5 problems

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…

Let $\Omega(t)$ be the evolving set with boundary parametrized by $x(t,\xi)$, and let the optimal control be the normal velocity control acting along this boundary. Denote the curv…

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow. Denote by $n_U(t)$ the number of critical points of its radial function relative to the fixed point…

Consider the evolution of an axisymmetric toroidal surface, with the singularity occurring when the toroidal surface closes up at the origin. Let $t$ denote time along the evolutio…

Let $M$ be the number of sides of the initial regular planar polygon, and let $t_{pq}=(2\pi/M^2)(p/q)$, where $p\in\mathbb Z$, $q\in\mathbb N$, and $\gcd(p,q)=1$. At this rational…