106 problems

Symplectic invariance conjecture. The free energies $\mathcal{F}^g$ are invariant under any symplectic transformation of $(x,y)$.

Higher BGW topological-recursion conjecture. The generating functions of the higher BGW models $\tau^{(m)}$ are given by a version of the Chekhov--Eynard--Orantin topological recur…

Functional-relations conjecture. For every $2g-2+n\geq 0$, these differentials satisfy the functional relations of Theorem, after those relations are expressed as relations between…

Let $\omega_{g,n}$ be the multidifferentials produced by topological recursion from a spectral curve with functions $x$ and $y$, and let $F_g$ denote the corresponding free energy.…

Let $\mathcal{S}$ be an admissible spectral curve with $x,y$ satisfying … A quantisation of $P$ is an operator $\hat{P}(\hat{x},\hat{y};\hbar)$ of the form … where…

Pole structure conjecture. All $\omega_{g,n}$ are meromorphic differentials on the spectral curve and have poles of finite degree only at the branch point $z_j=0$.

Taimanov's conformal invariance conjecture. The higher Willmore functionals and the spectral curve for tori in ${\mathbb R}^3$ are conformally invariant.

Let ${\cal D}$ be a double-periodic Dirac operator in a Weierstrass representation of a torus, and let $\Gamma$ be its spectral curve on the zero energy level. Let…

Quadratic growth conjecture. The quantities $\operatorname{l-ind}(\Sigma)$, $\operatorname{s-ind}(\Sigma)$ and $\operatorname{Area}(\Sigma)$ grow quadratically with $d$.

$L^q$-realization conjecture. All complex Fermi curves $\mathcal{F}(U,\overline U)$ are equal to the complex Fermi curve of some potential $U'\in L^q(\mathbb{R}^2/\Lambda)$, and th…

Finite-genus characterization conjecture. These conditions characterize the elements of $\overline{\mathcal{M}}_{\Lambda,\eta,\mathscr{W}}$ for all $\mathscr{W}>0$.

Let $\mathcal{W}$ be the Willmore functional of an immersed torus, and fix its conformal class. The spectral genus is the genus associated with the torus's spectral curve. Taimanov…

Let $S$ be a curve of bidegree $(k,k)$ in ${\mathbb P}_1\times {\mathbb P}_1$. Let $L^r(a,b)$ denote the line bundle used in the hyperbolic-monopole spectral-curve construction, re…

Let $N\geq 4$, let $\mathbf{T}_{m;n_1,n_2}(z)$ be the monodromy matrix, and let $L$ be the parity parameter appearing in its construction. Write $\vec{e}_1$ and $\vec{e}_N$ for the…

Let $V_n$ be the dynamical variables of the extended Lotka–Volterra lattice $\operatorname{LV}(N)$, let $\mathbf{M}_F(z)$ be the associated matrix, and let $B_F(z)$ be the relevant…

Scalar-product conjecture.

Approximation property. Any smooth potential can be approximated by algebraic-geometrical ones with the same periods.

Artemev's conjecture. Up to a suitable normalization, $A^{(g),\vee}_n(k_1,\dots,k_n)$ coincides with $A^{(g),\mathsf{sing}}_n(k_1,\dots,k_n)$ for $g\geq 0$, $n\geq 1$, and…

Artemev's conjecture. The resonance transformation is unnecessary if one swaps $x$ and $y$ in the spectral curve.

Degree-eight distinct-diagonal test case. Every reducible connected degree-eight spectral curve is obtained from a constant branch or from reversal symmetry. In particular, there i…

Primitive codimension-growth conjecture. One has

No-connected-hypersurfaces conjecture. For every $n\ge4$, every codimension-one component of the reducible locus is one of the cutting hyperplanes $b_i=0$. Equivalently, the reduci…

Amended reducibility conjecture. Every reducible factorization of $\chi_n$ is expected to be obtained by iterating the following operations on connected blocks: cutting the chain a…

Let $(\bar C,x,y)$ be a genus $g=0$ admissible spectral curve on $\bar C=\mathbb{P}^1$, with $dx$ and $dy$ having at most simple poles. Let the free energy be the quantity defined…

Let $(A,\phi,\psi)\in\mathcal{M}_\alpha^G$ be a solution and let $b=\mathrm{Hit}_\alpha^G(A,\phi,\psi)$. Denote by $\Sigma_b$ the associated spectral curve and by $L$ the correspon…