Overdetermined systems of partial differential equations may be processed using differential-elimination algorithms, and a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are effectively intractable by these methods due to the expression swell incurred in the intermediate stages of the calculations. This can happen when, for example, the input system depends on many variables and is invariant under a rotation group. Such a symmetry can mean that there is no natural choice of term ordering in the elimination and reduction processes.

In this talk, I describe how systems written in terms of the differential invariants of a Lie group action may be processed in a manner analogous to differential-elimination algorithms. We use the new, regularized moving frame method of Fels and Olver to write a differential system in terms of the invariants of a symmetry group. The methods described have been implemented as a package in Maple.

There are many theoretical challenges, and many potential applications, which I hope to describe.