Algebraic structures connecting Lie theory and many body Physics

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Physical Science can be thought of an attempt to understand, and make use of, the physical world. Part of the process of understanding any system is to make models of it. Such models can range from scaled-down versions, to mathematical simulations. This research proposal can be considered, from one motivating perspective, to be concerned with the mathematics needed in making useful mathematical models. (It is concerned, in particular, with the Algebra needed.)Modelling as an approach to understanding is not limited to physics. One can attempt to investigate abstract structures (such as groups of symmetries) by modeling. In this setting the process is called Representation Theory.We often find ourselves in the situation that a mathematical model of a physical system is still rather complicated, and itself benefits from modelling. Thus one aspect of this research proposal is concerned with the representation theory of structures which in turn model phenomena in physics (typically in Statistical Mechanics).A wonderful feature of this modelling chain is that the non-abstract end system is amenable to other approaches, such as experiment and physical intuition. Thus not only does the representation theory inform the physics, but also the physics informs the representation theory. By pursuing this line, many exciting new pieces of mathematics have been discovered. And many more remain to be discovered.Another great feature of this area of research is that mathematical models can be pushed into new generalisations and variations not obviously or intuitively indicated by (or easily available in) the physical context. This nicely underwrites the risks of research in an area where not every line of investigation is obvious or intuitive, even among those that do turn out to be physically important. In short, the methodology gives a way of sourcing radical new ideas in physical science.(In this specific proposal, however, the emphasis is primarily on the intrinsic interest of the mathematics, and on the contribution of physics `feeding back' into the study of challenging open problems in representation theory itself; in tandem with several proven technical devices from abstract algebra itself.)In summary then, this research proposal is concerned with the representation theory of structures (such as groups and algebras) used, or potentially used, in physical modelling. The strategy is to use a tight interplay between the (very distinctive) mathematical and physical contexts.