Abstract
The goal of this thesis is to study exceptional collections of modules over hereditary algebras and their relationship with other popular areas of research. In Chapter 1, we begin by discussing some of the fundamentals of representation theory of finite-dimensional associative algebras. In Chapter 2 we define exceptional sequences and a generalization called exceptional collections. In this chapter, we begin with generalizing a combinatorial model of Garver, Igusa, Matherne, and Ostroff that describes exceptional sequences in type $\mathbb{A}$, to a model that describes exceptional collections in type $\tilde{\mathbb{A}}$. We will use these models to place all exceptional collections in type $\tilde{\mathbb{A}}$ into finitely many families, discuss the algebraic consequences of Dehn twists of these diagrams in terms of the Auslander--Reiten quiver, and count how many families there are for one particular orientation of the quiver. Moreover, we will prove that the notion of relative projectivity and injectivity is well-defined in exceptional collections, and use the models for type $\mathbb{A}$ to classify relative projectivity and injectivity of modules in exceptional collections. In Chapter 3, we will study cluster theory. We will begin by providing a survey of cluster theory through the lens of representation theory. Throughout the survey, we will need the notion of a quiver Grassmannians to explain the coefficients of cluster characters. It is here in Section 3.3 that we will provide a four-fold equivalence of categories between compact Riemann surfaces, fields of transcendence degree one over $\mathbb{C}$, smooth projective curves, and a category of quiver Grassmannians inspired by works of Hille, Huisgen--Zimmermann, Reineke, and Ringel. After, we will study cluster algebras that arise from a surface, where we will compare the combinatorial models introduced in Chapter 2 to study exceptional collections in type $\tilde{\mathbb{A}}$ with the existing models used to study clusters in type $\tilde{\mathbb{A}}$. We will then place clusters in type $\tilde{\mathbb{A}}$ into finitely many families, discuss the algebraic consequences of performing Dehn twists of these diagrams in terms of the cluster category, and count how many families there are for any orientation. Finally, in Chapter 4, we will discuss the relationship between exceptional collections of modules over hereditary algebras and algebraic $t$-structures of the corresponding bounded derived category. In particular, we prove that all such $t$-structures can be attained from exceptional collections of modules by taking appropriate shifts.