9 problems

Reduced-coordinate conjecture. The variables $\mathbf{w}$ are cluster coordinates on $\mathcal{X}_{\mathrm{red}}$: every mutation in the $\mathbf{w}$ variables can be lifted to a s…

Let $u$, $\check{u}$, $t$, and $q$ be the parameters of the $c=1$ $q$-deformed conformal blocks, and let $\sigma$ and $\check{\sigma}$ be defined by $u=q^\sigma$ and…

D5 Weyl-group invariance conjecture. Since the Newton polygon of $\widehat H_{k,\bm M}$ is identical to that of the $D_5$ curve, the spectral operator $\widehat H_{k,\bm M}$ should…

Consider five-dimensional theories with physical moduli spaces containing highly fine-tuned regions and with underlying affine root lattices. A collimation chamber is a region in w…

The q-Painlevé equations describe cluster variables, while the TBA equations describe functions $Y_\gamma$ associated with the same quantum periods. The q-Painlevé–TBA corresponden…

Inaccessible-value conjecture. Points on $\mathcal{X}$ are related to inaccessible values on the initial value space of $q$-Painlevé VI, or, equivalently, to singular solutions of…

Let $\Xi_{{\mathbb F}_0}(\kappa,\hbar,\xi)$ be the spectral determinant of local ${\mathbb F}_0$, and let $\xi$ denote its mass parameter. A q-deformed Painlevé equation is a diffe…

Let $q_1$, $q_2$, $u$, and $Z$ be the parameters appearing in the 5d Nekrasov functions $\mathrm F^{(1)}$ and $\mathrm F^{(2)}$. For a choice of signs, interpret…

Fredholm determinant–q-Painlevé conjecture. The Fredholm determinant of the operator associated to the quantized mirror curve computes the $\tau$ function of the corresponding $q$-…