Prof. Peter Olver
According to Felix Klein, geometry is founded on group actions. Our focus is on the induced geometry of submanifolds, particularly equivalence, symmetry and invariants. In many classical geometries (Euclidean, affine, projective, etc.) the method of moving frames has long been recognized as an extremely powerful tool for resolving such issues. However, its extension to non-traditional geometries and group actions was hampered by misconceptions as to what a moving frame really is. Only a careful reading of the classic works of Elie Cartan reveals the correct framework for the method.
In this talk, I will describe the recent equivariant approach that enables one to construct moving frames for arbitrary transformation groups. Then I will survey a variety of new applications of moving frames, which include computer vision, numerical analysis, differential invariants, classical invariant theory, partial differential equations, and the calculus of variations.
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