Counting Periodic Orbits: Conley Conjecture for Lagrangian Correspondences and Resonance Relations for Closed Reeb Orbits

The thesis is centered around the theme of periodic orbits of Hamiltonian systems. More precisely, we prove that on a closed symplectic Calabi--Yau manifold every Lagrangian correspondence Hamiltonian isotopic to the diagonal and satisfying some non-degeneracy condition has infinitely many periodic orbits, and we give a new proof of the theorem that every contact form supporting the standard contact structure on \(S^3\) has at least two periodic Reeb orbits. The former result is obtained by considering the intersection Lagrangian Floer homology of suitable Lagrangians and estimating index growth for iterations, while the latter relies on a new homotopy invariant index which is in turn used to prove a new variant of the resonance relation for Reeb flows