Workshop Lecture Series: Moving Frames

Peter Olver (University of Minnesota)

The classical method of moving frames was developed by Elie Cartan into a powerful tool for studying the geometry of curves and surfaces under certain geometrical transformation groups. Recently, Mark Fels and I established a new foundation for the moving frame theory. Our method is completely algorithmic, and can be readily applied to completely general Lie group (and even pseudo-group) actions. The resulting theory and applications are remarkably wide-ranging, including geometry, differential equations, variational problems, numerical approximations, image processing and computer vision, and classical invariant theory, and others.

In this series of talks, I will introduce the basic ideas underlying the moving frame method, in a form amenable to direct application and explicit computation. The prerequisites are some familiarity with manifolds, differential forms, and the basics of Lie groups and their actions, as presented, for instance, in the first chapter of my Springer book. I will present a wide variety of new applications, including classification and syzygies of differential invariants and joint invariants, computation of invariant variational problems and invariant differential equations, equivalence, symmetry and rigidity properties of submanifolds, applications to polynomials in classical invariant theory, applications to object recognition in computer vision, and the design of symmetry-preserving numerical algorithms.