Mathematical Sciences: A Research Group in Representation Theory

这个奖项是在数学科学研究小组的倡议下颁发给利普斯曼、赫伯、库德拉和罗森伯格教授的。它将支持博士后、访问学者和研究生进行一系列表征理论的合作项目。具体的焦点是p进群、对称空间和群C*-代数的表示。该研究将涉及李群表示理论的各个方面，该理论以挪威数学家Sophus Lie的名字命名。李群的一个基本例子是球体的旋转群，其中群运算包括一个运动跟随另一个运动。关于这一群体的详细信息对于解决球面对称存在的数学或物理问题非常有帮助。其他组的运动捕捉到其他种类的对称性。李群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。给定大小的所有可逆实（或复）矩阵的群是李群，就像它的任何子群一样，可以用自然的方式来描述。我们希望能够在几何和代数的观点之间来回转换，例如考虑球体的旋转群可以被实现为一组可逆矩阵的许多方法。粗略地说，这就是表征理论。关于特定群体表征的事实倾向于非常经济地存储大量信息。根据所讨论的群体来自哪里，这些信息几乎可以影响从数论到数学物理的任何学科。

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.