Degenerate Arnol'd Conjecture on Lagrangian intersections

Prove that for a closed symplectic manifold (M, ω) and a Hamiltonian isotopic pair of Lagrangian submanifolds (L0, L1) where each Li is connected, closed, and satisfies ω|_{π2(M, Li)} = 0, the intersection cardinality satisfies |L0 ∩ L1| ≥ min{ |Crit(f)| : f ∈ C∞(Li) }, where Crit(f) denotes the set of critical points of f.

Background

A central problem in symplectic geometry is to obtain lower bounds on the number of intersection points between two Lagrangian submanifolds. In the transverse case, classical Floer theory provides bounds in terms of Betti numbers, while for possibly degenerate intersections, bounds have been given via cuplength and spectral refinements.

The Degenerate Arnol'd Conjecture strengthens these bounds by asserting that for Hamiltonian isotopic Lagrangians in a closed symplectic manifold with appropriate asphericity, the number of intersections is bounded below by the minimal number of critical points of a smooth function on the Lagrangian. The paper references known cases (such as the zero section in cotangent bundles) and situates its parameterized Floer homotopy construction as a tool toward stronger intersection bounds, but the general conjecture remains open.