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3. Symplectic topology and dynamics


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      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.


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          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.


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                  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.


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                              Problem 3.4.

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


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                                  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?


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                                      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.


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                                              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.


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                                                      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.