Abstract: In this talk, we will explore the topological, algebraic, and categorical structure of non-orientable surfaces! By decomposing a non-orientable surface S into triangles, we can associate different algebraic structures to S. Namely, we can associate a "cluster-like" algebra known as a quasi-cluster algebra or another algebra known as the Jacobian algebra that is constructed from a quiver with relations. We will explore both of these constructions, but will focus on the Jacobian algebra and its representation theory. In particular, we give examples of what modules of this algebra look like by using topology and combinatorics. Furthermore, we will ramp up this discussion and conclude the talk by defining an analogue of a cluster category in this setting. This is based on joint work with Vèronique Bazier-Matte and Aaron Chan and our preprint can be found here: https://arxiv.org/pdf/2211.15863.pdf
