9 problems

The paper considers two constructions of integrable systems associated with cluster structures whose quivers can be drawn on a torus: the Goncharov–Kenyon construction using the di…

Let $\Omega\subset\mathbb R^2$ be a bounded, strictly convex domain, and let $\mathbb T_t$ be the flow by curvature of an embedded triod in $\Omega$. Assume that the lengths of the…

Let $\mathbb T^\infty_{\mathfrak t}$ be an eternal evolution by curvature of an unbounded embedded triod, curve, or curve with a single endpoint. Translation conjecture. The triods…

Let $\mathbb T^\infty_{\mathfrak t}$ be an eternal evolution by curvature of an unbounded embedded triod, curve, or curve with a single endpoint, and let its curvature be defined a…

Let $\mathbb T_t$ be a smooth evolution of an embedded triod with fixed endpoints $P^1,P^2,P^3$ in a domain $\Omega\subset\mathbb R^2$. Denote by $L^i$ the lengths of its three cur…

Tropical multiple Horn conjecture. For sufficiently large networks $\Pi_1,\Pi_2,\Pi_3$, for example standard networks, these conditions are necessary and sufficient. Equivalently,…

Planar-network multiple Horn conjecture. For sufficiently large planar networks, these conditions are also sufficient for $(\lambda, \mu, \nu, \rho, \sigma, \tau)$ to be realized a…

Translation conjecture. The map $t$ is

No higher-multiplicity static-line tangent-flow conjecture. At a singular time of the planar network flow, no tangent flow that is a static line of higher multiplicity can develop.