This book presents an accessible approach to an emerging theory of picture groups. Intended for graduate students and researchers, it explains the connections between several branches of algebra and topology, and demonstrates how they interact. It begins with foundational material on modulated quivers and their representations, cluster categories, and semi-invariants. The text then develops virtual analogues of classical results, allowing dimension vectors with negative coordinates. Finally, it defines the notion of a picture group associated to a semi-invariant picture, also introducing picture spaces which are CW-complexes constructed from semi-invariant pictures. For quivers of type $A_n$ the key theorem draws on K-theory and states that the associated picture space is a $K(G(A_n) , 1)$ connected CW-complex for the corresponding group $G(A_n)$ associated with the same quiver.

This volume contains selected expository lectures delivered at the annual Maurice Auslander Distinguished Lectures and International Conference over the last several years. Reflecting the diverse landscape of modern representation theory of algebras, the selected articles include: a quick introduction to silting modules; a survey on the first decade of co-t-structures in triangulated categories; a functorial approach to the notion of module; a representation-theoretic approach to recollements in abelian categories; new examples of applications of relative homological algebra; connections between Coxeter groups and quiver representations; and recent progress on limits of approximation theory.

This book presents an accessible approach to an emerging theory of picture groups. Intended for graduate students and researchers, it explains the connections between several branches of algebra and topology, and demonstrates how they interact. It begins with foundational material on modulated quivers and their representations, cluster categories, and semi-invariants. The text then develops virtual analogues of classical results, allowing dimension vectors with negative coordinates. Finally, it defines the notion of a picture group associated to a semi-invariant picture, also introducing picture spaces which are CW-complexes constructed from semi-invariant pictures. For quivers of type $A_n$ the key theorem draws on K-theory and states that the associated picture space is a $K(G(A_n) , 1)$ connected CW-complex for the corresponding group $G(A_n)$ associated with the same quiver.

This volume contains selected expository lectures delivered at the annual Maurice Auslander Distinguished Lectures and International Conference over the last several years. Reflecting the diverse landscape of modern representation theory of algebras, the selected articles include: a quick introduction to silting modules; a survey on the first decade of co-t-structures in triangulated categories; a functorial approach to the notion of module; a representation-theoretic approach to recollements in abelian categories; new examples of applications of relative homological algebra; connections between Coxeter groups and quiver representations; and recent progress on limits of approximation theory.