UROP Proceedings 2021-22

School of Science Department of Mathematics 46 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: FOO Peace / MATH-PM Course: UROP2100, Fall UROP3100, Summer The first part of this brief overview covers the basis and elementary methods used in the study of quantum cluster algebras and surveys the parallels between classical and quantum forms of cluster varieties with concrete examples of the quantization process given in the form of quiver representations. This serves as a prelude to a discussion of the Quantum Lift Theorem examined in great detail by Goncharov and Shen as a synthesis of several different results relating to moduli spaces, cluster variety structures, and quantum cluster algebras. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: HO Sui Kei / COSC Course: UROP2100, Spring Plabic graphs and plabic networks are combinatorial objects that appears in a number of cluster algebra system. We start by reviewing the definitions and combinatorics of plabic graphs. Next, we review an application of plabic networks in representation of quantum group and compare with the quiver representation. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: KWOK Kin Ming / MATH-IRE Course: UROP2100, Fall Cluster algebra was introduced in S. Fomin and Zelevinsky, 2001 and Sergey Fomin and Zelevinsky, 2007, which also introduced the concepts of c-vectors, g-vectors and F-polynomials. Subsequently, from Nakanishi and Zelevinsky, 2011, c-vectors and g-vectors are collected as column vectors into matrices, known as Cmatrices and Gmatrices, allowing linear algebra method to be taken advantage of in the study of cluster algebra. Two main conjectures - the sign-coherent conjecture for C-matrices and the Laurent positivity conjecture has been proposed, which was proved in Gross, Hacking, Keel, and Kontsevich, 2016 using scattering diagrams and use of toric geometry. Subsequently, an alternative proof in Nakanishi, 2021, was presented using also scattering diagrams but without the usage of toric geometry. The result from Nakanishi, 2021 would be the main focus of this report .