Benson, Joseph Jonathan2015-02-242015-02-242014-10https://hdl.handle.net/11299/170102University of Minnesota Ph.D. dissertation. October 2014. Major: Mathematics. Advisor: Peter J.Olver. 1 computer file (PDF); vi, 74 pagesEvolution equations with infinite hierarchies of symmetries have been shown to nat- urally arise within the context of geometric, arc-length preserving flows of curves in the plane and in three dimensions. In the following work, a systematic investigation into this phenomenon is conducted for the case of group actions on planar curves. Equivariant moving frames and the variational bicomplex are used. A catalog of results is produced, connecting many invariant curve flows with integrable equations such as Burgers', KdV, mKdV, and Sawada-Kotera. In the last chapter, the techniques are extended to an investigation of the evolution of curvature of 2-dimensional surfaces in 4-dimensional Euclidean space under the Skew-Mean-Curvature flow.enIntegrableMoving framesVariational bicomplexMathematicsThesis or Dissertation