Geometry & Cluster Algebras: Finite Type Classification & Double Bruhat Cells

Abstract

In 2000, Fomin and Zelevinsky introduced a new language called cluster algebras for describing rings with certain combinatorial structures. Cluster algebras enjoy a variety of nice properties such as well-established collection of classification results and interesting geometric properties with the upper cluster algebra. We approach cluster algebras rst from the perspective of operations on quivers, then reacquaint ourselves with a more general definition. We then present the classification of cluster algebras of nite type and explore cluster algebra structures on the ring of regular functions of double Bruhat cells.