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2. Equivariant Stable Homotopy Theory

    1.     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 $G$.

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      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 Hill-Hopkins-Ravenel multiplicative norm.

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          Problem 2.2.

          [L. Hesselholt] If $\underline{R}$ is a commutative Green functor, what is a Witt-vector 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 $MU_{\R}$.

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              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 $\langle MU^{\Phi C_3} \rangle = \langle H\F_3 \rangle$?


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                  Problem 2.4.

                  [J. Greenlees] What should the correct notion of a Real spectrum be? Is it “more" than a genuine $C_2$-spectrum? (see [0908.3724] Sec B12)


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                      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.

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                          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


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                              Problem 2.7.

                              [L. Hesselholt] Can we construct a decompleted version of the Tate spectrum?


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                                  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 Milnor-Witt K-theory 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.

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                                      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.