10 problems

Vortex filament conjecture. Vorticity initially concentrated around a curve should remain close to a curve evolving according to the binormal-curvature equation above, to leading o…

For $N\geq 2$, let $\Omega_0$ be a bounded open subset of $\mathbb{R}^N$ with boundary $\Gamma_0$. Let $u$ solve the heat equation with initial data $\mathbf{1}_{\Omega_0}$, and de…

Let $\Gamma^0:=\bigcup_{i=1}^nS_i^0$ be an unbounded admissible polygonal curve, and let $\{\Gamma(t)\}_{t\ge0}$ be the globally defined elastic flow with finitely many restarts. W…

Finite-time continuation conjecture. Every finite-time singularity of an unbounded admissible polygonal crystalline elastic flow has the three properties above. The conjecture woul…

The Mullins–Sekerka equation describes the evolution of a phase boundary driven by mean curvature through a harmonic bulk field, with volume-preserving dynamics. Pre-singularity un…

Non-self-similar expanding breather conjecture. There exists at least one non-self-similar expanding breather of the mean curvature flow, as in the case of the Ricci flow.

Let a regular planar polygon evolve according to the binormal flow. At rational multiples of the time period, consider the resulting curve and the angle between consecutive sides.…

In the numerical experiment, let the initial curve be the union of a circle of radius $0.5$ and two quarter circles of radius $2$, and let $(\mathcal{W}_m)^h$ denote the finite ele…

For the axisymmetric surface diffusion flow, a pinch-off conjecture asserts that solutions developing singularities will have to go through a pinch-off. This conjecture concerns th…

Distance monotonicity conjecture. The Evans--Spruck distance monotonicity result holds for all manifolds of nonnegative sectional curvature, but fails for all manifolds of negative…