GRK 1052: Representation Theory and its Applications in Mathematics and Physics

GRK 1052：表示论及其在数学和物理中的应用

• 批准号: 375986
• 金额: --
• 依托单位: Bergische Universität Wuppertal
• 依托单位国家: 德国
• 项目类别: Research Training Groups
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2008-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/375986?language=en
• 关键词: GRK; 1052; Representation; Theory; its

Representation theory is a branch of mathematics that studies objects by investigating their symmetries. These symmetries can be rotations, reflections, translations or abstract generalisations thereof. Such considerations can have far-reaching consequences. For instance, they impose constraints which drastically reduce the number of candidates for various fundamental physical theories. This makes representation theory one of the most important tools in modern theoretical physics. What is more, representation theory also plays an important role in concrete calculations of experimentally observable quantities. By developing beyond the elementary, intuitive concept of geometric symmetry, representation theory has shown itself to be useful in more and more branches of mathematics and theoretical physics.Representation theory provides methods for classifying the complex spectra of atoms and molecules. It is used to analyse systems in both, solid state physics and the physics of elementary particles. One significant modern application is the study of interacting many-body systems via their correlation functions. Each symmetry imposes a constraint on the evolution in time, and the presence of a sufficiently diverse collection of symmetries means that the behaviour of the system is more or less uniquely determined. The possibility of handling quantum mechanical many-body systems in this manner is particularly significant, for the direct numerical approach fails because of its (non-polynomial) complexity. Similar effects occur in applications within mathematics, one example being the interplay between geometry and representation theory. On the one hand one often uses symmetries of well-understood geometrical objects to construct representations, translating geometrical knowledge to knowledge about representation theory. On the other hand, if one understands a representation, then the constraints it imposes provide a lot of information about the geometric object under consideration.Summing up, the scientific aim of the Research Training Group is the development of representation-theoretic methods and their applications in mathematics and theoretical physics.

表象理论是数学的一个分支，通过研究对象的对称性来研究对象。这些对称可以是旋转、反射、平移或其抽象概括。这些考虑可能会产生深远的后果。例如，他们施加的限制极大地减少了各种基本物理理论的候选人数。这使得表象理论成为现代理论物理中最重要的工具之一。更重要的是，表象理论在实验可观测量的具体计算中也发挥着重要作用。随着几何对称性的基本、直观概念的发展，表象理论已经在越来越多的数学和理论物理分支中发挥作用。表象理论提供了对原子和分子的复杂光谱进行分类的方法。它被用来分析固体物理学和基本粒子物理学中的系统。一个重要的现代应用是通过它们的关联函数研究相互作用的多体系统。每一个对称性都对时间演化施加了限制，而对称性集合的足够多样化的存在意味着系统的行为或多或少是唯一确定的。以这种方式处理量子力学多体系统的可能性尤其重要，因为直接数值方法由于其(非多项式)复杂性而失败。类似的效应也出现在数学中的应用中，其中一个例子是几何和表示理论之间的相互作用。一方面，人们经常使用众所周知的几何对象的对称性来构造表示，将几何知识转化为关于表示理论的知识。另一方面，如果一个人理解一种表示，那么它施加的约束就提供了关于所考虑的几何对象的大量信息。总而言之，研究训练组的科学目标是发展表示论方法及其在数学和理论物理中的应用。