Parking Cars, Rolling Balls, and Falling Cats: The Concept of Holonomy
The mathematical concept of ‘holonomy’ was gradually developed in the late 19th and early 20th centuries in a number of different contexts, and it was found to lie at the base of many everyday phenomena. Anyone who has ever tried to control the orientation of a 3-dimensional object on a computer screen using a trackball has dealt with the problem of trying to control three parameters (yaw, pitch, and roll) with an object that can only accept essentially two inputs (the direction and speed of rotation of the ball). That we can actually do this (and many other similar feats, such as parallel parking a car or a trailer or, for a cat, the ability to turn itself in the air so as to land on its feet) is due to the phenomenon of ‘non-holonomy’ of mechanical systems.
The problem of how to make this somewhat vague concept precise, so that it can be put to use, has occupied engineers and mathematicians for more than a century, and new things about it continue to be discovered today. It turns out to be deeply geometrical in nature. Even as simple a system as a ball rolling over a surface without slipping or twisting turns out to have surprising connections with other parts of mathematics, including the so-called ‘exceptional’ groups. ‘Holonomy’ is used to detect the curvature of space, and constraints on it are used to describe systems that are important in string theory and particle physics.
In this talk, I’ll describe some basic aspects of this concept, starting with simple physical problems and illustrating the ideas using familiar objects (including the Platonic solids).
About the Speaker
Robert Bryant is the Director of the Mathematical Sciences Research Institute in Berkeley, California. Born and educated in North Carolina, he has held positions in mathematics departments at Rice University, Duke University, and, currently, at the University of California at Berkeley. He is a member of the National Academy of Sciences and a Fellow of the American Mathematical Society.
Bryant’s research is in the area of differential geometry and its applications, particularly to the study of partial differential equations, control theory, and the calculus of variations. While continuing to develop techniques pioneered by Elie Cartan and Shiing-shen Chern in the early and middle 20th century for studying these problems, his work has produced results in the theory of holonomy of Riemannian manifolds (particularly, showing the existence of the so-called ‘exceptional’ holonomies that turn up in string theory), minimal surfaces, integrable systems, and several related areas.
