Estimating Reeb chords using microlocal sheaf theory

December 17, 2021
We show that, for closed Legendrians in 1-jet bundles, when there is a sheaf with singular support on the Legendrian, then (1) its self Reeb chords are bounded from below by half the sum of Betti numbers, and (2) the Reeb chords between itself and its Hamiltonian push off is bounded from below by Betti numbers when the C0-norm of the Hamiltonian is small. I will show how to visualize Reeb chords/Lagrangian intersections in sheaf theory, and then explain the duality exact triangle and the persistence structure used in the proof. If time permits, I will state a conjecture on the relative Calabi-Yau structure that arises from the duality exact triangle.