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1. Combinatorial data

In this section, we discuss ways of presenting the arboreal theory in a combinatorial way.

    1. Topological vs. Lagrangian arboreal singularities

          In analogy with the theory of ribbon graphs, the theory of arboreal singularities ought to allow us to express the data of a Weinstein skeleton in a completely combinatorial way: that is, as a topological space endowed with some extra discrete data. Such a topological space we will call a "topological arboreal singularity," as opposed to the "Lagrangian arboreal singularities" which are equipped with an embedding as Lagrangians in a symplectic manifold. (Think of this as the difference between the Thom-Mather and the Whitney theories of singular spaces). We want to relate these two concepts.

      Add remark

      Problem 1.1.

      What structure on a skeleton allows us to reconstruct uniquely its Weinstein neighborhood?


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          Problem 1.2.

          Can the structure of a topological arboreal space be globally upgraded to the structure of a Lagrangian arboreal space? And are all such Lagrangian arboreal realizations symplectomorphic?

            •     The symmetry groups of trees, posets, simplicial complexes, and arboreal Lagrangians may all be distinct. We would like to understand what symmetries actually act on the arboreal Lagrangian.

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              Problem 1.3.

              What is the maximal number of symmetries for a radial arboreal lagrangian in $\mathbb{C}^n$? When does this maximal symmetry occur? And does the ribbon of the arboreal link have this symmetry?

                • Extra combinatorial data

                      In order to grade the Fukaya category of a Weinstein manifold $W$, we need to choose a bicanonical trivialization of $W$.

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                  Problem 1.4.

                  How can we describe such a trivialization in terms of the skeleton of $W$? More basically, how can we describe the first Chern class of $W$ in terms of its skeleton?
                    1. Remark. More generally, begin with any question about manifolds (which we think of as smooth Lagrangians inside their cotangent bundles). Can you recover this question as a particular case of a more general question about arboreal Lagrangians?

                        • Combinatorial moduli of skeleta

                              The choice of an arboreal skeleton reduces a Weinstein manifold to a finite amount of data. But we would also like to understand relations among different skeleta in a combinatorial way. Ideally, this would be accomplished by developing a good theory of handle slides in terms of Weinstein skeleta.

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                          Problem 1.5.

                          Is there an explicit list of moves on a Lagrangian skeleton that preserve the Weinstein structure?
                            1. Remark. This is already interesting even in low dimensions.

                                •     From considerations in mirror symmetry, we are very interested in understanding "phase transitions" among different classes of skeleta.

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                                  Problem 1.6.

                                  Can one produce a finite list of all arboreal skeleta for a given Weinstein manifold? And is there a good notion of a wall-and-chamber structure separating these in the moduli space of skeleta?

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