18 problems

Tannakian variation conjecture. The variation of Hodge structure associated to $Gr_W H^i({\rm Bun}^s_{G,C})$ lies in the Tannakian subcategory generated by $R^1\pi_*{\mathbb Q}$ fo…

Let $G$ be a rank-one compact Lie group with maximal torus $T$, let $\mathfrak{M}$ be the moduli stack of $G$-bundles, and let $\mathfrak{M}_T$ be the corresponding moduli stack of…

Let $G$ be a compact Lie group, let $\Sigma$ be a closed surface of genus $g$, and let $\mathfrak{M}$ be the moduli stack of $G$-bundles on $\Sigma$. Let $\Pi:G^{2g}\to G$ be the p…

Let $G$ be a compact Lie group, let $\mathfrak{M}$ be the moduli stack of $G$-bundles over a genus-$g$ surface, let $\tau_0$ and $\sigma$ be the twistings appearing in the index co…

Full strange duality conjecture. The morphism $SD$ is an isomorphism. This is the global formulation of the full strange-duality statement for the generalized Verlinde bundles; the…

Göpel's orbital degeneracy-locus conjecture. Let $v$ be a general element of $\Delta$. Then:

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 $N_{r,d}$ be the moduli space of stable rank-$r$ bundles of degree $d$ on a smooth projective curve of genus $g$, and let $\Omega(N_{r,d},q,t)$ denote its refined Poincaré poly…

Let $N_{3,1}$ be the moduli space of stable rank-three bundles with fixed degree one on a smooth projective curve of genus $g$, and let $\Omega(N_{3,1},q,t)$ denote its refined Poi…

Let $\mathcal C$ be a smooth complete irreducible curve of genus $g\geq 2$, and let $\mathcal E$ be a stable bundle on $\mathcal C$ of degree $2g-1$. Define $F_{\mathcal E}$ to be…

Let $C$ be a smooth projective curve of genus $g\geq 2$, let $L$ be a line bundle of degree $1$ on $C$, and let $M(3,L)$ be the moduli space of rank-three stable vector bundles on…

Let $C$ be a smooth projective curve over an algebraically closed field of characteristic zero, let $L$ be a line bundle on $C$ of degree coprime to $r$, and let $M(r,L)$ be the mo…

Second analytic Langlands conjecture. The spectrum $\operatorname{Spec}_{\mathcal A}(\mathcal H)$ is discrete; equivalently, $\mathcal H$ is the completed direct sum of finite-dime…

Let $X_0$ be an irreducible nodal curve with exactly one node, and let $U_{X_0}(2,\mathcal{L}_0)$ be the moduli space of rank-two semistable sheaves on $X_0$ with determinant…

Let $G$ be a reductive group, let $X$ be the underlying smooth projective curve, and let $\bun_G$ be the moduli stack of $G$-bundles on $X$. Write $\bG$ for the Langlands dual grou…

Rigid special presentation conjecture. There exists an irreducible, regular component $\mathcal B\subseteq\widetilde{B_C^{k_i}}(d)$ such that

Let $G^{sc}$ be the simply connected cover of the derived group of a reductive group $G$, and let $\operatorname{Bun}_{G^{sc}}$ be the stack of $G^{sc}$-torsors on $X$. A smooth…

Let $X$ be a global curve, let $G$ and $G$ be Langlands-dual groups, and let $Bun_G$ and $Bun_{G}$ be their stacks of principal bundles on $X$. For twisting parameters $c$ and…