30 problems

Let a Gaudin model have a corresponding system of Bethe ansatz equations, with parameters in a generic regime, and let the model act on a representation of dimension $d$. Bethe ans…

Let $k\in\mathbb{N}$ and let $c_1,\dots,c_k$ be arbitrary constants. For each $i$, let $\{\vec{J}_i\}$ denote the set of vectors satisfying … For $\vec{J}_i$, write…

Non-vanishing conjecture. In the $sl_{N+1}({\mathbb C})$ Gaudin model, every Bethe vector is non-zero, for any $z$. For some values of $z$ the number of Bethe vectors, equivalently…

Affine geometric Langlands conjecture. The Geometric Langlands correspondence generalises to affine Gaudin models, so that their spectra are encoded in affine opers.

Let $\mathfrak{so}_{2n}$ and $\mathrm{O}_m$ act through the $(\mathfrak{so}_{2n},\mathrm{O}_m)$-duality on $\mathfrak{P}_{nm}$. On the $\mathrm{O}_m$ side, consider the cyclotomic…

Let $r=d_0-d_1$ and use the polynomials $W_r,U_r$ and the Hermite-Wronskian polynomial $V_\mu$ defined in the source. Unequal-degree large-$l$ conjecture. For generic $l,m$, the Ga…

Let $W_r(x)$ and $U_r(x)$ be the recursively defined polynomials associated with the unique exponential-Gaudin solutions indexed by $r\in\mathbb Z$. Polynomial-root conjecture. For…

Let the exponential Gaudin Bethe Ansatz system be as in the source, and let $d_0=r^2$, $d_1=r(r-1)$ characterize the Virasoro highest-weight locus. Exponential Gaudin nonexistence…

Let $d_0=d_1$ and let $\operatorname{Part}(d_0)$ denote the number of partitions of $d_0$. Large-$l$ Gaudin counting conjecture. For generic $l,m$, the Gaudin Bethe Ansatz system h…

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 $k,l$…

Completeness conjecture. The operator $\mathscr H(x)$ has simple spectrum on $(L_{\bm\Lambda}(\bm b))^{\mathrm{sing}}$. There is a bijection between the monic divisors $y$ of…

Polynomial cross-ratio conjecture. A set of cross ratios making all coefficients of the $N$-point comb channel Casimir operators polynomial in $d\geq N-2$ is obtained by complement…

Let $\mathfrak{g}$ be a Lie algebra, let $\mu\in\mathfrak{g}^*\cong\mathfrak{g}$, and choose $\mu\in\mathfrak b_- = \mathfrak n_-\oplus\mathfrak h$ after conjugation. Let…

Let $\mathfrak{g}$ be a Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Cartan subgroup $H$, let $M$ be the finite-dimensional representation under consideration, and let…

Let $\mathfrak{g}$ be a Lie algebra, let $M$ be the finite-dimensional representation under consideration, let $M^{\mathrm{sing}}$ be its singular subspace, and let…

For a Kac–Moody algebra ${\mathfrak g}$ of finite type, let ${{}^L\!{\mathfrak g}}$ be its Langlands dual. Frenkel–Feigin conjecture. Affine opers and affine Miura opers should enc…

Let $\boldsymbol\lambda$ be a sequence of arbitrary $\mathfrak{gl}_{1|1}$ weights, and let $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$ be its singular subspace. Let…

Consider a tensor product $L(\boldsymbol\lambda)$ of typical polynomial $\mathfrak{gl}_{1|1}$ modules, with singular subspace $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$, and let…

Let $\boldsymbol\lambda=(\lambda^{(1)},\dots,\lambda^{(n)})$ be a typical sequence of polynomial $\mathfrak{gl}_{M|N}$ weights, let $\boldsymbol z=(z_1,\dots,z_n)$ be a sequence of…

Let $M,N$ be nonnegative integers, and consider the $\mathfrak{gl}_{M|N}$ Gaudin Hamiltonians associated with tensor products of polynomial modules. A singular eigenvector means a…

Let $\mathfrak g$ be a Lie algebra with automorphism $\sigma$, let $\Gamma$ be the associated cyclotomic group, and let…

Let $\mathfrak{g}$ be a Lie algebra, let $\lambda_i\in\mathfrak{h}^*$ for $i=1,\ldots,N$ and $\lambda_\infty\in\mathfrak{h}^*$ be integral dominant highest weights, and let…

Let $\overline{M_{0,n+1}}(\mathbb{R})$ carry Speyer's covering and the covering constructed in the source, denoted the source's covering from Corollary. Speyer covering conjecture.…

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…