Title: Super Cluster Algebras from Surfaces Abstract: This talk will provide an introduction to cluster algebras from surfaces, and their generators, known as cluster variables. We will also describe how certain combinatorial objects known as snake graphs yield generating functions that agree with Laurent expansions of said cluster variables. In recent years, a lot of progress has been made on the problem of defining a super-commutative analogue of Fomin-Zelevinsky’s cluster algebras. In recent joint works with Nick Ovenhouse and Sylvester Zhang, we began the project of exploring the super cluster algebra structure from Penner-Zeitlin’s decorated super Teichmüller space, generalizing the notion of (classical) cluster algebras from triangulated surfaces. In this colloquium talk, I will survey our recent works on combinatorial and matrix formulas for super lambda-lengths, proving super-analogues of combinatorial features such as the Laurent Phenomenon and positivity. Such formulas again utilize snake graphs but using double dimer covers rather than single dimer covers. If time permits I will also discuss applications to super Fibonacci and super Markov numbers, as well as connections to super-frieze patterns. This talk is based on joint works with Nick Ovenhouse and Sylvester Zhang (arXiv 2102.09143, 2110.06497 and 2208.13664) and will not assume prior knowledge of cluster algebras.
