Mathematical Sciences: Local Harmonic Analysis With Applications to Automorphic Representations

Shelstad教授将继续研究内窥镜和扭曲内窥镜中轨道积分的转移，以及在局部谐波分析中的应用，以及与Arthur - Selberg迹公式比较相关的分布。这与李群的表示理论有关。这些基本的数学对象以几种方式自然产生。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。这种信息的传递（在两种不同的跟踪单个组表示的方法之间，或在两个不同但数学上相关的组之间）是Shelstad教授研究项目的潜在主题。

Professor Shelstad will continue her study of the transfer of orbital integrals in endoscopy and twisted endoscopy, along with applications in local harmonic analysis and to distributions associated with comparisons of Arthur - Selberg trace formulas. This has to do with the representation theory of Lie groups. These fundamental mathematical objects arise naturally in several ways. 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. Transfer of this information (between two different ways of keeping track of the representations of a single group, or between two different but mathematically related groups) is the underlying theme of Professor Shelstad's research project.