17 problems

The Garnier system has analogues of the Painlevé VI transformations $w_1,w_3,w_4,l_1,l_2,l_3$, defined by Tsuda, while $w_2$ denotes the remaining generator of the affine extension…

The solutions to isomonodromic deformation equations for two-dimensional Fuchsian systems with four singularities on the Riemann sphere $\mathbb{P}^1$ can be expressed through solu…

The isomonodromic tau-function is associated with semisimple Frobenius manifolds on Hurwitz spaces of branched coverings of genus $0$ and $1$. Existing formulae used auxiliary stru…

Kerz's commutativity conjecture. (i) For any $\lambda\in\mathbb R^*$, $\lambda\cdot\sigma_{\operatorname{Dol}}$ is also an isomonodromic deformation. (ii) For any…

Let $\mathcal T_g$ denote the Teichmüller space of compact Riemann surfaces of genus $g\geq2$, let $\mathcal X/\mathcal T_g$ be its universal family, and let…

Let $\mathcal T_g$ be the Teichmüller space of genus $g\geq 2$ compact Riemann surfaces, with universal family $\mathcal X/\mathcal T_g$. For an isomonodromic section…

Let $\sigma(0):=(E,\bar\partial,\theta)$ be a nilpotent Higgs bundle on $X_0$, and suppose that the non-abelian Kodaira–Spencer map $\Phi^{0,1}_{KS}$ is nonzero at $\sigma(0)$ for…

Let $\mathcal M_{\mathrm{Dol}}(\mathcal X/S)^{0}$ be an irreducible component of $\mathcal M_{\mathrm{Dol}}(\mathcal X/S)$. An irreducible-component triviality conjecture. The foll…

Jimbo–Miwa–Ueno differential conjecture. One has

Jimbo–Miwa–Ueno differential conjecture. The observation that the Jimbo–Miwa–Ueno differential is obtained from the Hamiltonian differential by formal evaluation at $\hbar=0$, up t…

Let the dimer model be coupled to a background connection, and regulate the divergent integrals arising from non-separated zippers as in the referenced section. Let the Jimbo–Miwa–…

Let projective connections on Riemann surfaces have apparent singularities, and consider their isomonodromic deformations. Bridgeland's Joyce-structure conjecture. These isomonodro…

Let $A_j$ and $G_j$ be the residue and diagonalizing data of the Fuchsian system, let $t_j$ be its pole positions, and let $\mathcal M$ be the monodromy manifold. Define the 1-form…

Connection-constant conjecture. The analog of the connection-constant product formula should hold: the product of two connection constants equals the ratio of three-point functions…

Let a linear system with isomonodromic deformations have monodromy data in a space $\mathcal{M}$ and isomonodromic times in a space $\widetilde{\mathcal{T}}$. Let $\omega$ be the a…

Conjectural conformal-block decomposition. Such a decomposition should also exist for the conformal blocks considered above. The claim extends the known decomposition of isomonodro…

Jimbo–Miwa–Ueno conjecture. The function $\tau_{\scriptscriptstyle{I}}$ is holomorphic everywhere in the space of deformation parameters outside the hyperplanes where the values of…