51 problems

Let $c$ be a symplectic capacity and let $X\subset\mathbb{R}^{2n}$ be a convex domain. Here $\operatorname{Vol}(X)$ denotes the $2n$-dimensional volume of $X$. Viterbo's conjecture…

Let $c$ be a normalized symplectic capacity and let $X\subset\mathbb{R}^{2n}$ be a convex domain. The Gromov width and cylindrical capacity are extremal among normalized symplectic…

Let $c_k^{GH}$ denote the $k$-th Gutt–Hutchings capacity and let $c_k^{EH}$ denote the $k$-th Ekeland–Hofer capacity. Gutt–Hutchings conjecture. For every $k=1,2,3,\ldots$, the two…

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 $\omega_{\mathrm{FS}}$ denote the Fubini–Study form on $\mathbb{CP}^n$. For a closed Lagrangian torus $L\subset(X,\omega)$, call $L$ extremal when…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)\subset\mathbb{C}^n$ be the ellipsoid defined by … For a closed Lagrangian torus $L\subset(X,\omega)$, set…

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…

Let $\bar c_L$ be the normalized Lagrangian capacity on $2n$-dimensional ellipsoids, let $P(1/n,\dots,1/n)$ denote the cube with all $n$ radius parameters equal to $1/n$, and let…

Let $\bar c_L$ be the normalized Lagrangian capacity on four-dimensional ellipsoids, and let $B(1)$ and $Z(1/2)$ denote the ball and cylinder of the indicated radii in…

Let $(X,\omega)$ be a rational algebraic manifold. Suppose there exists a surjective morphism $\pi:X\to\mathbb{CP}^n$ such that $\pi|_{X\setminus S}$ is one-to-one for some subvari…

Let $\Omega$ be a uniformly convex domain in $\mathbb{R}^{2n}$ with $n>1$, and let $c_k(\Omega)$ denote its Gutt--Hutchings capacities. The first capacity $c_1(\Omega)$ is the syst…

Let $B^{2n}(1)$ be the symplectic unit ball, with $n\ge2$. Consider the capacity obtained by taking the supremum of $A_{\min}(L)$ over all closed Lagrangian submanifolds $L$ in the…

For a star-shaped toric domain $X\subset\mathbb{C}^2$, let $C_d^{\mathrm{GH}}(X)$ denote its $d$-th Gutt–Hutchings capacity. The asymptotic existence conjecture for Gutt–Hutchings…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form $\omega_{\mathrm{std}}$. A Lagrangian torus…

Let $(M,\omega)$ be a closed symplectic manifold and let $K\subset M$ be a compact subset with non-empty interior, in the sense specified by the paper's section on symplectic-detec…

Let $(T^*\mathbb T^2,\omega_{\rm can})$ be the cotangent bundle of the two-torus, and let $\mathcal S$ denote the class of systolically convex domains defined in the source. A symp…

Let $n\geq 1$, let $p\in\mathbb{N}$ be odd, and let … be the corresponding lens space. Let $D_1L(p,1,\dots,1)$ denote its disk tangent bundle with symplectic form…

Let $L(p,q)$ be a lens space, and let $D_1L(p,q)$ denote its disk tangent bundle equipped with the symplectic form $\mathrm{d}\lambda$. Lens-space Hofer–Zehnder capacity conjecture…

Let $X\subset\mathbb{R}^{2n}$ be a convex domain, and let $\mathcal{A}_{\rm min}(X)$ denote its minimal action. Weak Viterbo conjecture. … This is a consequence of the strong Viter…

Viterbo's equality-case conjecture. Equality holds in this inequality only when $C$ is symplectomorphic to a ball.

Let $c$ be a symplectic capacity on subsets of $\mathbb{R}^{2n}$, and consider its values on ellipsoids and on convex domains. A normalization condition is imposed by requiring $c$…

Let $c$ be a symplectic capacity on subsets of $\mathbb{R}^{2n}$, and let the Gromov width denote the symplectic capacity obtained from the largest standard symplectic ball embeddi…

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…

Cube-normalized capacity uniqueness conjecture. All cube-normalized symplectic capacities coincide on $X$.

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.…