UROP Proceedings 2020-21

School of Science Department of Mathematics 55 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: KWOK Kin Ming / SSCI Course: UROP1100, Summer Cluster algebra was first introduced in S. Fomin and A. Zelevinsky. “Cluster algebras I: Foundations”. In: Journal of the American Mathematical Society 15 (2001), pp. 497–529, which was related to different area of Mathematics such as Lie theory and classification of Lie group. This report would first introduce cluster algebra from triangulation of convex polygon and Ptolemy’s theorem, and then touch on concept such as quiver, seed mutation, coefficient field and Laurent phenomenon. Then a result from Gregg Musiker and R. Schiffler. “Cluster expansion formulas and perfect matchings”. In: Journal of Algebraic Combinatorics 32 (2008), pp. 187– 209 on deriving the formula to represent any cluster variables in terms of Laurent polynomial of a any cluster using perfect matching would be discussed with an example. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: ZHANG Ke / MATH-PMA Course: UROP1100, Summer In this article, we summarize the study of cluster algebra in UROP1100 during summer 2021. We start with a summary on the lecture notes including the definition of cluster algebra, its relation to quiver mutation and Laurent Phenomenon. Next, we relate the triangulation of a surface to cluster algebra. We present a method turning a triangulation into a graph and build a map between certain paths in the triangulation and perfect matchings in the graph. Using this method, we obtain a formula for the cluster variable and its coefficients, which we illustrate with an example. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: ZOU Huaiyang / MATH-PMA Course: UROP1100, Summer This report first reviews the basics of cluster algebra. We recall the basic structure of cluster algebra like cluster variables, exchange matrix, mutation, coefficient and so on. And we also recall an important feature of cluster algebra, the Laurent Phenomenon, as well as Caterpillar Lemma which can prove the phenomenon. Secondly, the report talks about the content of the paper. We associate an unpunctured surface with a cluster algebra with principle coefficients and then use the theorem introduced in the paper to give the expansion formula of some cluster variables which are related to arcs on the surface. At the end, an original example of a decagon is given to illustrate the idea above.