12 problems

Finite-parameter optimal-profile conjecture. The profile $Q_phi$ is the minimizer of $D_lambda$, and the trajectory of the optimal profile $Q$ remains between the trajectories of t…

Optimal boundary-condition conjecture. For every $lambda\band [0,infty]$, the function $Q_phi$ minimizes

Meta-state minimiser conjecture. For odd $K$, the $Meta$ states are the $z$-invariant energy minimisers, with the splay vertices furthest apart.

BD minimiser conjecture. The $\mathbf{P}^b$-$BD$-$\mathbf{P}^t$ configuration is the unique reduced Landau-de Gennes minimiser for small $h$ and remains potentially stable for all…

Index comparison conjecture. The index of $p_u$ is always higher than the index of the mixed three-dimensional critical point with a $p_u$-type interior profile.

Planarity conjecture. After the field is removed, the director remains in the plane of the polygon.

Velocity blow-up rate conjecture. The velocity field satisfies

Shadow-vortex profile conjecture. Minimizers of the Gross–Pitaevski energy in the intermediate regime may also have a similar profile near the zero level set of $a(x)$.

Competing-map conjecture. There are at least two competing limiting harmonic maps for this two-dimensional problem, namely

Let $d$ be the director field in the three-dimensional nematic liquid crystal flow, and define the stress tensor … Under the same assumptions as the finite-time singularity theorem…

Let $R$ and $r$ be the major and minor radii of the torus, and set $\mu=R/r$. Let $\alpha_p\equiv\pi/2$ denote the constant critical configuration, and let the nonconstant equilibr…

Let $Q^{(L_k)}$ be global minimizers of the Landau–De Gennes energy converging to the limiting harmonic map … where $Q^{(0)}$ may have singularities. Optical defects are regions of…