Gonality of metric graphs and Catalan-many tropical morphisms to trees

This thesis consists of two points of view to regard degree-(g′+1) tropical morphisms Φ : (Γ,w) → Δ from a genus-(2g′) weighted metric graph (Γ,w) to a metric tree Δ, where g′ is a positive integer. The first point of view, developed in Part I, is purely combinatorial and constructive. It culminates with an application to bound the gonality of (Γ,w). The second point of view, developed in Part II, incorporates category theory to construct a unified framework under which both Φ and higher dimensional analogues can be understood. These higher dimensional analogues appear in the construction of a moduli space Gtrop/g→0,d parametrizing the tropical morphisms Φ, and a moduli spaceMtrop/g parametrizing the (Γ,w). There is a natural projection map Π : Gtrop/g→0,d →Mtrop/g that sends Φ : (Γ,w) → Δ to (Γ,w). The strikingly beautiful result is that when g = 2g′ and d = g′+1, the projection Π itself is an indexed branched cover, thus having the same nature as the maps Φ that are being parametrized. Moreover, fibres of Π have Catalan-many points. Each part has its own introduction that motivates and describes the problem from its own perspective. Part I and its introduction are based on two articles which are joint work with Jan Draisma. Part II contains material intended to be published as two articles. There is also a layman summary available at the beginning