Group actions in function approximation spaces

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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.