27 problems

For $a\ge1$, define the embedding capacity … Here $E^4(1,a)$ and $B^k(r)$ are the standard symplectic ellipsoid and ball, respectively. The Lagrangian embedding capacity identity.…

Let $X$ and $X'$ be $2n$-dimensional Liouville domains, and consider the stabilized ellipsoid embedding problem of determining when … Let $\mathfrak{g}_{\mathfrak{b}}$ denote the c…

Unobstructedness conjecture. The function $c_b(z)$ is unobstructed for $z<z_[\infty^b$ if and only if $b=m/d$ for some $(d;m;p,q)\in (H)$.

Let $X_{\Omega}$ be a convex toric domain, and suppose its weight expansion is infinite. An infinite staircase is an infinite sequence of distinct embedding obstructions below the…

Let $(X_1,\a0\omega_1)$ be a bounded domain in $\mathbb{R}^{2n}$. Packing stability means that sufficiently many disjoint copies of $(X_1,\omega_1)$ fully fill every target in the…

Let $a$ be the ellipsoid parameter, and let $a_{\operatorname{acc}}$ be the accumulation point of the corresponding staircase. Consider the symplectic embeddings constructed in Pro…

Stabilized four-ball sharpness conjecture. This embedding is sharp for all $a>\tau^4$. Proving sharpness can be reduced to an existence problem for $(p,q)$-sesquicuspidal symplecti…

Let $p,q,d in \mathbb{Z}_{\geq 1}$ satisfy … and let $[L]\in H_2(\mathbb{CP}^2)$ be the line class. The ellipsoidal superpotential nonvanishing conjecture. One has…

Let $n\leq 8$ and let ${\mathbf c}=c_1\geq\cdots\geq c_n>0$ be admissible capacities. Write $\Im\!\operatorname{Emb}_n({\mathbf c},{\mathbf C}{\mathbf P}^2)$ for the space of sympl…

Let $H_b:=\mathbb{C}P^2_1\#\overline{\mathbb{C}P}\!\,^2_b$ be the one-fold blow-up of $\mathbb{C}P^2$, with line area $1$ and exceptional divisor area $0\leq b<1$. For $z\geq 1$, d…

Let $X\subset\mathbb{R}^4$ be a bounded domain, and define … where the infimum is over symplectic embeddings $\Phi\colon RX\to\mathbb{R}^4$ with Lipschitz constant at most $L$, and…

Volume-loss scaling conjecture. For any $\ell,r>0$, the limit

Let $R_{\mathrm{sup}}\in (1,\infty]$ be the supremum of the radii $R$ such that there is a codimension $2$ subset of $B(\pi R^2)$ whose complement can symplectically embed into…

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…

Let $E(a,b)$ denote the four-dimensional symplectic ellipsoid, let $B^4(c)=E(c,c)$ be the round four-ball, and for $N\in\mathbb{Z}_{\geq 1}$ define … Also let $f_0(x)$ be the corre…

Higher homotopical stabilization conjecture. For every $m\in\mathbb{Z}^+$, there exists an $\epsilon(m)>0$ such that, whenever $c_i<\frac{\epsilon(m)}{k}$, the map $\sigma$ is $m$-…

Let $X$ be a symplectic manifold of dimension $4$, let $\operatorname{Emb}(c_i)$ be the space of ball packings with sizes $c_i$, and let…

Let $(M,\omega)$ be a compact symplectic manifold. For each ball capacity $c$, let $\operatorname{pb}(c)$ denote the Poisson-bracket invariant associated with the corresponding con…

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 $c_b(a)$ be the infimum of the $lambda>0$ for which the ellipsoid $E(1,a)$ symplectically embeds into the polydisc $P(\lambda,\lambda b)$, and let … … The associated exceptiona…

Let $a,b\geqslant 1$ be real numbers, and for $N\geqslant 3$ let $c_b^N(a)$ denote the stabilized symplectic embedding capacity, while $c_b(a)$ denotes the corresponding four-dimen…

Let $(M,\omega)$ be a closed symplectic manifold and let $b>0$. Denote by $\operatorname{Emb}\left(B^{2n}(b),M,\omega\right)$ the space of symplectic charts…

Let $P_n$ and $H_n$ be the Pell numbers and half companion Pell numbers defined by … and … Define … Let $E(1,a_2,\dots,a_n)$ denote the symplectic ellipsoid and let…

Let $d(a,b)$ denote the embedding function for a four-dimensional ellipsoid into a polydisc. For $b\geq6$, let $k$ be an integer satisfying $0\leq k<\sqrt{2b}+\{b\}$, and define ……

Let $b>1$ and define … where $d(a,b)$ is the symplectic embedding function for four-dimensional ellipsoids into polydiscs. Sharp threshold conjecture. … The preceding proposition e…