36 problems

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

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 $X$.

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…

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…

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…

Structural decomposition conjecture. The stack $\operatorname{LS}_G^{\mathrm{restr}}(D^{\circ})$ is a disjoint union of formal completions of a collection of pairwise non-intersect…

Category-theoretic and quiver-bundle moduli stack equivalence conjecture. There is an equivalence of stacks

Condensed and bornological geometry model conjecture. The models of geometry in condensed mathematics, such as analytic stacks, and the models of geometry based on bornological rin…

Derived-stack universal vector bundle conjecture. These choices satisfy the assumptions introduced for the moduli stack of vector bundles and its universal vector bundle.

Rational cohomology conjecture. The rational cohomology of $\mathscr{B}un_{G,X}^0$ is

Let $\mathfrak{M}_{Dol}(r,0)$ and $\mathfrak{M}_{B}(r,0)$ be the stacks of semistable Higgs bundles and semisimple representations, respectively. Let $W_\bullet$ be the weight filt…

Representability conjecture. The fpqc sheaf $\prescript{}{1}{\mathcal{C}_1^{\leq\mu}}$ is represented by a normal $k$-scheme. This predicts that the indicated truncated double quot…

Let $r$ be a positive integer and $d$ a positive odd number. The classes $4c_1,c_24$ generate the polynomial ring describing the presentation, and let…

Pointed Shafarevich conjecture. The set of morphisms $f:C\to U$ satisfying $f(c)=u$ is finite. This predicts finiteness of families over a curve after fixing one fibre over one mar…

Uniformizability conjecture. The stack $\mathcal{M}_h$ is uniformizable by a quasi-projective scheme over $\mathbb{Q}$. Uniformizability would allow finiteness results proved for s…

Smoothness conjecture. The morphism

Chiral D-affineness conjecture. The stack quotient $(R\oplus\Pi V)/G$ is chiral $D$-affine.