## Geometric Numerical Integration

A branch of numerical analysis that emerged in the 1990s, geometric integrators are methods for simulating dynamical systems that preserve key properties of the system*exactly*; sometimes this leads to startling improvements in performance. The research combines a study of the geometric properties of the system with numerical analysis and simulation, and the methods are widely used in computational physics and chemistry. For example, modern geometric integrators were used in 2004 in a large simulation of the solar system that resulted in a major re-calibration of the geological epochs of the Earth. Wikipedia entry on geometric integration.

## Euler Equations

In 1966 the Russian mathematician 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.## Image Registration

Bringing images into alignment with each other, so that matching structures between them can be clearly identified, is known as image registration. It has applications in many areas, but the most common one is medical image analysis, where it can be used to assist in the diagnosis of degenerative diseases. Images from multiple patients — or the same patient over time — are brought into alignment be warping one of them so that its appearance more closely matches another (there are movies of this happening on the 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.## Symmetries and Reduction

Symmetry has been a driving and unifying force in physics for more than a century. In 1918 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.