Algebraic structures connecting Lie theory and many body Physics

物理科学可以被认为是理解和利用物理世界的一种尝试。理解任何系统的过程的一部分就是为它建立模型，这些模型可以是缩小的版本，也可以是数学模拟。从一个激励的角度来看，这个研究建议可以被认为是与制作有用的数学模型所需的数学有关。(It特别是与代数有关）。建模作为一种理解方法并不局限于物理学。人们可以尝试通过建模来研究抽象结构（如对称群）。在这种情况下，这个过程被称为表示论。我们经常发现自己处于这样一种情况，即物理系统的数学模型仍然相当复杂，而且它本身也受益于建模。因此，这个研究计划的一个方面是与结构的表示理论有关，而结构的表示理论又反过来对物理学中的现象进行建模（通常在统计力学中）。这个建模链的一个奇妙特征是，非抽象的最终系统可以服从其他方法，例如实验和物理直觉。因此，不仅表示论告知物理学，而且物理学也告知表示论。通过沿着这条线走下去，人们发现了许多令人兴奋的数学新成果。还有更多尚待发现。该研究领域的另一个重要特征是，数学模型可以被推入新的概括和变化中，而这些概括和变化并不明显或直观地由物理环境指示（或在物理环境中容易获得）。这很好地保证了研究的风险，在这个领域，并非每一条研究路线都是显而易见或直观的，即使在那些确实被证明是物理上重要的领域。简而言之，方法论提供了一种在物理科学中寻找激进新思想的方法。(In然而，这个具体的建议，重点主要是数学的内在兴趣，以及物理学对表征理论本身中具有挑战性的开放问题的研究的贡献;与抽象代数本身的几个已证明的技术设备相结合。总之，这项研究计划是关于结构（如群和代数）的表示理论，或可能用于物理建模。策略是使用（非常独特的）数学和物理环境之间的紧密相互作用。

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.