Noncommutative Resolutions of Singularities

Abstract

Singularities in algebraic varieties present profound challenges in both classical and modern geometry. While Hironaka’s resolution of singularities provides a powerful tool in the commutative setting, many situations in higher-dimensional geometry and representation theory call for alternative approaches. This paper explores noncommutative resolutions—particularly noncommutative crepant resolutions (NCCRs)—as a means to “smooth out” singular spaces via noncommutative algebras. We review the theoretical foundations, provide key examples illustrating the construction of noncommutative resolutions, discuss their applications in fields such as string theory and representation theory, and outline open problems and future directions.