Specialization of Cluster Variables - Khrystyna Serhiyenko

A quiver Q is a directed graph without loops or oriented 2-cycles.   In addition, we decorate the quiver by placing a positive integer at every vertex of Q.   There is a simple combinatorial operation called mutation that produces a new decorated quiver by making some local changes at a given vertex.  Then starting with Q, we can recursively perform mutations one after another, resulting in infinity many new quivers.  It follows from the properties of Cluster Algebras that the values at the vertices are positive and rational, but not necessarily integers.   

The goal of the project is to find conditions on the quiver Q such that the values remain integers after all possible sequences of mutations.   Computing various examples, we want to formulate conjectures and then prove them.