Curvature and Computation

Abstract

When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which certain computations can be efficiently carried out. In short, I describe the close but surprising connection between negative curvature and efficient computation. This theorem was one of the clearest early indications that applying a metric perspective to traditional group theory problems might lead to new and important insights.

10 pages, expository article; version 2 clarifies some ambiguous language on computational complexity and corrects a passing musical reference from Stravinsky to Prokofiev