320 problems

Let $d>3$ and let $f$ be the function appearing in the discrete-time $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra system. Let $C$ denote its Casimir function. An invariant is called t…

Let $a_n$ be the cluster variables associated with the extending vertex, and let $\tilde{K}_n$ denote the quantity whose period-two behavior is being conjectured. Period-two formul…

Let $\tilde{K}_n$ be the determinant introduced above, namely … Here the variables $a_n$ and $h_n$ are the cluster variables associated with the extending vertex and its neighborin…

Identify the root lattice of $A_{N-1}$ with … For $j\in\mathbb{Z}/N\mathbb{Z}$, define the Fourier-transformed Nekrasov functions by … where $\omega_0=0$ and $\omega_j$ are the fun…

Let $k=2m+1\geq 7$ be odd, and define rational functions $y_n\in\mathbb{Q}(y_0,\dots,y_{2m-3},\alpha,\beta,\gamma)$ by the difference equation specified in the source. Odd-dimensio…

Let $m$-periodic variables $\theta_i^t$ satisfy the generalized discrete Toda equations referred to as, and define $q_{j,i}^t$ from them via. Theta-to-q conjecture. The quantities…

Classification conjectures. (1) Every Hamiltonian operator is equivalent to one whose leading coefficient is $1$. For $N=2n+3\geq 3$, this is equivalent to an operator with leading…

Integer-spectrum conjecture. The algebraic eigenvectors span the three lowest levels of the spin chain. When $M\teq1$, the fourth algebraic energy, $\mathsf{E}=3$, is the fifth low…

Let $\mathcal{T}$ be the phase space of the ultra-discrete $(g+1)$-periodic Toda lattice, let $\mathcal{C}$ be the moduli space of compact tropical curves $\Gamma_C$, and let … be…

Let $n$ be a positive integer, let $g$ and $ell$ be parameters, and let $\lambda=(\lambda_1,\ldots,\lambda_n)$. Write $(y;q)_m$ for the $q$-Pochhammer symbol and…

Differential Galois integrability conjecture. The identity component of the differential Galois group of the $k$-th order variational equations is Abelian for every…

Integrability conjecture. These equations are integrable in the Jacobi sense with polynomial first integrals $I_1$ and $I_2$, where

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…

Let $M$ be a manifold with compatible Poisson structures $\{,\}_{1}$ and $\{,\}_{2}$. Let $L$ be a finite set with $r$ elements, and let $F_{l,\lambda}$, for $l\in L$ and…

The odd-dimensional open Volterra system, the even-dimensional periodic Volterra system, the full Toda lattice, the multidimensional Euler top, and the regular case of the cited bi…

Let the DSII hierarchy be the integrable hierarchy governing deformations of surfaces immersed in $\mathbb{R}^4$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$ via generalized Weierst…

Zero-location conjecture. For $n<p_0$, $T^{(3)}_{n-1}(u,v)$ has two real zeros $\pm\zeta^{(3)}_{n-1}$ for some $\zeta^{(3)}_{n-1}\in\mathbf R_{>0}$, while for $n=p_0$ it has a doub…

Zero-location conjecture. All the zeros of $T^{(1)}_{n-1}(u,v)$ are located on the almost straight lines

Let $\mathcal{C}_{k,m}$ be the orbifold obtained from $\mathbb{P}^1$ by replacing neighborhoods of $0$ and $\infty$ with the orbifold discs $D_1/\mathbb{Z}_k$ and…

Nonexistence conjecture. There is no global analytic first integral for this system on $\mathbb{R}^2$. The claim concerns global analytic integrability of planar Liénard systems; t…

Let $X$ be a toric Fano surface. Let $D$ be a toric invariant divisor, and let $D^+$ and $D^-$ be disjoint toric invariant divisors. Consider the suitably defined generating series…

Transfer-matrix q-character conjecture. The identification

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 $p\in P(m)$ be a state with soliton content $m$, and let $(T^{(r)}_l)^t(p)\neq0$ for every $t\geq0$. Denote by ${\mathcal P}^{(r)}_l$ the least common multiple defined earlier…

Cohn polynomial conjecture. As a polynomial in $J$ with integer coefficients, no Cohn polynomial has a nontrivial factor, except for the Type I Cohn polynomials with…