Wikipedia entry on geometric integration.
Vladimir Arnold showed how the Euler equations of a spinning top and the Euler equations of a fluid were both the equations of geodesics on an appropriate group. Since then many other equations of mathematical physics — collectively known as (generalized) Euler equations — have been found to share this structure. We study the structure of the equations, their dynamics, and their applications.
images page). If the warps are chosen to be diffeomorphic then the problem reduces to one of solving the generalised Euler equations on the full diffeomorphism group. We have proposed rapid methods of numerically integrating these warps and demonstrated them on medical images. We have also considered the problem of computing statistics on the space of images.
Emmy Noether proved that a conservation law is associated with any smooth symmetry (like a rotation) of a differential equation. Our understanding of the geometry of this situation became much deeper in the 1970s and 80s with the development of the so-called Marsden-Weinstein reduction, which often allows one to solve differential equations or to analyze the stability of their solutions. Our work combines geometry and analysis to study generalizations of this theory, and to understand applications to fluid and other equations.