67 problems

The geometric Langlands correspondence. There exists a canonical equivalence of categories

Let $r$ be a positive integer, and consider the moduli stack of smooth $r$-spin $1$-pointed curves of genus $1$. Jarvis–Picard finiteness conjecture. The Picard group of this modul…

Let $C$ be a smooth projective geometrically connected curve over $k$, of genus $g$, and let $G$ be a split semisimple connected algebraic group over $k$. Denote by…

Harder's conjecture. If ${\mathcal G}$ is split, the Tamagawa number of $G$ equals the number of connected components of the moduli space of ${\mathcal G}$-torsors on $X$.

Geometricity conjecture. The derived stack $\mathbb{R}\underline{Vect}_{n}(X)$ of vector bundles on $X$ is geometric.

Let $X$ be a smooth projective curve over a ground field $k$, let $E$ be an $m$-dimensional local system on $X$, and let $\Bun_m$ be the moduli stack of rank $m$ vector bundles on…

Let $f:X\to S$ be a projective morphism of smooth varieties, and let $\underline{{\cal U}}$ be a (linearized) pre-vnamhs over $X$. Relative sections conjecture. There is a (lineari…

Let $G$ be a quasi-split reductive group over $E$, let $\operatorname{Bun}_G$ be its moduli stack of $G$-bundles, and let $\operatorname{Bun}_G^b$ denote the stratum indexed by…

Let $M_N$ be the KSBA moduli stack of index one covers, let $M_N^{\operatorname{Bub}}$ be the moduli stack of bubble-tree index one cover DM stacks, and let…

Correspondence conjecture. The smooth loci of ${\mathcal U}$ and ${\mathscr U}_{\mathbb C}$, respectively the normal loci of ${\mathcal U}$ and ${\mathscr U}_{\mathbb C}$, correspo…

Let $S$ be the surface and let $Coh^{ss,0}_c(S)$ denote the category of parabolic semistable coherent sheaves on $S$ of parabolic degree zero with pure one-dimensional compact supp…

Very good implies $L^2$. If $\mathcal X$ is very good, then $\mathcal X$ is an $L^2$-stack.

Let $(G,\mu)$ be a Shimura datum, let $E$ be the completion at a place above $p$ of its reflex field, and let $K=K_pK^p\subset G(\mathbb{A}_f)$ be a compact open subgroup. Write…

Categorical non-Abelian Hodge conjecture. There are mappings

Cartier–Witt divisor conjecture. An $R$-point of $\widehat{\bbF}_1^\mathrm{Syn}$ is given by a formal group over $R$, a Cartier–Witt divisor on this pullback, and an isomorphism ov…

Let $\bbS^{\widehat{\mathbbl{\Delta}}}$ be the complement of the zero section in the universal formal group $\widehat{\bbG}_\mathrm{univ}=\bbS^{\widehat{\calN}}$, regarded as an an…

The paper considers a maximal finite subgroup $G'\subseteq \mathbb{G}_h$, a minimal-genus curve $X$ with automorphism group $G'$, the deformation stack $\operatorname{Def}_{(X,G)}$…

Let $X$ be a smooth complete curve over a field $k$, let $G$ be a reductive group, and let $\Shv_{\Nilp}(\Bun_G)$ be the category of $\ell$-adic sheaves on $\Bun_G$ with nilpotent…

Várilly-Alvarado and Vakil's density-function conjecture. The quantities $\rho^{\textnormal{geo}}(x)$ and $\rho^{\textnormal{gr}}(x)$ exist and are independent of the choice of…

Várilly-Alvarado and Vakil's Grothendieck-ring measure conjecture. For all $B$, the limit $\lim_{g\to\infty}\pi^{\textnormal{gr}}(B,g)$ converges to a measure…

Várilly-Alvarado and Vakil's limiting conjecture. For all $B \subseteq \mathbb{R}^{d-1}$, the limit $\lim_{g \to \infty}\pi^{\textnormal{geo}}(B,g)$ exists and depends only on the…

Let $\mathscr{S}$ be a stratum of a birationally admissible stratification of $\mathscr{M}_{\rm slc}(d, \Phi_c, \Gamma, \sigma)$. Logarithmic uniformisability conjecture. The pair…

Let $X$ be a curve over a field $k$ of characteristic $0$, let $\mathcal{G}$ be the parahoric group scheme under consideration, and let $L$ be a line bundle on the moduli stack…

Let $\mathcal{B}$ be a hereditary $\mathcal{O}_X$-order that is Iwahori at each $y\in Y$, and let $I_\bullet=(I)$. Write $\widetilde{\mathbf{w}}_{\operatorname{bsc}}$ for the basic…

Let $X$ be a smooth complex projective curve of genus $g$, let $G$ be a connected semisimple complex algebraic group of rank $r$ with exponents $d_1,\ldots,d_r$, and let $Z_C(t)$ d…