Wakimoto Modules, FMS Bosonization and Integrable Hierarchies

Partial differential equations (PDEs) are of central importance to many areas of mathematics, physics and other disciplines. Solving them is an extremely difficult and often wholly intractable task. In the past fifty years, the systematic algebraic study of a certain special class of equations called integrable PDEs has assumed a prominent position in mathematics and physics. More recently, the representation theory of infinite dimensional Lie algebras has been discovered to be intimately connected to so called soliton solutions of certain hierarchies of integrable PDEs.;In this thesis, we investigate at the algebraic level some of this representation theory in the form of vertex algebras and their structure theory. We prove certain uniqueness results and give a unified treatment of a number of previous results. We then use one of these results to produce a new hierarchy of integrable, non-autonomous PDEs. The appearance of non-autonomous PDEs appears to be a novel result. This is due to the fact that we use the boson-boson correspondence, where previously the boson-fermion correspondence was always employed.