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Lie groups, differential equations and moving frames

Peter J. Olver

University of Minnesota

In this series of lectures, I will begin with an introduction to the theory of Lie groups and differential equations, following chapters 2 & 3 of my Springer text. Applications include computation of symmetries and conservation laws, integration of ordinary differential equations, and determination of explicit solutions of partial differential equations, of use in checking numerical algorithms. Then I will discuss a new, general approach to the geometric method of moving frames. The method is surprisingly elementary, completely algorithmic, and can be readily applied to completely general Lie group (and even pseudo-group) actions. The resulting theory and applications lead to a remarkably wide range of applications in geometry, algebra, differential equations, the calculus of variations, computer vision and geometric integration. Particular topics include 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.

Prerequisites: Basic knowledge of Lie groups, manifolds and vector fields.

Key references:

Olver, P.J.; Applications of Lie Groups to Differential Equations; Second Edition, Graduate Texts in Mathematics, vol. 107, Springer--Verlag, New York, 1993

Olver, P.J.; Moving frames --- in geometry, algebra, computer vision, and numerical analysis; Foundations of Computational Mathematics; R. DeVore, A. Iserles and E. Suli, eds., London Math. Soc. Lecture Note Series, vol. 284, Cambridge University Press, Cambridge, 2001, pp. 267--297

Olver, P.J.; Geometric foundations of numerical algorithms and symmetry; Appl. Alg. Engin. Commun. Comput.; 11 (2001) 417--436