14 problems

A Hamiltonian diffeomorphism is a time-one map of a Hamiltonian flow on a symplectic manifold. For the standard symplectic $2$-torus, a periodic point is a point fixed by some posi…

$C^0$-flux conjecture. The Hamiltonian diffeomorphism group $\operatorname{Ham}(M,\omega)$ is $C^0$-closed in $\operatorname{Symp}_0(M,\omega)$.

Let $(M,\omega)$ be a symplectic manifold. Write $\operatorname{Symp}_0(M)$ for the identity component of the symplectomorphism group, and let ${\rm BI}(M)$ be the normal subgroup…

Let $L$ be a displaceable Lagrangian in a symplectic manifold $M$, and let $d_{C^0}$ be the $C^0$ distance on Hamiltonian diffeomorphisms. Local Hamiltonian nondisplacement conject…

Let $(M,\omega)$ be a closed and connected symplectic manifold, and let $\operatorname{Ham}(M,\omega)$ be its group of Hamiltonian diffeomorphisms equipped with Hofer's metric…

Let $X$ be a closed symplectic manifold. Define … where $d_H$ is Hofer's metric and $\operatorname{Aut}(X)$ denotes the autonomous Hamiltonian diffeomorphisms. Unbounded autonomous…

Let $\Sigma_g$ be a closed surface of genus $g$, let $T^2$ be the two-torus, and equip the Hamiltonian diffeomorphism groups with the positive Hofer length functional. A metric spa…

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…

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…

Banyaga's conjecture. The restriction of Banyaga's Hofer-like norm to the group of Hamiltonian diffeomorphisms is equivalent to Hofer's norm.

Let $(M,\omega)$ be a symplectic manifold. For each integer $k\geq 0$, let ${\rm BI}_{k,0}(M)$ be the subgroup of $k$-bounded symplectomorphisms in $\operatorname{Symp}_0(M)$, and…

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 a closed symplectic manifold. Write $Ham(M,\omega)$ for the group of Hamiltonian diffeomorphisms, and let $e$ be the norm on $Symp(M,\omega)_0$ defined by sympl…