School of Science Department of Mathematics 49 Quantum Groups Supervisor: IP, Ivan Chi Ho / MATH Student: CHUNG, Soobeom / PHYS Course: UROP4100, Fall In this report, we construct the positive principle series representation, or simply positive representation, involving complex powers of generators of quantum groups ( (3, ℝ)) considered as positive and selfadjoint operators on a Hilbert space. The concept of modular double for quantum groups by Faddeev in the case of ( (2, ℝ)) is extended to the case of ( (3, ℝ)) for the construction of a discrete series of representations in the region |τ| = 1. This region corresponds to the strongly coupled region 1 < c < 25 for the central charge of Virasoro algebra in Liouville model. In addition, these generators are represented graphically on a quiver diagram using the language of quantum cluster algebra. Quantum Groups Supervisor: IP, Ivan Chi Ho / MATH Student: ZHANG, Yitong / MATH-IRE Course: UROP1100, Spring UROP2100, Summer This is the report for the UROP2100 program, which is considered to be the continuation of the former UROP1100 program. In the program, three papers by N.Reshetikin and V.G.Turaev are read, which, in chronological order, introduced topological invariants of links, ribbons (or, framed links) and 3-manifolds (and more generally, 3-cobordisms) generate by algebras satisfy certain different structures, with (special) quantum groups as examples. This report would be the summary of those three papers, with the pathway: 1). the topological background; 2). the relation between topological categories and certain algebra structures; 3). the properties and categorification of such algebra structures. Efficient Numerical Methods for Dynamic Interface Supervisor: LEUNG, Shing Yu / MATH Student: CHEN, Jhao Song / SSCI Course: UROP1100, Summer This report summarizes the problems associated with mean curvature motion and presents numerical algorithms to simulate the evolution of a closed curve based on these motions.

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