Tropical Techniques in Cluster Theory and Enumerative Geometry

Résumé du livre

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 $\gamma$.