School of Science Department of Mathematics 50 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: DINH Vu Tung Lam / MATH-PMA Course: UROP 2100, Fall UROP 3200, Spring The previous paper has already shown main ingredients in quiver for general -triangulated unpunctured surface, with the formulas for different value of and the triangulations of polygon up to hexagon. In this paper, we shall introduce several algebraic concepts of moduli space of local systems which allows us to see the connection with the cluster realization ( +1). Based on corresponding sequence of mutations for each flip from the given triangulation, we will give the general recurrence relation in order to calculating the resulted diagonal of the general flip, then we shall show an example of calculating case by case of the general surface. This project aim to discover the properties of expansion formula of the cluster algebra structure of the moduli space +1, and attempt to find the general formula by cases corresponding to the standard case -triangulated -gon. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: HO Chiu Ming Chatwin / MATH-IRE Course: UROP 1100, Fall Quantum groups is a type of noncommutative algebraic structure which has found profound applications in theoretical physics and quantum statistics. It is an algebraic structure which can be utilized to generate nontrivial solutions of interest to the Yang–Baxter equation, which describes the scattering of particles in quantum many-body systems. In this report, we give a definition of the algebraic structure and a description of report in preparation of a detailed analysis of the subject. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: SOMER Uras / MATH-IRE Course: UROP 2100, Spring A topological invariant of closed connected 3-manifolds is constructed using modular Hopf algebras and ribbon graphs. Ribbon Hopf algebras and ribbon graphs are defined and their relationship is shown. From ribbon Hopf algebras, modular Hopf algebras are constructed. An example modular Hopf algebra generated by the quantum group derived from the deformation of the universal enveloping algebra of 2 is presented and shown to satisfy the necessary axioms. The invariant is constructed and Kirby moves are used to argue its existence.
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