15 problems

Hofer's conjecture. If the path generated by $H$ is not length minimizing in its homotopy class, then the Hamiltonian flow has a nonconstant contractible periodic orbit with period…

Polterovich–Shelukhin conjecture. For every closed symplectic manifold $(M,\omega)$,

Let $(M,\omega)$ be a closed symplectic manifold, let $k\geq 2$ be an integer, and equip the group of Hamiltonian diffeomorphisms with Hofer's metric. Hofer-distance conjecture. Th…

Linear growth conjecture on the universal cover. The limit exists and satisfies

Let $G_\omega(M)$ be the symplectomorphism group equipped with the Hofer-like norm $\|\cdot\|_{HL}$. For a geodesic $\{\phi_t\}_{t\geq 0}$ in this metric with nonzero flux, write…

Let $L_0$ be a single circle in the disc $mathbb{D}$ bounding a disc of area $Afrac{1}{2}$. For each integer $igeq 2$, embed the disc into a sphere of area $(i+1)A$ and let…

Let $X=(X,\omega)$ be a closed symplectic manifold. The group of Hamiltonian diffeomorphisms of $X$, denoted by $\mathrm{Ham}(X)$, and its universal cover…

Let $(M,c\bfomegae)$ be a symplectic manifold and let $\operatorname{Ham}(M,\omega)$ denote its Hamiltonian diffeomorphism group equipped with the bi-invariant Finsler Hofer metric…

Unboundedness conjecture. For every closed symplectic manifold $(M,\omega)$,

Main conjecture on $g$-areas. The $g$-area of any Hamiltonian diffeomorphism $\phi$ is equal to the positive Hofer distance between $\phi$ and the subspace of Hamiltonian diffeomor…

Let $B_H(\rho)$ be the space of Hamiltonian diffeomorphisms of the two-sphere $S^2$ having Hofer norm at most $\rho$, viewed as a subspace of ${\rm Ham}(M)$. Weak small Hofer-ball…

Let $(M,\omega)$ be a closed symplectic manifold, let ${\rm Ham}(M)$ be its group of Hamiltonian diffeomorphisms, and let $B_H(\rho)$ denote the subspace of elements whose Hofer no…

Let $(M,\omega)$ be the symplectic manifold under consideration and let $f\in \operatorname{Ham}(M,\omega)$. Define the positive total Hofer norm as the function assigning to each…

Let $(S^2,\omega_{st})$ be the symplectic two-sphere normalized by … Let $inj(M,\omega)$ be the pseudo-injectivity radius, namely the supremum of the radii $r$ such that…

Let $(M,\omega)$ be a closed symplectic manifold, and let $\gamma$ be a Hamiltonian circle action generated by a Morse Hamiltonian. The Hofer index $I^{H}(\gamma)$ is the maximal d…