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1. Commutative ring spectra

    1. Equivariant chromatic support of $N_{\infty}$-rings

          The chromatic support of a ($p$-local) $\mathbb{E}_\infty$-ring $R$ are the $n$ for which $R\otimes K(n)$ is nonzero. A theorem of Jeremy Hahn [arXiv:1612.04386] classifies the chromatic supports of an $\mathbb{E}_\infty$-ring.

      There is a notion of chromatic support in equivariant homotopy theory coming from the fact that we know the Balmer spectrum (as a set) of equivariant homotopy theory.

      Add remark

      Problem 1.05.

      [Tomer Schlank] Given an $N_{\infty}$-operad, and a $R$ an algebra over it, what are the possible chromatic supports of it?

        • Odd primary actions on Lubin–Tate theories

              A wonderful feature of the prime $2$ is that there is a great model for the action of $C_2$ on Lubin–Tate theories, coming from quotients of $\mathrm{MU}_{\mathbb{R}}$.

          Add remark

          Problem 1.1.

          [Dylan Wilson] For an odd prime $p$, can one produce an understandable model for the $C_{p^n}$-action on a $E_{(p-1)mp^{n-1}}$?

            • $\mathbb{E}_{\infty}$-total power operation maps for Lubin–Tate theories and Morava $K$-theories

                  Any $\mathbb{E}_{\infty}$-ring $R$ has a Nikolaus–Scholze Frobenius map $R \to R^{tC_p}$, which is a map of $\mathbb{E}_{\infty}$-rings. In the case $R=\tau_{\geq 0}E_1$, the map $R \to \tau_{\geq 0}R^{tC_p}$ on $\pi_0$ is the map encoding the total power operation of $E_1$. In this way, the map can be considered an $\mathbb{E}_{\infty}$-refinement of the total power operation.

              Add remark

              Problem 1.15.

              [Allen Yuan] Let $\tau_{\geq0}^{G}$ denote the connective cover in $G$-spectra which is the connective cover on each categorical fixed point. Let $R$ be $E_n$ or $K(n)$, and equip $R$ with the trivial $(C_p)^m$ action. Then compute $(\tau_{\geq0}^{(C_p)^{m}}R)^{\Phi (C_p)^{m}}$.

                • Multiplicative structure on equivariant algebraic $K$-theory


                  Add remark

                  Problem 1.2.

                  [Andrew Blumberg] How does equivariant algebraic K-theory behave with respective to multiplicative structure? For example, if I have an $N_{\infty}$-algebra or an $\mathbb{E}_{V}$-algebra $A$ in $G$-spectra, what structure does $K_G(A)$ have?

                    • Self-injectivity of equivariant ring spectra


                      Add remark

                      Problem 1.25.

                      [Clover May] Given $R$ a $G$-ring spectrum, when is the $\mathrm{RO}_G$-graded Mackey functor valued homotopy ring of $R$ self-injective?

                        • $RO_G$-graded homotopy via formal groups

                              Given a complex oriented ring $R$, $\pi_{2*}R$ can be interpreted as the line bundle on the associated formal group law.

                          Add remark

                          Problem 1.3.

                          [Mike Hill] Given a complex orientable $G$-ring spectrum $R$ (in a suitable sense), is there an interpretation of the $RO_G$-graded homotopy groups $\pi_VR$ in terms of formal groups?

                            • A $K$-theoretic criterion for maps to even rings.

                                  The $\mathbb{E}_{\infty}$-ring $\mathbb{F}_p^{tC_p}$ has two special properties that are related to each other. It admits no nontrivial $\mathbb{E}_{1}$ maps to an even ring, and its rational $K$-theory vanishes.

                              Add remark

                              Problem 1.35.

                              [Ishan Levy] Given an $\mathbb{E}_{1}$-ring $R$ with the unit nontrivial in rational $K$-theory, does $R$ admit an $\mathbb{E}_{1}$ map to a nonzero even ring?

                                • Spin orientation from equivariant homotopy theory


                                  Add remark

                                  Problem 1.4.

                                  [Mike Hill] Can the spin orientation $\mathrm{MSpin} \to \mathrm{ko}$ be recovered from the map $\mathrm{MU}_{\mathbb{R}} \to \mathrm{ku}_{\mathbb{R}}$?

                                    • Detecting Hopf invariant $1$ elements in fixed points of Lubin–Tate theories


                                      Add remark

                                      Problem 1.45.

                                      [Hood Chatham] Do the Hopf invariant one elements that don’t exist in the sphere also not exist in $E_n^{hG}$ if the group $G$ is sufficiently large?

                                        • Blueshift for $\mathbb{E}_{\infty}$-rings

                                              Blueshift suggests that taking $tC_p$ lowers height. However given an $\mathbb{E}_n$-ring $R$, if $n<\infty$ of height $h$, $R^{tC_p}$ may not be of height $h-1$.

                                          Add remark

                                          Problem 1.5.

                                          [Allen Yuan] Given an $\mathbb{E}_{\infty}$-ring $R$ that is $K(n)$-acyclic, is $R^{tC_p}$ $K(n-1)$-acyclic?

                                            • Multilinear relations for 1-semiadditive power operations

                                                  Given a $G$ acting on an $X$ in a $1$-semiadditive stable symmetric monoidal category $C$, the $1$-semiadditive structure gives rise to a power operation $\theta_{G,X}$. In the case $C$ is height $1$ (such as $K(1)$-local spectra), this operation satisfies a bilinear relation with respect to products corresponding to the fact that the $\theta$ operation gives rise to a ring homomorphism at height $1$.

                                              Add remark

                                              Problem 1.55.

                                              [Tomer Schlank] At height $n$, does the operation $\theta_{G,X}$ satisfy an $n+1$-multilinear equation with respect to products?

                                                • $\mathrm{THH}$ for structured $G$-ring-spectra


                                                  Add remark

                                                  Problem 1.6.

                                                  [Mona Merling] Is is possible to build a version of $\mathrm{THH}$ for $G$-ring-spectra that is $G$-symmetric monoidal?

                                                      Cite this as: AimPL: Equivariant techniques in stable homotopy theory, available at http://aimpl.org/equivstable.