77 problems

Consider a strictly convex closed planar curve $C$ and the billiard in the bounded domain it encloses. The billiard is Birkhoff caustic-integrable if a topological annulus adjacent…

Błocki's conjecture. Every $m$-subharmonic function should be locally $L^p$ integrable for every

Blocki's integrability conjecture. In the flat case, every $m$-subharmonic function belongs to $L^p$ for every

Let $\Omega$ be a convex planar billiard with smooth boundary. It is polynomially integrable if its billiard flow near the unit tangent bundle of the boundary has a first integral…

Let ${\cal F}^{(r)}(q,p)$ and ${\cal F}^{(2r-2)}(q,p)$, for $r=3,4,\ldots$, be perturbed harmonic oscillator Hamiltonians with potentials $V^{(r)}$ and $V^{(2r-2)}$ of degrees $r$…

Let $X$ be an analytic vector field on the plane or sphere, and let $C(X)$ be the real vector space whose elements are analytic vector fields associated with $X$. Infinite-dimensio…

Zakharevich's conjecture. If $w(t)$ is integrable at $n+3$ distinct values of $t$, then the Veronese curve of distributions is integrable.

Gelfand–Zakharevich conjecture. Locally, the Veronese web is complete: it determines the bihamiltonian structure up to an isomorphism.

Nonexistence conjecture. There are no values of $n$ for which a solution exists.

Yang–Baxter integrability conjecture. For two types of inhomogeneities, a circuit is Yang–Baxter integrable if the bulk gate is a solution of the Yang–Baxter equation, the boundary…

Consider the kernel … where $\delta,\varepsilon>0$, and let $d^{(n)}$ denote the $n$-fold quantity appearing in the condition for strong sharpness, with norm…

Let $\Omega\subset\mathbb{C}^n$ be an open set, let $\omega$ be a Hermitian $(1,1)$-form on $\Omega$, and let $u$ be an $m$-subharmonic ($m$-sh) function, meaning that the eigenval…

Let $\Gamma$ be a bounded strictly convex billiard table with smooth boundary, and let $\mathcal{C}(\Gamma,\ell)$ denote the associated coin billiard of height $\ell>0$. Integrabil…

Haagerup–Izumi fusion categories of higher rank are expected to exhibit the same pattern as the Haagerup spin chains discussed here: the analogue of $\mathsf{P}_\rho$ inherits crit…

Consider the polynomial system denoted by (gs), its Bautin ideal $B$, and the associated parameter-space vector field denoted by (Vvfsh). Bautin-variety invariance conjecture. The…

Let ${\bf W}$ be a component of the integrability variety ${\bf V}(B)$ of the system under consideration. Let $\mathcal A|_{\bf W}$ denote the restriction to ${\bf W}$ of the opera…

Consider the $1:-1$ resonant polynomial system … Let $I_S$ be its Sibirsky ideal and let $B$ be the Bautin ideal, with ${\bf V}(I_S)$ and ${\bf V}(B)$ their varieties. Sibirsky ide…

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain, with $n \geq 2$ and $1 \leq m \leq n$, and let $u \in \mathcal{SH}_m(\Omega)$ be an $m$-subharmonic function. Błocki's conjec…

Let $\Omega$ be a strictly convex $C^r$ domain in $\mathbb{R}^2$ with $r>3$. An integrable billiard domain is one whose billiard dynamics has the relevant integrability property; i…

Let $h:(0,\infty)\rightarrow(0,\infty)$ be a log-convex function satisfying … Assume that $h^2(x)/h(2x)$ is integrable on $(0,\infty)$. Handelman's integrability conjecture. Then…

Let $(X,J)$ be an almost complex M-polyfold that is locally sc-diffeomorphic at every point $p$ to some $\Gamma^{k(p)}_{n(p)}$ equipped with the complex structure defined in the so…

Non-integrability conjecture. There are no other integer solutions of the displayed relation with $\lambda_i\in {\mathscr J}_+\cup {\mathscr J}_-$ for $i=1,\ldots,k$ beyond the one…

Integrability conjecture. It seems premature to conjecture that only systems with radial potentials and potentials depending on one variable are integrable.

Binary black hole (BBH) merger waveforms are gravitational-wave signals produced during the coalescence of two black holes; integrability means that the underlying dynamical system…

The quantum spectral curve (QSC) is a set of equations proposed for the AdS$_3$/CFT$_2$ integrable system. At finite volume, the spectrum receives contributions from both massive a…