14:15
The relationship between the topological or homotopy-invariant properties of a symplectic manifold X and the set of possible immersed or embedded Lagrangian submanifolds of X is rich and mostly mysterious. In 2020, D. Alvarez-Gavela, Y. Eliashberg, and D. Nadler proved that any Weinstein manifold (e.g. an affine variety) admitting a Lagrangian plane field retracts onto a Lagrangian submanifold with arboreal singularities (a certain class of singularities which can be described combinatorially). I will discuss work in progress with D. Alvarez-Gavela and T. Large investigating the other direction, in which we prove a partial converse to the AGEN result and show that most Weinstein manifolds do not admit such skeleta. This suggests that the Floer-theoretic invariants of some well-known open symplectic manifolds may be more complicated than expected.