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2. Quantitative Heegaard Floer theory and ECH

Heegaard Floer theory constructs powerful invariants for $3$-manifolds and links therein. Recently, such a theory has found applications in the study of Hamiltonian diffeomorphism groups of surfaces. There is a parallel theory from embedded contact homology. This section records questions not yet answered by current techniques.

    1. Asymptotics of spectral invariants

          Lagrangian link spectral invariants constructed via quantitative Heegaard Floer theory and ECH spectral invariants exhibit remarkable asymptotic behaviors. It is natural to ask how to get a refined understanding of the asymptotic behavior of these spectral invariants.

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      Problem 2.1.

      [Ivan Smith] For a symplectic surface $(\Sigma, \omega)$ and a Hamiltonian $H \in C^{\infty}([0,1] \times \Sigma, \mathbb{R})$, investigate the subleading asymptotic of $c_d(H) \rightarrow \text{Cal}(H)$, where $c_d$ is the $d^{\text{th}}$ link spectral invariant associated to $H$ and $\text{Cal}(H)$ is the Calabi invariant.
        1. Remark. [Ivan Smith] See [arXiv:2206.10749]. It is known that the highest-order subleading asymptotic recovers a form of Ruelle invariant when $H$ is autonomous. Is it true for all $H$?


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              Problem 2.2.

              [Dan Cristofaro-Gardiner] The same question as the previous one for periodic Floer homology (PFH) spectral invariants.

                • Further developments of quantitative Floer theory in low dimensions


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                  Problem 2.3.

                  [Dan Cristofaro-Gardiner] What is the relationship between the link and PFH spectral invariants?
                      See the work of Guanheng Chen [arXiv:2111.11891], [arXiv:2209.11071] for current status.


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                      Problem 2.4.

                      [Dan Cristofaro-Gardiner] Can quantitative Heegaard Floer invariants be used to give quantitative invariants of contact forms on $3$-manifolds related to ECH spectral invariants by analogy with the previous problem?


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                          Problem 2.5.

                          [Dan Cristofaro-Gardiner] If one could provide a positive answer to the previous question, can it be used to better understand the subleading asymptotics of ECH spectral invariants?
                              Cf. conjectures of Hutchings [MR4441526].
                            1. Remark. So far, subleading asymptotics have been best understood for link spectral invariants, so a Heegaard Floer counterpart of the theory should be helpful.


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                                  Problem 2.6.

                                  [Robert Lipshitz] What more can the computational apparatus in Heegaard Floer theory tell us about dynamics?

                                    • Miscellaneous


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                                      Problem 2.7.

                                      [Peng Zhou] To what degree can link spectral invariants and their applications be generalized to higher dimensions?

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