Mathematical Colloquium Geometric surface models from representation theory

Derived categories are ubiquitous in mathematics and often serve as a go-between of different areas of mathematics such as representation theory, algebraic and symplectic geometry. In general, however, they are difficult to parse. But, in certain cases, combinatorial models can be constructed which give a good picture of these categories. One such case are the bounded derived categories of so called gentle algebras, where their derived categories can be represented in terms of curves and crossings of curves on surfaces. Behind the seemingly more simple surface combinatorics, lies the deep theory of Fukaya categories and homological mirror symmetry.  In that case the gentle algebras and their derived categories serve as a go-between of the A- and B-side of homological mirror symmetry. We will show in this talk how the resulting surface combinatorics gives new insights into the related areas and can be used to answer open questions and construct counter-examples.