14 problems

Let $X^{2n}_{\Omega}\subset \mathbb{C}^n$ be a convex toric domain, with standard symplectic form $\omega_{\mathrm{std}}$. Let $\operatorname{diagonal}(X^{2n}_{\Omega})$ denote the…

Let an extremal Lagrangian torus in $(\bar{B}^{2n}(1),\omega_{\mathrm{std}})$ mean a Lagrangian torus whose symplectic energy attains the relevant Lagrangian capacity. Cieliebak–Mo…

Let $S$ be a three-dimensional simplex and let $P$ be any one of the five three-dimensional parallelohedra, namely the cube, hexagonal prism, elongated dodecahedron, rhombic dodeca…

For $\Omega\in{\mathscr{S}}^2$, let $X_\Omega\subset{\mathbb C}^2$ be the associated star-shaped toric domain, and let $c_k(\Omega)$ be the chain-complex capacities defined from…

Let ${\mathscr{S}}^2$ denote the class of star-shaped regions used to define the filtered chain complex $C^\Omega_*$. For $\Omega\in{\mathscr{S}}^2$, set … For $0<a<b\leq\infty$, l…

Let $X_{\Omega}$ be a convex or concave toric domain, let $c_L$ denote the Lagrangian capacity, and let $\delta_{\Omega}$ be the parameter associated with the domain in the source.…

Tightly-constrained domain conjecture.

Reflexive polygon conjecture. The embedding function $c_\Omega(a)$ has infinitely many singular points only if some scaling of $\Omega$ is reflexive.

Hardy–Littlewood discrepancy conjecture. The function $C_{a_0}(n)+C_{1/a_0}(n)$ is bounded if and only if $a_0+1/a_0\in\mathbb{N}$. If it is bounded, it is identically zero. If it…

Reflexive polygon conjecture. If the ellipsoid embedding function of a rational convex toric domain has an infinite staircase, then its moment polygon is a scaling of a reflexive p…

Weight-gcd conjecture. If

Hutchings's conjecture. The full statement of this bound could be proven using the Hutchings criterion. Hutchings had proved it for $2\leq a\leq 2.4$, while the cited theorem estab…

Let $X_\Omega$ be a smooth convex toric domain with boundary $Y$. Let $C_*^{T^3}$ be the chain complex freely generated over $\mathbb{Z}/2$ by torus generators, with differential g…

Hutchings's formula conjecture. Hutchings's formula should remain true in most, and probably all, cases where $\mu(U)$ touches one or both axes.