Arnaldsson, Orn2017-10-092017-10-092017-06https://hdl.handle.net/11299/190473University of Minnesota Ph.D. dissertation.June 2017. Major: Mathematics. Advisor: Peter Olver. 1 computer file (PDF); iii, 164 pages.This thesis combines two major equivalence methods, Cartan's equivalence method and the method of the equivariant moving frame for pseudo-groups, to obtain a more efficient tool with which to tackle a large class of equivalence and symmetry problems. These include, for example, all equivalence problems for sections of tensor bundles under change of variable and those arising in the calculus of variations. Furthermore, once the connection between the two original equivalence methods has become clear, we provide a proof of termination of Cartan's method in these cases. To obtain this termination result, we develop a novel algorithm for the computation of Pommaret bases for polynomial modules.enCartan-Kuranishi completionCartan's equivalence methodDifferential geometryMoving framesPommaret basesThesis or Dissertation