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Wednesday, October 21, 2015

**Abstract:** In groundbreaking work from a few years ago, K. Lee, L. Li, and A. Zelevinsky constructed a basis for any rank 2 cluster algebra that consists of a special family of indecomposable positive elements. They coined the term Greedy Basis for this construction and illustrated a combinatorial interpretation of the Laurent expansions of its elements using Dyck paths. In the last year, M. Gross, P. Hacking, S. Keel, and M. Kontsevich used algebraic geometry, as inspired by mirror symmetry and tropical geometry, to define the Theta Basis for any cluster algebra. The construction of the theta basis can be described in terms of the machinery of broken lines and scattering diagrams. In this talk, these two bases will be constructed, assuming no prior knowledge, and compared. In particular, I will discuss joint work with M. Cheung, M. Gross, G. Muller, D. Rupel, S. Stella, and H. Williams started at an AMS Mathematical Research Community, in which we equate these two bases in the rank two case.