| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

3. Symplectic topology and dynamics


    1. Add remark

      Problem 3.1.

      [Egor Shelukhin] Given a closed symplectic manifold $(M, \omega)$, consider $\text{Ham}^{\text{aut}}(M, \omega) \subset \text{Ham}(M, \omega)$ the subset of autonomous Hamiltonian diffeomorphism groups. For $\phi \in \text{Ham}(M, \omega)$, can the Hofer distance from $\phi$ to $\text{Ham}^{\text{aut}}(M, \omega)$ be arbitrarily large?
          Yes when $M$ is a surface other than $S^2$, unknown for $S^2$.


        • Add remark

          Problem 3.2.

          [Ivan Smith] Develop a toolkit for producing Lagrangian links in symplectic manifolds, and study its relationship to questions such as the Lagrangian packing problem and problems from dynamics.
            1. Remark. A Lagrangian link in a symplectic manifold is a tuple of disjoint embedded Lagrangians.


                • Add remark

                  Problem 3.3.

                  [Dusa McDuff] Is the volume preserving diffeomorphism group of $\mathbb{R}^2$ with respect to the standard volume form $dx \wedge dy$ perfect?
                    1. Remark. McDuff showed the corresponding statement is true for $\text{Diff}(\mathbb{R}^n, dx_1 \wedge \cdots \wedge dx_n)$ for $n \geq 3$.
                        • Remark. $\text{Diff}(\mathbb{R}^2, dx \wedge dy)$ is not simple because the compactly supported volume-preserving diffeomorphisms form a proper normal subgroup.


                            • Add remark

                              Problem 3.4.

                              [Cheuk Yu Mak] How about the perfectness for volume-preserving diffeomorphism groups of surfaces of infinite type?


                                • Add remark

                                  Problem 3.5.

                                  [Sheel Ganatra] Do existing techniques say anything about the structure of $\overline{\text{Ham}}(M, \omega)$ for $\text{dim} M \geq 2$?


                                    • Add remark

                                      Problem 3.6.

                                      [Sobhan Seyfaddini] Is the group of $C^2$ diffeomorphisms of $S^1$ simple?
                                        1. Remark. Classical works in dynamics leave open the simplicity question for $C^{k+1}$ diffeomorphism groups for $k$-dimensional manifolds.


                                            • Add remark

                                              Problem 3.7.

                                              [Cheuk Yu Mak] Can one better/explicitly understand the abelianization of the $\overline{\text{Ham}}(\Sigma, \omega)$ for $(\Sigma, \omega)$ a symplectic surface? More generally, how about quotients of $\overline{\text{Ham}}(\Sigma, \omega)$ by other natural normal subgroups?
                                                1. Remark. [Sobhan Seyfaddini] If the abelianization is both torsion-free and divisible, it is isomorphic to $\mathbb{R}$ in a non-canonical way.


                                                    • Add remark

                                                      Problem 3.8.

                                                      [Tsuboi-Polterovich] It’s known that the group of volume-preserving diffeomorphisms on $S^3$ is perfect. Is the commutator length uniformly bounded?
                                                        1. Remark. Cf. [arXiv:0710.1412] for related questions.

                                                              Cite this as: AimPL: Floer theory of symmetric products and Hilbert schemes , available at http://aimpl.org/floerhilbert.