholt] If $\underline{R}$ is a commutative Green functor, what is a Wittvector model for $\pi_0(N^{S^1}_{C_p} H\underline{R})^{C_{p^n}}$? 
In order to solve the $3$primary Kervaire invariant one problem using analogous methods to those in [0908.3724], one would need a $C_3$equivariant version of $MU_{\R}$.
Problem 2.3.
[D. Ravenel] Can we describe such a $C_3$commutative ring spectrum $MU_{\R}$ so that its underlying Bousfield type is $MU$ and $\langle MU^{\Phi C_3} \rangle = \langle H\F_3 \rangle$? 
Problem 2.4.
[J. Greenlees] What should the correct notion of a Real spectrum be? Is it â€śmore" than a genuine $C_2$spectrum? (see [0908.3724] Sec B12) 
Problem 2.5.
[J. Greenlees] Can we understand the thick subcategory of $Sp^G$ generated by representation spheres? 
In [MR1337494], Thomason showed that connective spectra can be modeled by symmetric monoidal categories.
Problem 2.6.
[M. Merling] Is a similar statement true equivariantly? That is, is there an equivalence between $G$symmetric monoidal categories and the homotopy category of connective $G$spectra? 
Long Term
Problem 2.7.
[L. Hesselholt] Can we construct a decompleted version of the Tate spectrum? 
Problem 2.8.
[K. Ormsby] Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6) 
In [MR2061856] Morel showed that MilnorWitt Ktheory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.
Problem 2.9.
[J. Heller] Can we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.
Cite this as: AimPL: Equivariant derived algebraic geometry, available at http://aimpl.org/equideralggeom.