53 problems

Let $V=\bigoplus_i V_i$ be a finite-dimensional vector space, with each $V_i$ equipped with a non-degenerate symmetric bilinear pairing, and let $K\subset V$ be a subspace such tha…

Parabolic h-positivity conjecture. The Frobenius character

Let $\Delta$ be a complete fan of dimension $n$, and let ${\mathcal A}(B\Delta)$ be its associated graded algebra with elements $L_i\in {\mathcal A}(B\Delta)$ of degree $e_i$, for…

Let $S$ be a noncompact smooth curve with smooth compactification $\overline S$, let $j:S\hookrightarrow\overline S$ be the inclusion, and let $L_{\rho}$ be the local system associ…

Let $M$ be a manifold with boundary fibration … and suppose that the metric on $M$ is quasi-isometric near the boundary to … where $R\in[1,\infty)$, $-1\leq c<0$, and $k$ is a symm…

Let $X^n$ be a pseudomanifold and let $Y\subset X$ be a compact codimension-$1$ submanifold such that … where $Y\subset\subset X^{\mathrm{reg}}$. For a perversity function…

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety, and let $R(\mathcal{A})$ be the ring generated by the elements…

Fix a cocompact lattice $J\subset Z(\mathbbA)/Z(F)$ and an admissible $n$-tuple $\overline{\omega}$. Let … be the cuspidal intersection cohomology representation, and let…

Intersection-cohomology stalk conjecture. Its stalk at $(\phi,\underline{c},\underline{s})$ is isomorphic to

Let $\Delta\subset\mathbb{R}^d$ be a polytope, let $\Phi$ be its dual fan, and let $\overline{M}_\Phi$ be the quotient combinatorial intersection cohomology module defined from the…

Let $\Phi$ be a complete fan in a vector space $V$ of dimension $n$. Let $\Omega^1_\Phi$ be the sheaf with stalk $\Omega^1_{\Phi,\sigma}=\operatorname{Span}(\sigma)^\perp$, and let…

P=W and PI=WI conjectures. There exists a real Lagrangian fibration $h$ such that

Cappell–Shaneson conjecture.

Let $\Sigma$ be a rational fan defining a projective toric variety $X_\Sigma$. The combinatorial intersection cohomology is denoted $IH_{\mathrm{comb}}^*(\Sigma,\mathbb{Q})$, and f…

Let $C$ be a smooth projective curve with $g(C)\geq 2$, let $G$ be a connected semisimple group, and let $G^{\vee}$ be its Langlands dual. Let ${\mathcal L}$ be a line bundle on…

Let $G=\mathrm{SL}_n$, let $\overline{\mathrm{Bun}}_{B^-}$ be the compactification of the moduli stack of $B^-$-bundles, and let $\mathrm{IC}_{\overline{\mathrm{Bun}}_{B^-}}$ denot…

Let $G$ be a reductive group, let $\overline{\mathrm{Bun}}_{N^-}$ be the compactification used in the paper, and let $\mathrm{IC}_{\overline{\mathrm{Bun}}_{N^-}}$ denote its inters…

Intersection-cohomology ordinariness conjecture. For every prime number $\ell$, there exists an infinite set of maximal ideals $\mathfrak{p}$ of $\mathscr{O}_F$ with residue charac…

Katz's intersection-cohomology Newton–Hodge inequality. There exists a finite set $S=S(X,n)$ of maximal ideals of $\mathscr{O}_F$ such that for every prime number $\ell$ and every…

Let $(S,H)$ be a polarized surface, let $\beta$ be an effective divisor, and let $M_{\beta,\chi}$ denote the moduli space of polystable one-dimensional sheaves on $S$ with determin…

Let $G$ be a reductive group with maximal torus $T$, let $V$ be a weakly symmetric representation of $G$, and let $\lambda\in X_*(T)$. Write $G^{\lambda}$ for the centralizer of th…

Let $b\in B(G)$ be basic, let $\mu_\bullet$ be a finite collection of cocharacters, and let $\rho$ be an irreducible admissible representation of $G_b(\bbQ_p)$ whose parameter…

Hodge–Zucker conjecture. The isomorphism in Theorem 2.8 is an isomorphism of $\mathbb{R}$-Hodge structures. The question refines Zucker's conjecture by asking whether its cohomolog…

Let $r,n$ be integers with $r>0$, not necessarily coprime, and let $\mathcal{M}_{B}(r,n)$ and $\mathcal{M}_{Dol}(r,n)$ be the Betti and Dolbeault moduli spaces. Let…

Billey and Postnikov's conjecture. If $X_w\subset G/B$ is singular, then