27 problems

Convergent matrix-integral decomposition conjecture. The convergent matrix integral $Z_{\rm conv}$ is a finite linear combination of convergent formal solutions of the loop equatio…

One-dimensional search-space conjecture. The matrix model has search-space dimension one: all moments in the large-$N$ limit can be written in terms of the coupling constant $g$ an…

Equality-of-solutions conjecture. The $(g,g,g)$ and $(g,-g,g)$ two-matrix models have the same unique solution satisfying the positivity constraints.

Square-root critical behavior conjecture. The formal matrix model has the asymptotic expansion

Hyperbolic matrix-model conjecture. The disk partition function admits an analytic continuation in $q$ near $q=4$ such that, with

Let $D$ be the Dirac operator in a formal Dirac ensemble of type $(1,0)$ or $(0,1)$, with coupling constants $t_j$ and partition function … Here $\mathcal{G}$ is the relevant Dirac…

Bipartite quadrangulation conjecture. For $g>0$, the topological recursion for this spectral curve gives

Let $Z_{\rm KP}$ and $Z_{\rm ABT}$ denote the partition functions associated with the Kontsevich–Penner and Alexandrov–Buryak–Tessler constructions. KP–ABT equality conjecture. One…

Consider the deformed generalized Kontsevich matrix model (GKM) with potential given by equation, and allow a shift of its variables. Deformed GKM conjecture. The shifted deformed…

Let $R:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be the ramified covering defined by the model, let $\beta_1,\ldots,\beta_{2d}$ be its ramification points, and let $\sigma_i$ be…

Let $\mathcal G$ be a connected vacuum Feynman graph, let $\alpha(\mathcal G)$ denote its large-$D$ exponent, and let $P_{\mathcal G}$ be the associated polynomial. Write…

Let $M$ be a symplectic surface and let $S_{cl}$ be the classical functional for maps from $M$ to a flat torus $(S^1)^n$. For each $N$, let $\Phi_1,\dots,\Phi_n\in U(N)$ and let…

Let a Hadamard matrix model be a matrix model arising from a complex Hadamard matrix, and call it stationary on its image when its associated integration state has the stationarity…

Let a Weyl matrix model be a matrix model constructed from Weyl matrices, and let $PE$ denote the associated quantum group appearing in the untwisted comparison model. Weyl-model t…

Let $G\subset S_N^+$ be quasi-transitive, with all orbits of size $K$, and let … be its universal quasi-flat model. The uniformity conjecture. If $G$ additionally satisfies a suita…

Let $G\subset S_N^+$ be quasi-transitive, with all orbits of size $K$, and write $\Gamma=\widehat{G}$. Let $X_G$ be the universal quasi-flat model space and let … be the correspond…

Extended refined Kontsevich–Penner conjecture. For any $N$ we have

The exact two-Schur correlator conjecture. The average of the two Schur-polynomial insertions satisfies

Let $\mathcal{M}_{(g,b),k,l}$ be the moduli space of Riemann surfaces with boundary of genus $g$, with $b$ boundary components, $k$ boundary marked points, and $l$ interior marked…

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 $z_j=0$.

Projection trace inequality conjecture. Under these hypotheses,

$F_p$-monotonicity conjecture. For every $x\in U_N^N$ and every $p\geq2$,

Let $S_N^+$ be the quantum permutation group, let $X_N$ be the associated matrix-model space, and let … be the universal $N\times N$ flat matrix representation. Flat matrix represe…

Convergence conjecture. The maps $\Phi$ and $\Psi$ increase the volume and, after an infinite number of steps, respectively land in $K_N$ and $X_N$. This conjecture would imply con…

Q-graded correlator conjecture. These correlators are the $Q$-graded correlators for open intersection numbers. The proposed grading is intended to separate contributions by bounda…