Mathematical Sciences: Local Harmonic Analysis With Applications to Automorphic Representations

谢尔斯塔德教授将继续研究 在内窥镜检查和扭转内窥镜检查中，沿着 局部调和分析和分布的应用 与亚瑟-塞尔伯格迹公式的比较有关。 这与李群的表示论有关。 这些基本的数学对象自然出现在几个 的方式李群的一个基本例子是旋转群 一个球体，其中的组操作包括以下一个 另一种运动。关于这个群体的详细信息是 对解决数学或物理问题很有帮助， 存在球面对称性。其他类别的动议 捕捉到其他类型的对称性。一个更代数的（相对于 李群的例子的来源来自于 矩阵的乘法所有可逆的真实的（或 一个给定大小的（复数）矩阵是一个李群，因为 关于其任何亚组，可以在自然界中描述， 方式希望能够在 几何和代数的观点，例如， 考虑到许多方式，其中旋转组的 球可以被实现为一组可逆矩阵。这个， 粗略地说，就是表征理论。Facts about the 一个给定群体的表征往往会存储大量的 信息经济。这些信息的转移 （在两种不同的跟踪方法之间， 一个单一的群体，或两个不同的，但 数学相关的群体）是 谢尔斯塔德教授的研究项目

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.