Representations of Real Lie Groups, Symmetry Breaking, and Automorphic Forms

研究人员将继续对对称性及其应用进行长期研究。 对称性是一个数学概念，直接应用于其他科学，如化学，物理学，生物学和工程学。 代数的研究，其中这项研究的一部分，是在许多方面的对称性的研究。 本研究课题的目的是为了提高对重要的代数结构的理解，并将其应用于数论和理论物理学。本研究课题的研究内容是对称破缺，即从大的约化半单李群的表示得到小的李群的表示。 大群的表示通常是无限维的，而较小群的表示与大群的原始表示的子项有关。 表示的对称破缺还远未得到很好的理解，因此实例的构造和理解非常重要。这项研究可以应用于数论，也可能应用于理论物理。 这些构造中使用的许多技术都是分析性的，因此在分析中出现了一些新的、有趣的和具有挑战性的问题，需要解决。

The investigator will continue long-term research into the study of symmetries and some of their applications. Symmetry is a mathematical concept with direct applications in other sciences such as chemistry, physics, biology, and engineering. The study of algebra, of which this research is a part, is in many ways the study of symmetries. This project aims to advance understanding of important algebraic structures, with potential applications to number theory and theoretical physics.This research project is concerned with symmetry breaking, that is, obtaining representations of a smaller Lie group from the representations of a large reductive semi-simple Lie group. The representations of the large group are usually infinite dimensional, and the representations of smaller groups are related to quotients of the original representation of the large group. Symmetry breaking of representations is far from well understood, and so the constructions and the understanding of examples are very important. This research has applications to number theory and possibly to theoretical physics. Many of the techniques used in these constructions are analytic, and thus some new, interesting, and challenging problems in analysis arise and need to be solved.