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Faculty of Arts and Science - Department of Mathematics and Statistics
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Extension de l'homomorphisme de Calabi aux cobordismes lagrangiens Theses and supervised dissertations / 2019-09
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.