Effective Field Theories and PDE

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In many problems in theoretical physics it is possible to make progress by exploiting a hierarchy of scales between 'heavy' and 'light' excitations. When considering excitations of the light fields, there is a well-understood process to formally remove the heavy fields from the model, leaving field equations for the light excitations which are formally given by an infinite expansion - the Effective Field Theory expansion - whose terms typically have higher and higher derivatives. This has presented a puzzle, since naively truncating such a series gives rise to pathologies. Recently progress has been made in understanding such situations rigorously. This project will bring the techniques of modern PDE analysis to bear on understanding how Effective Field Theories can arise as limits of classical PDE. In particular this will involve a detailed examination of averaging techniques for nonlinear wave equations, with applications in General Relativity and Cosmology.This falls naturally within the remit of EPSRC's Mathematical Physics and Mathematical Analysis Research Areas.