Part I: What is tilting theory?
Part II: Convex geometry for fans of tilting theory.

In Part I, we will use modules over a polynomial ring in one variable to illustrate how quiver representations provide a concrete method for understanding the structure of modules over algebras and for viewing the representation theory of an algebra. This will provide us with elementary examples with which to illustrate tilting theory, which allows us to compare the representation theory of different algebras. Using these examples, we will illustrate a conceptual explanation of why two algebras related by tilting have similar representation theory.

In convex geometry, a fan is a set of cones such that all intersections are common faces of cones in the fan. In Part II, we will discuss how to obtain a fan, the « heart fan », of an abelian category (such as representations of a quiver) from tilting theory. We will describe the relationship with the g-vector fan coming from cluster theory and a convex-geometric object associated to c-vectors, which are related by a Galois connection. We will describe how the heart fan simultaneously completes the g-vector fan and provides a construction of a virtual g-vector fan in cases where the “projective minded” objects used to construct g-vectors need not exist.