5. Structures in arboreal geometry
The theory of arboreal spaces (or "arborifolds") offers a rich new geometry with many interesting questions. The theory also offers us a new language for presenting already useful or important structures from symplectic, differential, and complex geometry.
By splitting a closed $(n+1)$dimensional arboreal space along an $n$dimensional arboreal space, we can treat it as a relation between two $n$dimensional arboreal spaces. This geometry motivates the following question:
Problem 5.1.
Is there a theory of cobordism for closed arboreal spaces? 
The data of a vector bundle with flat connection on a smooth manifold $M$ is the same as the data of a microlocal sheaf on $T^*M$ microsupported on $M$. This suggests that there should be a notion of "microlocal vector bundles" on an arboreal space $L$ so that "microlocal vector bundles with flat connection" are precisely microlocal sheaves on $L$.
Problem 5.2.
Give the correct definition of "microlocal vector bundles" on an arboreal space.
Remark. The resulting category should contain an object called the "microlocal tangent bundle" of $L$, which is the same thing as a ribbon for $L$. We should be able to recover invariants like Chern classes from index theory.


Boundary conditions in Floer theory are given by smooth Lagrangians equipped with local systems. As in the above question, the analogous data on an arboreal Lagrangian is a microlocal sheaf. We might hope to understand this analogy in a Floertheoretic way.
Problem 5.3.
Is it possible to set up Floer theory with boundary on an arboreal skeleton? What are arboreal Lagrangian boundary conditions? 
Problem 5.4.
Suppose a Weinstein manifold $W$, with arboreal skeleton $L$, has a holomorphic symplectic structure. How can we use this structure to enhance the arboreal space $L$ with extra data?
Cite this as: AimPL: Arborealization of singularities of Lagrangian skeleta, available at http://aimpl.org/arborlagrange.