Non-commutative crepant resolutions: scenes from categorical geometry

TL;DR

This paper explores non-commutative crepant resolutions, their theoretical foundations, and their applications in algebraic geometry, string theory, and representation theory, highlighting recent research developments.

Contribution

It contextualizes Van den Bergh's definition within broader mathematical and physical frameworks and discusses current research directions in the area.

Findings

Connections to tilting theory and McKay correspondence

Applications to string theory and representation theory

Current research trends in non-commutative resolutions

Abstract

Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.