The talk is split into two parts (each 45 mins with a 30min break)

Part 1: Introduction to cluster algebras and cluster categories of surfaces

Cluster algebras of surfaces are fundamental examples in the theory of cluster algebras. They can be understood very explicitly in terms of combinatorial geometry of surfaces. After an introduction to these, we will gently explain what a categorification of a cluster algebra in terms of a triangulated category (the so-called cluster  category) is.

Part 2: Topological Fukaya categories and the Higgs category

We first recall the construction of the topological Fukaya category of a surface in terms of a constructible cosheaf of dg-categories. We then present a description of the relative version of the cluster category of a marked surface (called the Higgs category) in terms of a topological Fukaya category valued in a 1-CY cosingularity category.