Doctor of Philosophy (PhD)
Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work along with previous work by others seeks to establish a base case in answering the question of whether a given finite mutation type cluster algebra exhibits a maximal green sequence. In this paper we will provide a triangulation for orientable surfaces of genus n with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.
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Bucher, Eric, "Cluster Algebras and Maximal Green Sequences for Closed Surfaces" (2016). LSU Doctoral Dissertations. 205.