The concept of holonomy arose in the 19th century in the study of mechanical systems subject to `rolling constraints’, such as an object rolling on a surface without slipping or twisting. It quickly found application in differential geometry and mathematical physics, in the analysis of parallel translation associated to a connection and its relation with the curvature of the underlying geometric structures. The subject continues to develop in a number of areas, including Riemannian, sub-Riemannian, and parabolic geometries, where questions of existence and uniqueness turn out to devolve on solving systems of differential equations that often do not fit into the standard types, but that can be approached using Cartan’s theory of exterior differential systems.
In this series of lectures, I will introduce the ideas of the subject, illustrated by a series of examples drawn from mechanics and geometry, and discuss some of the interesting problems from differential equations that these examples raise as well as recent progress on their solutions.
No notes or recordings available for these lectures.