Geometric models associated to triangulations of Riemann surfaces arose in the context of cluster algebras and have since been used as an important tool to study representation theory of algebras and provide connections with algebraic geometry and symplectic geometry.
Significant applications of geometric models include a description of extensions and a classification of support tau-tilting modules over gentle algebras. Gentle algebras are a particular subclass of string algebras, which are of tame representation type, meaning it is often possible to get a global understanding of their representation theory.
In this talk I will describe the module category of a gentle algebra via partial triangulations of unpunctured surfaces and explain how to extend this model to a geometric model of the module category of any string algebra. This is based on joint work in progress with Karin Baur.