Affine cluster algebras as dynamical systems, surface triangulations, quiver representations and friezes

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The research of the year April 2021 - March 2022 has resulted in the two paperstitled ``~A and ~D type cluster algebras: Triangulated surfaces and friezes'' and ``Period 2 quivers and their T- and Y-systems''. The first of those has been submitted to the Journal of Algebraic Combinatorics, and the corrections have been completed.In the first paper we were able to classify cluster variables for cluster algebras of affine ~A and ~D type. This was done by looking at these cluster variables as arcs of triangulations for appropriate surfaces, a method pioneered in [Fomin, Shapiro and Thurston 2008]. By first identifying the T-system cluster variables and periodic quantities studied in [Fordy and Hone 2014] and [Pallister 2020] as arcs (cluster variables) we were able to identify all other cluster variables in terms of these. We were also able to prove that these cluster variables can be arranged as friezes. Our results agree with similar results found via the representation theory of these quivers. In the second paper we were interested in finding all period 2 quivers. These are quivers that, if mutated twice, look essentially the same. This turned out to be a difficult problem, instead we found all period two quivers with a low number of vertices, of which there are surprisingly many. We then used results of [Nakanishi 2011] to write the T- and Y- systems for these quivers, which here are systems of 2 recurrence relations. We looked at some interesting examples of these which had periodic quantities.