26 problems

Let $C$) be a Riemann surface with a compatible $\mathbb{CP}^1$-structure and a spin structure. Let $u=(u_2,\ldots,u_n)\in\bigoplus_{i=2}^n \mathrm{H}^0(C,K^{\otimes i})$, let…

Generic-oper contractibility conjecture. The space of generic oper structures on a given irreducible local system is homologically contractible.

Let $W_r$ be the polynomial defined from the unique exponential-Gaudin solution indexed by $r$, and consider the monodromy system in the variables $w_i$ displayed in the source.…

Let $L_0$ be the displayed BLZ-type oper, with distinct points $t_j\notin\{0,1\}$, satisfying the condition $b_0+b_1+\sum_iq_i=0$ and having monodromy-free singularities at $x=1$ a…

Let $L_1$ be the displayed scalar oper with distinct nonzero points $s_i$, integer $r$, and parameters $k,l,n_1$ as in the source. Gaudin no-monodromy conjecture. For generic $k,l$…

Torelli conjecture for opers. The following conditions are equivalent:

Generic transversality conjecture. The two real submanifolds $\mathrm{Op}_P(\mathscr{X},G)$ and $\mathrm{Rep}_P^0(\Gamma,G(\mathbb{R}))$ intersect transversally at every point. Equ…

Let $G$ be a connected complex semi-simple group with real structure, let $\mathcal C$ be a compact complex Riemann surface with antiholomorphic involution $\tau$ and no real point…

Let $\mathcal C$ be a compact complex curve, let $G^{\vee}$ be the Langlands dual group, and let $o$ be a real $G^{\vee}$-oper. For a dominant coweight $\lambda$, let $V_\lambda$ b…

Let $\operatorname{Oper}_{G^{\vee}}(\mathcal C)^{\mathbb R}$ denote the real locus of $G^{\vee}$-opers on a compact complex curve $\mathcal C$, and let $L^2(\operatorname{Bun})_o$…

Let $\chi$ be an $SL_n$-oper with real monodromy. Let $(\mathcal V_{\omega_1},\nabla_\chi)$ be its associated flat vector bundle, with determinant identified with the trivial flat…

Frenkel–Gaitsgory's main conjecture. The induced functor

Beilinson–Drinfeld regular-singularity conjecture. The $D$-module $\Delta^0_\lambda$ is irreducible and has regular singularities on each connected component of…

Basic spectral decomposition conjecture. The Hilbert space has an orthogonal decomposition

Let $M_{\rm dR}(X,r)$ be the de Rham moduli space with its stratification, and let $G_u^1$ denote the oper stratum. The source identifies $G_u^1$ as the highest stratum. Simpson's…

Let a Higgs bundle lie in the Hitchin component, and let its conformal limit be the flat connection obtained from the conformal-limit construction. An oper is a flat bundle satisfy…

Let $P_n(N)$ be the number of $n$-coloured partitions of $N$, let $n=\operatorname{rank}\mathfrak{g}$, and let $\operatorname{systemonc}$ denote the displayed system of equations f…

Let a quantum $\widehat{\mathfrak{g}}$-KdV oper be an oper of the form … with the additional regular singularities chosen to have trivial monodromy for every value of $\lambda$. Le…

Let $d$ be a positive exponent with $d>1$, let $\gamma$ be a Pochhammer contour, and let $\Psi^{\mathrm{on}}_{\bm c}(\bm w)$ be an on-shell Bethe state in $H_{\bm\lambda}$. Associa…

Affine Gaudin eigenvector conjecture. For every exponent $i$ and every permitted contour $\boldsymbol{\beta}$,

Gaiotto's conjecture. The scaling limit

Let $\lambda_0\in\mathfrak{h}^{*,\nu}$ be the dominant $\nu$-invariant weight satisfying $\langle\check\alpha_i,\lambda_0\rangle\in\mathbb{Q}$ for all $i\in I$. Let…

Let $M_{\l,\mu,\nu}$ be the set of monodromy-free ${}^L G$-opers on $\mathbb{C}\mathbb{P}^1$ with regular singularities of residues $\l,\mu,\nu$ at $0,1,\infty$, respectively. Defi…

Let $\operatorname{LocSys}_\cG$ be the stack of $\cG$-local systems and let … be the map from global opers with singularities parametrized by the Ran space. Ran-space oper contract…

Let $\operatorname{LocSys}_\cG^{\operatorname{irred}}$ be the open substack of irreducible $\cG$-local systems. For every finite set $I$ and map $\lambda^I:I\to\Lambda^+$, let…