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4. Geometric applications

The theory of arboreal Lagrangian skeleta should have many applications to classical questions in symplectic and contact geometry. We name a few of these here.


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      Problem 4.1.

      What are the Reeb dynamics on the boundary of a tubular neighborhood of a conical arboreal Lagrangian in $\mathbb{C}^n$?

        •     Given a Lagrangian sphere $S$ in a symplectic manifold, we can perform a Dehn twist about $S$. More generally, the theory of tête-à-tête monodromies associates symplectomorphisms to certain singular Lagrangians. The theory of arboreal singularities gives us control over many other singular Lagrangians.

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          Problem 4.2.

          Generalizing Dehn twists and tête-à-tête symplectomorphisms, can we associate symplectomorphisms to more general classes of arboreal Lagrangians?

            •     Suppose that we had access to the "arboreal symplectomorphims" described above. We might hope to use these symplectomorphism to build arboreal open books. This is one way we might hope to approach the following question:

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              Problem 4.3.

              Is there a notion of arboreal skeleton for a contact manifold?


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                  Problem 4.4.

                  Prove without using holomorphic curves that an exotic parametrization of $S^n\subset\mathbb{R}^{n+1}$ is not regularly homotopic to the standard embedding without dangerous self-tangencies.


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                      Problem 4.5.

                      Can the tête-à-tête automorphism be expressed as a product of right-handed Dehn twists?

                          Cite this as: AimPL: Arborealization of singularities of Lagrangian skeleta, available at http://aimpl.org/arborlagrange.