FRG: Collaborative Research: Atlas of Lie Groups and Representations: Unitary Representations

这个项目有两个主要目标。首先是解决幺正对偶问题：描述实约李群的不可约幺正表示。主要工具是计算任意给定群的酉对偶的算法，我们在“atlas”软件中实现了这个算法。我们计划用这些信息来证明关于酉对偶的结果，从亚瑟的无幂表示的酉性开始。第二个主要目标是通过软件、网站、公共研讨会和其他方式，使非专业人员能够获得关于真实群体的表示理论的信息。地图集软件在地图集网站上免费提供，并将继续在那里无限期地维护。利用对称性来研究数学和科学问题的想法可以追溯到近200年前傅立叶对热的研究。在赫尔曼·魏尔、尤金·维格纳和安德烈·魏尔的研究中，对称性在量子力学和数论中发挥了核心作用。李群是以挪威数学家索菲斯·李命名的，它是对称背后的数学对象。表征理论研究给定对称性或李群的所有表现方式。理解所有“酉”表示（其中对称操作保持长度）的问题是该主题中最重要的未解决问题之一，并且在许多领域具有潜在的应用；例如，它是这个问题的抽象版本，“什么样的量子力学系统可以承认某种对称性？”

This project has two primary goals. The first is to solve the problem of the unitary dual: to describe the irreducible unitary representations of real reductive Lie groups. The primary tool is an algorithm to compute the unitary dual of any given group, which we are implementing inside the "atlas" software. We plan to use this information to prove results about the unitary dual, beginning with the unitarity of Arthur's unipotent representations. The second primary goal is to make information about representation theory of real groups accessible to non-specialists, via the software, a web site, public workshops, and other means. The atlas software is freely available on the atlas web site, and will continue to be maintained there indefinitely.The idea of using symmetry to study problems in mathematics and science dates back to Fourier's work on heat nearly two hundred years ago. In the hands of Hermann Weyl, Eugene Wigner, and Andre Weil, symmetry has come to play a central role in quantum mechanics and in number theory. Lie groups, named after the Norwegian mathematician Sophus Lie, are the mathematical objects underlying symmetry. Representation theory studies all of the ways a given symmetry, or Lie group, can manifest itself. The problem of understanding all "unitary" representations (in which the symmetry operations preserve lengths) is one of the most important unsolved problems in the subject, and has potential applications in many areas; for example, it is an abstract version of the question, "what quantum mechanical systems can admit a certain kind of symmetry?"