29 problems

Viterbo's conjecture.

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 $\Omega$ 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…

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

Let $\omega_{\mathrm{FS}}$ 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 $(T^*\mathbb T^2,\omega_{\rm can})$ 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 $X\subset\mathbb{R}^{2n}$ be a convex domain. A symplectic capacity is normalized if it takes the same value on the unit ball and the unit symplectic cylinder. The strong Viter…

Let $c$ be a symplectic capacity and let $X\subset\mathbb{R}^{2n}$ be a convex domain. Write $\operatorname{Vol}(X)$ for the Euclidean volume of $X$. Viterbo's conjecture. For ever…

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

Let $\Sigma \in \mathcal{K}^{2n}$ be a convex domain in $(\mathbb{R}^{2n},\omega_0)$, let $c$ be a symplectic capacity, and let $\operatorname{vol}(\Sigma)$ denote its symplectic v…

The higher-capacity p-product conjecture. The displayed formula should hold for every such pair of star-shaped domains and every admissible $p$. The conjecture generalizes the know…

Let $A$ be a convex body in $\mathbb R^{2m}$, and let $c$ be a symplectic capacity. Capacity uniqueness conjecture. All symplectic capacities coincide on convex bodies. The claim i…

Let $U$ be a bounded star-shaped domain, and let $c_k^{\text{EH}}(U)$, $c_k^{\text{GH}}(U)$, and $c_k(U)$ denote the Ekeland–Hofer, Gutt–Hutchings, and capacities considered in the…

Let $D$ be a convex domain in $\mathbb{R}^{2n}$ invariant under a fixed linear anti-symplectic involution. Capacity coincidence conjecture. All normalized symplectic capacities and…

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…

Let $K_1,\ldots,K_m$ be convex bodies in $T^*\mathbb R^n$. Hofer-Zehnder subadditivity conjecture. The Hofer-Zehnder capacity of their union should satisfy … This is stronger than…

Let $K,K_1,\ldots,K_m$ be convex bodies in $T^*\mathbb R^n$. Akopyan–Karasev–Petrov conjecture. If … then … The conjecture is a subadditivity principle for the Ekeland-Hofer-Zehnde…

Let $M$ be a smooth affine variety with $\kappa(M)=-\infty$, and let $c_1^{\mathit{GH}}(M)$ denote its first Gutt–Hutchings capacity. Gutt–Hutchings capacity conjecture. … The sour…

Let $K\subset \mathbb{R}^{2n}$ be a convex body covered by a finite set of convex bodies $\{K_i\}$. Subadditivity conjecture. One has … This is a special case of the subadditivity…

Let $X$ be a stably non-displaceable compact subset of a closed symplectic manifold $(M,\omega)$. Kawasaki's conjecture. For any $R>0$ and $\ell\in\mathbb{Z}$, one has … This conje…

The capacity conjecture. The equality

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…