School of Science Department of Mathematics 48 Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: PHAN, Nhat Duy / MATH-PMA Course: UROP2100, Fall UROP3100, Summer We study cluster algebra structures on coordinate rings of Schubert cells and partial flag varieties. The cluster algebra on coordinate rings of Schubert cells can be constructed from regular functions on G called generalized minors. The initial seed of coordinate rings of Schubert cells can be lifted to an initial seed of coordinate rings of partial flag varieties. We further apply these constructions to group SL3(ℂ) and SL4(ℂ). Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: TAI, Sung Chit / COSC Course: UROP2100, Fall This study aims to study the saturation and emptiness of Newton polytopes of cluster algebras from a marked surface. Much has been conjectured on the saturation of Newton polytopes of cluster algebras, but little has been done on investigating the general case. By introducing a saturation and emptiness score of the Newton polytope, it is hoped that more insights and information on the Newton polytopes can be found by investigating the equivalent perfect matching polytopes. It is hypothesized that duplicate weight vectors of snake graphs induced by an arc passing through triangulation may contribute to the non-emptiness of certain Newton polytopes. Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: ZHANG, Zhang / SSCI Course: UROP1100, Summer This article reports the contents that I learnt during this summer term, including the basic knowledge about cluster algebra (definition, theorems, etc.) and some related knowledge, which are necessary for introducing how to associate a cluster algebra to a surface, and hence calculating the cluster expansion for an arbitrary arc on the surface by a formula related to the perfect matchings of a graph constructed from the arc, which was introduced in the paper Cluster Expansion formulas and perfect matchings. Finally, I worked on the interesting example that a torus with a disk subtracted with one marked point on the boundary. Quantum Groups Supervisor: IP, Ivan Chi Ho / MATH Student: CHEN, Ziming / MATH-PMA Course: UROP1100, Fall In this report, we try to give a concrete example of the quantum Clebsch-Gordan formula: Vq,2⊗Vq,2 ≅ Vq,4⊕Vq,2⊕Vq,0. Especially, we focused on the detailed correspondence between the direct product and the direct sums.

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