Mathematical Sciences: A Research Group in Representation Theory

这一奖项授予利普斯曼、赫布、库德拉和罗森博格教授，是根据数学科学研究小组的倡议颁发的。它将在表征理论的一系列合作项目中支持博士后助理、访客和研究生。特定焦点是p-Addi群、对称空间和群C*-代数的表示。这项研究将涉及李群表示理论的各个方面，李群以挪威数学家索菲斯·李的名字命名。李群的一个基本例子是球体的旋转群，其中群运算包括跟随一个运动接着另一个运动。关于这个群的详细信息对解决存在球对称的数学或物理问题非常有帮助。其他运动组捕捉到了其他类型的对称性。李群的例子的一个更代数的(相对于几何的)来源来自于矩阵的乘法。所有给定大小的可逆实(或复)矩阵组成的群是李群，就像它的任何可以用自然方式描述的子群一样。希望能够在几何和代数观点之间来回移动，例如考虑可以将球体的旋转群实现为一组可逆矩阵的多种方式。粗略地说，这就是表象理论。关于给定群体的陈述的事实往往会非常经济地存储大量信息。根据所讨论的群体来自哪里，这些信息可能会影响到从数论到数学物理的几乎任何学科。

This award to Professors Lipsman, Herb, Kudla, and Rosenberg is made under the Mathematical Sciences Research Groups initiative. It will support postdoctoral associates, visitors, and graduate students in a series of collaborative projects in representation theory. Specific foci are representations of p- adic groups, symmetric spaces, and group C*-algebras. The research will deal with various aspects of the representation theory of Lie groups, which bear the name of the Norwegian mathematician Sophus Lie. One basic example of a Lie group is the group of rotations of a sphere, where the group operation consists of following one motion by another. Detailed information concerning this group is very helpful in solving mathematical or physical problems in which spherical symmetry is present. Other groups of motions capture other kinds of symmetry. A more algebraic (as opposed to geometric) source of examples of Lie groups comes from the multiplication of matrices. The group of all invertible real (or complex) matrices of a given size is a Lie group, as is just about any subgroup thereof that can be described in a natural manner. It is desirable to be able to go back and forth between the geometric and algebraic points of view, for instance to consider the numerous ways in which the rotation group of the sphere can be realized as a group of invertible matrices. This, roughly, is representation theory. Facts about the representations of a given group tend to store a lot of information very economically. Depending on where the group in question comes from, this information can impinge on almost any subject from number theory to mathematical physics.