Arnol'd's Lagrangian intersection conjecture

Let LL be a closed smooth manifold and let TLT^*L denote its cotangent bundle. For every compactly supported Hamiltonian diffeomorphism ϕ\phi of TLT^*L, consider the intersection points of the zero section LL with its image ϕ(L)\phi(L). Arnol'd's conjecture. The number of intersection points ϕ(L)L\phi(L)\cap L is bounded from below by the minimal number of critical points of a smooth function on LL. This is the Lagrangian version of Arnol'd's conjecture for cotangent bundles and motivates lower bounds for Lagrangian intersections; the source does not provide enough information here to determine whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Habib Alizadeh, Marcelo S. Atallah and Dylan Cant, “Lagrangian Intersections and the spectral norm in convex-at-infinity symplectic manifolds”, arXiv:2312.14752 (2023).

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