On geometric models in representation theory

TL;DR

This paper reviews how geometric surface models, especially for gentle algebras, have advanced the understanding of their derived categories and connections to Fukaya categories, highlighting recent theoretical progress.

Contribution

It provides an overview of recent theoretical advances linking geometric surface models to derived categories and Fukaya categories in the context of gentle algebras.

Findings

Surface models elucidate the structure of derived categories of gentle algebras.

Connections between gentle algebra models and Fukaya categories are established.

Theoretical frameworks enhance understanding of algebraic and geometric interplay.

Abstract

Geometric models have emerged as an important tool in the representation theory of algebras. Surface models associated to gentle algebras have been particularly fruitful in advancing our understanding of their module and derived categories. We give an overview of some of the theoretical advances that geometric surface models for the derived categories of graded gentle algebras and their connections to Fukaya categories of surfaces have made possible.