This dissertation explores the quantitative and qualitative properties of the cotangent bundles T ∗M of a closed smooth manifolds M , from the symplectic point of view. Quantitative aspects involve packing the open neighborhood W of the zero section with symplectic balls. We introduce a distance-like function ρW on the zero section M using the symplectic packing of two balls. In the case when W is the unit disc-cotangent bundle associated to the Riemannian metric g, we show how to recover the metric g from ρW . As an intermediate step, we construct a symplectic embedding from the ball B2n(2/√π) of capacity 4 to the product of Lagrangian unit discs Bn(1) × Bn(1). Such a construction implies the strong Viterbo conjecture for Bn(1) × Bn(1). We also give a bound on the relative Gromov width Gr(M, W) when M admits a non-contractible S1-action. The bound is given in terms of the symplectic action of the lift of non-contractible orbits of the S1-action. We also provide examples of when such a bound is sharp. This result is part of the joint work with Dylan Cant. The second part of this joint work is related to the qualitative aspects. We show the existence of periodic orbits of Hamiltonian systems on T ∗M for a large class of Hamiltonians. Another qualitative aspect is proof of the Arnol’d chord conjecture for conormal Legendrians in the co-sphere bundle S∗M . This part of the dissertation is joint work with Dylan Cant and Egor Shelukhin. We show that for a given closed submanifold N ⊂ M there exists a non-constant Reeb chord in (S∗M, α) with endpoints on ΛN := ν∗N ∩ S∗M, for arbitrary contact form α on S∗M which induces standard contact structure.

In this master's thesis, we construct a new invariant of Lagrangian cobordisms. This construction is inspired by the recent works of Solomon in which an extension of the Calabi homomorphism to exact Lagrangian paths is given. Solomon's extension was motivated by the fact that the graph of any Hamiltonian isotopy is an exact Lagrangian path. We use the Lagrangian suspension construction, which associates to every exact Lagrangian path a Lagrangian cobordism, to extend Solomon's invariant to Lagrangian cobordisms. In the first chapter, we give a brief introduction to the elementary properties of symplectic manifolds and their Lagrangian submanifolds. In the second chapter, we present an introduction to the group of Hamiltonian diffeomorphisms and discuss the fundamental properties of the Calabi homomorphism. Chapter 3 is dedicated to Lagrangian paths, Solomon's invariant and its critical points. In the last chapter, we introduce the notion of Lagrangian cobordism and we construct the new invariant. We analyze its critical points and evaluate it on the trace of the Lagrangian surgery of two curves on the torus. In this setting we further bound the new invariant in terms of the shadow of the cobordism, a notion recently introduced by Cornea and Shelukhin.