Surface multi-space

Abstract

Multi-space for curves was defined by Peter Olver to provide geometric foundationsfor symmetry methods in the numerical analysis of ordinary differential equations. Here we define multi-space for surfaces, thereby extending the framework to partial differential equations in two independent variables. The main technical tool is the Hilbert scheme of points on a surface, which we adapt to the context of smooth manifolds. We begin the study of prolongations of group actions, their invariants, and the process of extending a differential equation to multi-space, whereby the resulting equation encompasses both the original differential equation and a family of difference equations approximating it.