29 problems

Let $G$ be a reductive group with Borel subgroup $B$, unipotent radical $U$, maximal torus $T=B/U$, and quotient map $\pi:G\to G/U$. Let $\Phi_{G,\rho,\psi}$ be the $\rho$-Bessel s…

Admissibility conjecture. The assumption that $G$ satisfies the two-three condition is only technical and is forced by the techniques used to prove admissibility of compact groups.

Admissibility conjecture. It should be possible to remove the two-three condition using other techniques.

Let $X$ be a manifold, and for each open subset $U\subset X$ let ${\cal S}_i(U)$ be the space of continuous valuations on $U$. These spaces form a presheaf ${\cal S}_{i,X}$ on $X$.…

Let $E$ be a rank $2$ torsion-free sheaf on $\mathbb{P}^k$ that is the cohomology of a special monad. Special-monad semistability conjecture. Then $E$ is semistable. This extends t…

Let Setup be fixed. For a weight $v \colon P \to \mathbb{R}_{>0}$, let $C_v>0$ be a constant, and let $\mathcal{E}$ be a $\mathbb{G}$-equivariant torsion-free sheaf. The equivarian…

An almost basic set is a set of the type defined in the paper, and $H^k(U,\mathcal{LIP})$ denotes its degree-$k$ cohomology with coefficients in the sheaf of locally integer-valued…

Let $G$ be a group, let $\Shv(G)^{\on{q-mon}}$ denote the quasi-monodromic sheaf category, and write $G\mmod$ for the category of categorical representations of $G$. Let … be the i…

Let $\varphi\colon W_F\to{{}^LG}({\overline{\mathbb{Q}}_\ell})$ be a semisimple $L$-parameter of Langlands–Shahidi type, and let ${\mathcal{F}}_\varphi$ be the corresponding Hecke…

Let $X$ be a $t$-dimensional cubical complex and let $\mathcal{F}$ be a sheaf satisfying the requirement in Dinur (2024). For integers $i,j,k,l$ satisfying … let…

Let $k$ be a field of positive characteristic, let $G$ be a reductive group, and let $\Bun_{G,\sk}$ be the moduli stack of $G$-bundles in the positive-characteristic setting. Write…

Let $X_\Sigma$ be the toric variety with Givental–Hori–Vafa mirror $W_\Sigma$, and let $\widehat{\Lambda}_{\geq}$ denote the Novikov ring. Write…

Let $\mathcal M$ be the monad on probability sheaves defined by … Let $\operatorname{Kl}(\mathcal M)$ denote its Kleisli category, and let $\mathbb{{C}}$ be the underlying Markov c…

Let $A$ be a homotopical ring and let ${\mathrm{Spec}^{\hspace{-0.4mm}\tiny{\mathtt{NC}}}\hspace{-0.5mm}}(A)$ be its noncommutative spectrum equipped with the structure pre-sheaf.…

Equivalence conjecture. The functor $\bar\xi_{C,\lambda}$ is an equivalence.

The filtration–singular-support conjecture.

Let $G$ be a profinite group acting on a $G$-space $X$. Suppose that $X$ has finite Cantor--Bendixson rank and non-empty perfect hull. Equivariant Sugrue conjecture. The homologica…

Let $\Gamma$ be a finite group and $(E,X,\pi)$ be a principal $\Gamma$-bundle, where $(X,\widetilde{\mathbb{Z}}_X)$ is a ringed, connected, compact, Hausdorff, and PCVP space. Supp…

Sphericality conjecture. For any good stratification $A$, this pullback-limit adjunction is spherical and arises as the restriction of a spherical adjunction involving the sheaf in…

Let $G$ be a reductive group over $\mathbb F_q$, let $\rho$ be a representation as in the Braverman–Kazhdan construction, and let $\Phi_{G,\rho,\psi}$ be the corresponding $\rho$-B…

Let $G$ be a reductive group with Borel subgroup $B$, unipotent radical $U$, maximal torus $T=B/U$, and Weyl group ${\mathrm W}$. Let $\mathcal{F}\in\mathscr{D}_{\mathrm W}(T)$ be…

Let $X$ be a topological space with finite Cantor–Bendixson rank and non-empty perfect hull. Consider sheaves of $\mathbb{Q}$-modules on $X$. The conjecture. The injective dimensio…

Sheaf-theoretic conjecture. The abelian model $\mcA(G)$ is conjectured to be a category of sheaves over the space of closed subgroups of $G$, with the stalk over a subgroup $K$ giv…

Let $n\geq 1$, let $\mathbbm{k}$ be a field, and let $\Lambda\subset\mathbb{R}^3$ be a Legendrian link. The representation–sheaf conjecture. The $A_\infty$ category…

Higher singularity conjecture for sheaf lines. The order of singularity of a sheaf line is higher than that of other normal jumping lines.