13 problems

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

Let $C$ be a strictly convex closed curve equipped with a projective billiard structure, and call the billiard Birkhoff integrable when its boundary and closed caustics form a foli…

Let $f$ be a planar homeomorphic mapping of finite distortion satisfying the hypotheses of Theorem $\mathrm{}$, except that radiality is not assumed. The conclusions of Theorem…

Let $k$ be a positive integer, and consider an additive $2k$th-order difference equation … Here $\mathbf{x}_{n}^{(m,l)}=(x_{n+m},\dots,x_{n+k})$ as defined in the source, and $f$,…

Let $M$ be a differentiable manifold and let $\boldsymbol{A},\boldsymbol{B}:TM\rightarrow TM$ be two $(1,1)$ tensor fields. Assume that…

Let $Q$ be a recurrent quiver and let $\mu_-\mu_+$ be its associated mutation map. Let the associated upper cluster algebra be the intersection of the Laurent polynomial rings of i…

Let $Q$ be a recurrent quiver, meaning that its underlying graph is bipartite and mutating at its black vertices followed by its white vertices returns the same quiver. A positive…

Let $Q$ be a recurrent quiver: its underlying graph is bipartite, and mutating at its black vertices followed by mutating at its white vertices returns the same quiver. A positive…

Let $(X_t)_{t\in(1-\delta,1+\delta)}$ be a quadratic harness on $(1-\delta,1+\delta)$ for some $\delta>0$, with parameters $(\eta,\theta,\sigma,\tau,\gamma)$. For a fixed $t$ in th…

Let … be the Hamiltonian of the minimally coupled scalar-field system. Assume $\Lambda\ne0$, let $n$ be an integer satisfying … and consider a generic energy level $E\ne0$. Integra…