This talk will introduce cluster algebras, with an emphasis on their combinatorics, and describe a recent joint result with Vincent Pilaud and Sibylle Schroll. At the heart of a cluster algebra is a complicated, branching recursion that defines cluster variables (certain rational functions organized into finite sets called clusters). The recursion looks bizarre at first glance (and at subsequent glances), but has surprisingly close connections to various areas of mathematics. For a given cluster algebra, one typically needs a combinatorial model before even formulating the question of how to solve the recurrence. One class of cluster algebras is modeled by surfaces with a finite set of distinguished “marked points”, where the cluster variables are indexed by tagged arcs (curves connecting the marked points) and the clusters are collections of arcs that cut the surface into triangles. In this model, solving the recursion means finding a formula for the cluster variable associated to a given tagged arc.  Earlier solutions hint at the importance of the Fundamental Theorem of Finite Distributive Lattices (FTFDL).  We take the FTFDL as a guiding principle to give a new formula for cluster variables, with simpler combinatorics and with simpler proofs that leverage the hyperbolic geometry of the surfaces.  The talk will assume no prior knowledge of cluster algebras, marked surfaces, or distributive lattices.

Speaker’s webpage: https://nreadin.math.ncsu.edu