Group actions in function approximation spaces

In 1987, Britain experienced a hurricane that was famously not predicted by the UK Met Office.The PI heard that mathematicians working at the Met Office decided, after due deliberation, that the reason was that the numerical code used to predict extreme weather had a particular subtle flaw: that a conservation law, known as potential vorticity, was not built into the computer code. This kind of conservation law is guaranteed by a famous theorem, proved by Emmy Noether in the early 1900's, which links the symmetries of the physical model directly to the conservation laws.The long term aim of the line of research of the project is to build such laws, exactly, intonumerical code that models physical systems. This involves first understanding how the physical symmetries, which are understood in terms of smooth actions on smooth spaces,are mapped into actions in discrete or digital spaces; actually, onto the approximate functions on such spaces used to model the smooth functions. The second part involves investigatinghow Noether's theorem transfers to the discrete case in a concrete and practical way.

Variational problems are endemic throughout physics and engineering. Principles of optimisation such as minimum time, minimum flow resistance and minimum power expenditure have been invoked through the history of science and common speech... The deterministic success of these principles has become so routine that success is taken for granted. (from A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, 2000.) Variational problems with a continuous symmetry group have guaranteed conservation laws. The PI has been concerned about automatically embedding conservation laws into numerical code ever since speaking to a Met Office mathematician. She was told that the reason the Met Office failed to predict the 1987 hurricane was decided, after due investigation, to be that the relevant numerical models did not preserve potential vorticity. This is a conservation law arising via Noether's Theorem from the particle relabelling symmetry; a rather subtle and seemingly non physical pseudogroup action. As computers become faster and more efficient, it becomes feasible to incorporate properties of the original physical models into their associated numerical models, and indeed it is philosophically attractive to do so. This is the premise of the `hot' field of geometric integration. The possible impact of this proposal is high because 1) the range of physical phenomena that are modelled by a variational principle and have a continuous symmetry group is broad, and 2) the approach is algorithmic, that is, is centered on general methods with proven properties of outputs. While incorporating potential vorticity exactly into numerical code is a ``holy grail , along the way there should be a wide range of applications whose impact is assured although the specifics can only be conjectured at present.