School of Science Department of Mathematics 47 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: DINH Vu Tung Lam / MATH-PMA Course: UROP 1100, Spring The first part of the paper shows the basic notations and terminology of quiver and cluster algebra, which leads to one of the most important results, the Laurent Phenomenon. The next part is the relation between applying the three different methods to verify the expansion formula in the An-case: the perfect matching method, the crossing path method, and the quiver mutations sequence method. The Positivity Theorem is verified on the An-case, as all coefficients are either 0 or 1. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: KWAN Cheuk Yin / MATH-PM Course: UROP 2100, Spring This report aims to study generalized associahedra associated to finite root systems. We first give a description of root systems, and its close relationship with cluster algebras of finite type in that they can be classified by the classical Dynkin diagrams. We then introduce the concept of generalized associahedra, provide an algorithm for constructing one corresponding to a given root system, and show how root subsystems corresponds to cross-sections of polytopal realizations of generalized associahedra. A few examples for rank 3 and 4 root systems are also provided. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: PHAN Nhat Duy / MATH-PMA Course: UROP 4100, Spring We study the relationship between cluster algebras and Schubert varieties. It proves that the coordinate ring of an (open) Schubert variety coincides with a cluster algebra, resolving a long-standing conjecture. The authors of further generalize this result to skew Schubert varieties, constructing a cluster structure that differs from the conjectures in prior work. The proof strategy involves relating Leclerc’s cluster structure on Schubert varieties to the combinatorially-defined cluster structure arising from plabic graphs. A key technical tool is a construction by Karpman used to bridge the two perspectives. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: WANG Daci / MATH-GM Course: UROP 1100, Fall We study the relation between the snake diagram and the folded graph of ̅ , , or path diagram, and hence show the number of terms of cluster variables on a disk is Fibonacci. We establish the notion of point resolutions, which extracts the information of the triangulation into combinatorics. We construct a generating set of , for any arc γ given a triangulation T on some unpunctured orientable surface and give a simplest solution.
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