Manchester Geometry Seminar 2014/2015

Consider an algebraic pseudogroup transitively acting on a smooth manifold (more generally on a geometric structure, more generally on an algebraic differential equation). This action naturally extends to the space of infinite jets (partial case: actions by Lie groups). By differential invariant we will understand a rational function on this space, which is invariant with respect to the prolonged action. Alternatively this is a non-linear scalar differential operator (defined globally as a rational function). The main theorem states that the algebra of all scalar differential invariants is generated by a finite number of differential invariants and invariant derivatives. This is the base for solution of the equivalence problem. A number of examples from geometry, algebra and physics will be presented. Some applications to local moduli count and integrability of differential equations will be discussed.