UC San Diego

There are three parts in this thesis. First, we generalize the class of tropical

curves from trivalent to 3-colorable which can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $\Gamma$.

Second, we prove the equality of two canonical bases of a rank 2 cluster

algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.

Third, we link up scattering diagrams D with quiver representations of

corresponding quivers Q. We define a notion of good crossing of broken lines $\gamma$ on D. Then we show if $\gamma$ has good crossing over D, then it goes in the opposite direction of the Auslander-Reiten quiver of Q. Then we give a stratification of quiver representations by the bendinga of $\gamma$.

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