34 problems

Let $X$ be a smooth connected projective algebraic curve over $\mathbb{C}$, let $G$ be a reductive group over $\mathbb{C}$, and let…

Let $\mathcal{N}$ be the moduli space of stable vector bundles, and let $\mathcal{W}$ denote the locus of wobbly bundles, namely stable bundles that are not very stable. A bundle i…

Let $X$ be the curve and let the base locus of the generalized theta divisor refer to the locus of points where the associated linear system fails to be generated. Base-locus dimen…

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…

Geometric Langlands conjecture. If $E$ is an absolutely irreducible $n$-dimensional local system on $X$, then there exists a perverse sheaf ${\mathcal S}_E\in P(\Bun_n)$ that is a…

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…

Let $G$ be a compact Lie group, let $\Sigma$ be a closed surface of genus $g$, and let $\mathfrak{M}$ denote the moduli stack of $G$-bundles on $\Sigma$. Let $\tau$ be an admissibl…

Let $X$ be the underlying smooth projective curve, let $\bar R$ be the coefficient ring with local homomorphism $\bar R\to\mathcal O$, and let $\bar E$ be a smooth $\bar R$-sheaf o…

Let $X$ be a smooth projective curve of genus $g$, let $SU_X(r)$ denote the moduli space of semistable rank-$r$ vector bundles with trivial determinant, and let $U_X(k,0)$ denote t…

Let $\mathcal{V}ec^\mathcal{H}(r,d,g)$ be the universal moduli stack of vector bundles of rank $r$ and degree $d$ over hyperelliptic curves of genus $g$. For a hyperelliptic curve…

Let $C$ be a smooth projective curve, let $S$ be a finite set of points containing all bad points of the group scheme $\mathcal{G}$, and let $q^*$ be the pullback map from the Pica…

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…

Lange's conjecture. A generic choice of stable bundles $E_{r_1,d_1}$ and $\bar E_{r-r_1,d-d_1}$, together with a generic choice of the extension between them, gives rise to a stabl…

Let $\operatorname{Bun}_n$ be the moduli stack of bundles over the $n$th infinitesimal reduction of the curve, let $\mathcal S_{cusp}(\operatorname{Bun}_n)_\chi$ be the cuspidal su…

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 $G$ be a split connected semi-simple group, let $\mathcal C$ be a smooth complete irreducible curve of genus $g\geq 2$, and let $\operatorname{Bun}=\operatorname{Bun}_G$ be the…

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…

First analytic Langlands conjecture. The pair $(\mathcal A_{\mathbb R},S(\mathcal A))$ is a strongly commuting family of unbounded essentially self-adjoint operators on…