121 problems

Let $G$ be a compact Lie group. Write ${{\mathsf{Sp}}}^G_\mathbb{Q}$ for the category of rational $G$-spectra, and let $\EuScript{A}(G)$ be an algebraic category. Greenlees's algeb…

Algebraic model conjecture. For each compact Lie group $G$, there is an abelian category $\mathcal{A}(G)$ and a Quillen equivalence

Rubin's saturation conjecture. Cyclic groups are saturated as long as the prime factors of their order are sufficiently large.

Blumberg–Hill's classification conjecture. $N_\infty$-operads are completely classified by their associated $G$-transfer systems.

Let $G$ be a finite group. The Segal conjecture. After completion at the augmentation ideal, the Burnside ring of $G$ and the cohomotopy of $BG$ should agree. This conjecture relat…

In the Morel--Voevodsky model, let $X$ be a finite ${\rm C}_2$-space, and let … be the natural map from fixed points to homotopy fixed points induced by coarse sheafification. The…

Mahowald's conjecture. There is a $2$-adic equivalence

Equivariant model conjecture. The fixed subalgebra $A^{\Phi}$ is a model for $X$.

Let $G$ be a finite group. A $G$-transfer system is a partial order on the set of subgroups of $G$ that refines the closed finite-index subgroup relation and is closed under conjug…

Proper stable module category equivalence. There should be an equivalence of symmetric monoidal stable $\infty$-categories

Intermediate-differential conjecture. For $j\gg k>0$,

Global ultracommutative structure conjecture. Equivariant $\operatorname{TMF}$ admits the structure of a global ultracommutative ring.

Let $Q_8$ be the quaternion group, let $C_2$ be its indicated subgroup, and let $N_{C_2}^{Q_8} MU\mathbb{R}$ denote the quaternionic norm of real bordism. For a subgroup $L$ of…

Let $G$ be a compact Lie group. Write $\mathcal{A}(G)$ for an abelian category and let $\mathrm{DG}\text{-}\mathcal{A}(G)$ denote the category of differential graded objects of…

Disklike characterization conjecture. The maximal compatible transfer system is

Let $G$ be the group acting in the hyperreal setting, let $C_2$ be its indicated subgroup, and let $E_{2+\sigma}$ and $BBU_{\mathbb R}$ denote the spaces appearing in the source. W…

Non-abelian generalization conjecture. The assumption that $G$ is abelian can be removed from the theorems identifying $\mathrm{Hull}(\mathcal{D}(\mathcal{U}))$ with…

Realization conjecture for compatible pairs. Every compatible pair of transfer systems can be suitably realised as an operad pair.

Abelianness conjecture for saturated groups. If $G$ is a saturated finite group, then $G$ is abelian.

Let $C_2$ be the cyclic group of order two, let $\operatorname{MSpin}^c_{\operatorname{\mathbb{R}}}$ denote Real spin bordism, let $\mathcal{P}$ be the indexing set for the Anderso…

Let $\Gmr$ be a finite group and let $p$ be a prime. For each $i$, let $\Ecal^{(p^i)}_{\Gmr,p}$ be the collection of integral $\uprho_{\Gmr}$-stable $\Hmr\underline{\FF}_p$-Euleria…

Let $G$ be a finite group. A pair of $G$-transfer systems is called compatible when it satisfies the compatibility conditions for a pairing of $N_{\infty}$-operads, and it is reali…

Let $G$ be a compact Lie group and let $\mathfrak{X}$ be a Zariski clopen subset of the space $\operatorname{Sub}(G)/G$ of conjugacy classes of subgroups. Write…

Malkiewich–Ponto's conjecture. The fiberwise Fuller trace

Let $G$ be a compact Lie group and let $\mathfrak{X}$ be a collection of conjugacy classes of closed subgroups of $G$. Write \text{G-\bf spectra}\langle \mathfrak{X} \rangle fo…