UROP Proceedings 2022-23

School of Science Department of Mathematics 47 Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: FOO, Peace / MATH-PM Course: UROP4100, Spring We review recent developments in the topic of super cluster algebras as constructed by Ovsienko–Shapiro and Shemyakova–Voronov using different methods and a super cluster algebraic interpretation of the decorated super-Teichmüller spaces of Ip–Penner–Zeitlin, and endeavor to highlight links between these different super versions. Representations of super cluster algebras via quivers and triangulations will be examined and examples given, in the context of the super analogues of the well-known Plücker and Ptolemy relations. Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: KWAN, Cheuk Yin / MATH-PM Course: UROP1100, Summer This paper mainly aims to study cluster algebras with principal coefficients which are associated to unpunctured surfaces. We first give a brief description of cluster algebras, as introduced by Fomin and Zelevinsky in 2002, and a proof of the Laurent phenomenon. We then investigate cluster algebras from unpunctured surfaces and their relations to perfect matchings of the graph GT,γ, i.e., the cluster expansion formula in terms of the initial cluster derived from the snake graph generated from a given triangulation T and an arc γ that is not part of T. We then end with demonstrations of the formula with an octagon and some patterns found between neighbouring arcs. Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: LI, Shujian / MATH-PMA Course: UROP1100, Spring UROP2100, Summer In [4], the authors show that the embedding of ( ) into quantum cluster algebra ( ) can be concisely described using quivers and planar directed networks, and this embedding is consistent under mutation. This report first uses Coxeter moves and some identification operations to fold the graph of type A2n−1 into the graph of type Cn; then explores similar planar networks to describe the embedding of ( ) of type Cn.