Tropical techniques in cluster theory and enumerative geometry
Man-Wai Cheung · eScholarship (California Digital Library) · 2016
There are three parts in this thesis. First, we generalize the class of tropicalcurves 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 clusteralgebra, 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 ofcorresponding 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 $\\gamma$..