17 problems

Integrability conjecture. Any system of Lagrangians given by such a 1-form is integrable in the sense of multidimensional consistency, equivalently in the sense of commuting flows.…

Let $\mathcal{Z}_{\alpha}(\boldsymbol{z},u)$ and $\mathcal{Z}_{\alpha}(\boldsymbol{x})$ be pure partition functions, and let $\Phi(\boldsymbol{x},\boldsymbol{\xi},u)$ and…

Spectral-polynomial identity. The two polynomials are equal:

Algebraic Calogero–Moser classification conjecture. The algebraic Calogero–Moser equations exhaust, up to a gauge equivalence, all possible second-order elliptic PDEs whose solutio…

Let ${{\cal A}_n(m)}$ and ${{\cal C}_{n+1}(m,l)}$ be the deformed configurations with multiplicities $m$ and $(m,l)$, respectively, and let $P_{stable}^{{{\cal A}_n(m)}}$ and…

Let the deformed quantum Calogero–Moser operators be generalized by replacing the trigonometric interaction function $1/\sin^2 z$ with the Weierstrass elliptic function $\wp(z)$. T…

Let $G$ be a Coxeter group with multiplicity function $m$. Let $Q_m$ be the ring of $m$-quasiinvariant polynomials, let $H_m\subset Q_m$ be the space of $m$-harmonic polynomials, a…

The discussion concerns the twisted potential for non-simply laced algebras, with Langlands duality sending $\tau$ to $-\frac{1}{s\tau}$; the self-dual cases $B_2$, $G_2$, and…

Calogero's conjecture. The classical Calogero–Moser system is completely integrable.

An algebraically integrable operator with rational coefficients and $b$-symmetry is being considered, where $b$ denotes the cyclic group of order $m$. Its poles are viewed as point…

Heckman–Opdam normalization conjecture. With a specific normalization, the function satisfies

For $N\geq 1$, let … be the configuration space, and let ${\cal F}_N(b;x,y)$ be the joint eigenfunction. Let ${\cal N}_N$ be a normalization constant. Orthogonality and completenes…

Let $b\in(0,2a)$, let $\alpha$ be the spectral parameter, let $u$ be the scattering function, and let $C_N$ be a constant. For the joint eigenfunction ${\cal F}_N(b;x,y)$, consider…

Let $a=a_++a_-$, let ${\cal J}_N(b;x,y)$ be the modified joint eigenfunction, and define … The Hamiltonians are invariant under the substitution $b\mapsto 2a-b$. Parameter-symmetry…

The relativistic Calogero–Moser systems of hyperbolic type describe particles whose momenta are conserved and whose scattering factorizes. Soliton scattering conjecture. The partic…

Let $L$ be an algebraically integrable differential operator on an elliptic curve whose coefficients satisfy the symmetries under consideration. Symmetric-operator conjecture. In t…

Total symmetry conjecture. The regular eigenfunctions of the system will be totally symmetric functions under permutation of the particles for all values of $N$ and $a$.