School of Science Department of Mathematics 48 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: ZOU Huaiyang / MATH-PMA Course: UROP2100, Fall This report introduces some results of Nakanishi's “Cluster Patterns and Scattering Diagrams” [Nak21] about the proof of sign-coherence and Laurent positivity. The major ideas originate from the work of Gross, Hacking, Keel, and Kontsevich using scattering diagrams and toric geometry. The paper of Nakanishi translates some results into pure cluster algebra languages but gives new representations and logic of the proof. In this report, we introduce some of the results which are still based on scattering diagram method. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: CHAN Hong Ming / COSC Course: UROP3100, Summer In this report, an elementary way to construct Lusztig’s Basis from Poincar´e–Birkhoff–Witt(PBW) Basis in Quantum Group of A, D, E type pointed out by Peter Tingley in  is accessed. In section 1, definition of Quantum group, a set of automorphisms, bar involution and PBW basis will be introduced. In section 2, explicit calculation have been done on the well know A2 case. First, introducing Lusztig’s Basis in A2. Then, the transforming matrix between PBW Basis and Lusztig’s Basis is calculated for A2 case. And is expressed as combination Lusztig’s Basis. This expression is claim to be useful in constructing Lusztig’s Basis according to . In section 3, some reviews and observations towards  will be given.  Peter Tingley. Elementary construction of Lusztig’s canonical basis. 2016. doi: 10.48550/ARXIV.1602.04895. url: https://arxiv.org/abs/1602.04895. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: CHUNG Soobeom / PHYS Course: UROP3100, Summer In this paper, the nineteen vertex model and its associated R-matrix will be introduced. It is related to the square lattice model with a spin-1 representation on each edge obeying the ice rule. After that, by using the Boltzmann weights of the vertex model, FRT construction will be used to show the algebra associated with the six-vertex model first and subsequently the 19-vertex model. The result is that they share an identical algebraic structure. Using different representation and parameters, one will be able to produce a corresponding algebra much richer in terms of the structure and with many different applications.