The geometry of cluster varieties from surfaces

Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented surfaces with boundary. The main original contribution of this thesis is to develop the properties of a particular kind of cluster variety called the symplectic double. We show that the symplectic double is birational to a certain moduli space of local systems associated to a doubled surface. We define a version of the notion of measured lamination on such a surface and prove that the space of all such laminations is a tropicalization of the symplectic double. We describe a canonical map from this space of laminations into the algebra of rational functions on the symplectic double. The second main contribution of this thesis is a proof of Fock and Goncharov's duality conjectures for quantum cluster varieties associated to a disk with finitely many marked points on its boundary. These duality conjectures identify a canonical set of elements in the quantized algebra of functions on a cluster variety satisfying a number of special properties.