School of Science Department of Mathematics 47 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: PHAN Nhat Duy / MATH-PMA Course: UROP1100, Summer We study cluster algebra with principal coefficients that are associated to unpunctured surfaces. We recall the formulas in Musiker and Schiffler that used perfect matchings of the graph to give the specific formula for the Laurent polynomial expansion of cluster variables in these cluster algebras. We then apply that result to generalize Laurent polynomial expansions over an annulus and a pair of pants. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: TAI Sung Chit / COSC Course: UROP1100, Summer This study aims to study the properties of the snake graphs on the once-punctured torus, and an equivalence relation is defined on the set of snake graphs corresponding to the arcs of the torus. Consequently, a unique decomposition theorem is proposed, where it is argued that a snake graph can be uniquely decomposed into 2 smaller snake graphs. The decomposition ensures that there is a surjection between positive coprime pairs of integers and the Markov numbers. However, to solve the Markov Unicity Conjecture (MUC), which states that every Markov triple has a unique maximal element, the surjection needs to be a bijection. Therefore, a recurrence relation is proposed for computing the number of perfect matchings on a given snake graph by defining a state space on the snake graph. A few examples are provided at the end of the paper as an attempt to demonstrate the use of the recurrence formula. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: ZHANG Ke / MATH-PMA Course: UROP2100, Fall In this article, we summarize the study of cluster algebra in UROP2100 during fall 2021. In this semester, we went through the artical “Cluster Algebras and Scattering Diagrams” by Tomoki Nakanishi, where we learnt about the proof of sign-coherence conjecture related to scattering diagrams. We start with reviewing some definitions in cluster algebra which will be used in the later part. Next, we introduce the definition scattering diagram and wall-crossing automorphisms. At last, we relate scattering diagram with cluster algebra and give an example.