24 problems

Let $\cat{C}$ be a $tt$-category. The nerves of steel condition holds when the map … is a bijection. Nerves of Steel Conjecture. Every $tt$-category satisfies the nerves of steel c…

Let $\mathcal K$ be a finite tensor category, let $\mathsf{C}$ denote its categorical center, and let … be the comparison map. The comparison-map homeomorphism conjecture. For ever…

Let $A$ be a finite-dimensional unipotent Hopf algebra, let $A^{\operatorname{env}}$ be its enveloping Hopf algebra, and let … be the comparison map. The comparison-map conjecture.…

Let $\mathscr{K}^c$ be a rigid tensor triangulated category. Its homological spectrum is $\operatorname{Spc}^{\text{h}}(\mathscr{K}^c)$, its Balmer spectrum is…

Let $(\mathfrak{f},\mathfrak{f}_{\bar 0})$ be a detecting subalgebra of a classical Lie superalgebra, and let $\operatorname{Stab}(V{C}_{(\mathfrak{f},\mathfrak{f}_{\bar 0})})$ den…

Let $\mathsf{T}$ be a tensor-triangulated category with compact objects $\mathsf{T}^{\mathrm{c}}$. Write $\mathsf{Spc}^{\mathsf{h}}(\mathsf{T}^{\mathrm{c}})$ for the space of homol…

Property PB conjecture. All restricted Lie algebras satisfy Property PB regardless of representation type.

Property PC conjecture. For any restricted Lie algebra ${\mathfrak g}$ over $k$, ${\mathfrak g}$ satisfies Property PC if and only if ${\mathfrak g}$ is of tame or finite represent…

Central cohomological support conjecture. The central cohomological support satisfies these conditions, and consequently

Stratification conjecture. The ring $R$ stratifies $\mathbf{K}$.

Let $\mathbf T$ be a finite tensor category, let $\underline{\mathbf T}$ be its stable category, and let … be the surjective continuous map from Theorem B. Vashaw–Yakimov's conject…

Let $\mathbf T$ be a finite tensor category and let $\underline{\mathbf T}$ be its stable category. Write $C^\bullet_{\underline{\mathbf T}}$ for the categorical center of its coho…

Let $0 \le i \le n+1$. For $i \le n$, let $\mathcal D_i$ be the full subcategory of dualizable objects $X$ in $\Sp_{k,n}^{\dual}$ that are retracts of $Y \wedge Z$ for some dualiza…

Let $\cat K$ be an essentially small tensor triangulated category. Write $\Spch(\cat K)$ for its homological spectrum, $\Spc(\cat K)$ for its Balmer spectrum, and … for the canonic…

Equivariant E(n)-local spectrum image conjecture. The map $\varphi_G$ is a homeomorphism onto its image.

A small quantum Borel algebra is associated to a complex simple Lie algebra and a choice of group-like elements; its cohomological support map assigns support data to objects in th…

Let $K(n)$ be the Morava $K$-theory spectrum at height $n\geq 0$, and let $\operatorname{Sp}_{K(n)}^{\operatorname{dual}}$ denote the category of $K(n)$-locally dualizable spectra.…

Let $\FF$ be a general real closed field, and consider the canonical comparison map … The fiber above the generic point of $\Spec(\bbZ)$ is the fiber whose structure is undetermine…

Let $(\mathcal{T},\otimes,\mathbb{1})$ be a tensor-triangulated category, with compact objects $\mathcal{K}=\mathcal{T}^c$. Let $\operatorname{Spc}(\mathcal{K})$ denote the Balmer…

Let $\mathcal{T}$ be an arbitrary essentially small tensor triangulated category. Write … for its graded-commutative central ring, and let … be Balmer's continuous map, defined by…

Let $SH(\mathbb{F}_q)^c_\mathbb{Q}$ be the compact rational stable motivic homotopy category over $\mathbb{F}_q$, and let $\operatorname{Spc}$ denote its tensor-triangular spectrum…

The $\log_p$-conjecture. For all $n\geq r$, one has

Let $\mathscr{T}$ be a compactly generated $\otimes$-triangulated category and let $\mathscr{U}$ be a compactly generated triangulated category, both with all coproducts. Suppose ……

Spectrum conjecture. For every finite group $G$, the natural map