Incidence algebras have first appeared in combinatorics, but have proved to be of crucial importance in other instances, and especially in the representation theory of finite dimensional algebras. Their combinatorial nature makes them important examples of a particular type of representations, thin representations, and also of distributive modules. A finite dimensional module is thin if the multiplicity of any simple in its composition series is at most one. Classically, (commutative) rings whose lattice of ideals is distributive have played an important part; these are the Prüfer rings. 

Our main results show that incidence algebras are the counterpart of these in the finite dimensional world. Furthermore, we show that every thin representation comes from an incidence algebra, in the following sense: if M is a thin finite dimensional module over a (f.d.) algebra A, then A/ann(M) is isomorphic to an incidence algebra. We give a method to classify thin representation via simplicial cohomology. On the way, we give characterizations and classifications incidence algebras, and their deformations, which is one of the main tools introduced. We show these are precisely acyclic algebras with distributive (equivalently, finite) lattice of ideals. Time permitting, we will show how these will be used to study representation type of algebras, and give other applications to a conjecture of Bongartz and Ringel.

The methods involve a blend of non-commutative and homological algebra, combinatorics and algebraic topology, but in depth familiarity with any topic is not required.