Mathematical Sciences: Representation Theory of Infinite Dimensional Classical Groups

Boyer教授将研究与算子代数有关的经典群的表示理论。重点将放在紧群的归纳极限上，紧群是仅次于有限维群的下一阶复杂性。分析将使用近似有限维C*-代数的框架(特别是K-函子及其序结构)、动力系统及其遍历度量以及逼近群的组合学。一个特定的目标是对归纳极限酉群的表示进行分类，它推广了著名的正则(反)对易关系的准自由表示。这个项目是目前许多领域正在进行的一项努力的一部分，目的是将李群(以挪威著名数学家索菲斯·李的名字命名)研究中长期熟悉的概念和结果扩展到更具异国情调的物体。通常意义上的李群的一个例子是球体的所有旋转的群，其中群运算包括跟随一个运动接着另一个运动。旋转群的详细知识对分析球对称存在的任何数学或物理情况都非常有帮助。这个组的一个重要特征是，我们只需要有限数量的参数(在本例中是三个)来指定它的任何成员。同样，我们可以考虑更高维度的旋转群。只要维度仍然是有限的，就不会出现真正的新现象，但如果我们让它无限增加，也就是说取极限，我们就得到了一种无限维李群。对于这个归纳极限旋转群来说，许多相同的问题和普通的三维极限旋转群一样有意义。博耶教授将致力于研究在这种更困难的情况下回答这些问题所需的新方法。

Professor Boyer will investigate the representation theory of classical groups associated to operator algebras. The emphasis will be on inductive limits of compact groups, which are the next order of complexity after finite dimensional groups. The analysis to be undertaken will use the framework of approximately finite- dimensional C*-algebras (especially the K-functor and its order structure), dynamical systems and their ergodic measures, and the combinatorics of the approximating groups. One specific objective is to classify the representations of the inductive limit unitary group which generalize the well-known quasi-free representations of the canonical (anti-) commutation relations. This project is part of an effort now underway in many quarters to extend concepts and results long familiar in the study of Lie groups (named after the eminent Norwegian mathematician Sophus Lie) to objects of a much more exotic sort. An example of a Lie group in the usual sense is the group of all rotations of a sphere, where the group operation consists of following one motion by another. Detailed knowledge of the rotation group is extremely helpful in the analysis of any mathematical or physical situation where spherical symmetry is present. It is an important feature of this group that we need only a finite number of parameters (in this case three) to specify any of its members. Likewise, we can consider the rotation group in higher dimensions. No truly new phenomena occur as long as the dimension remains finite, but if we let it increase without bound and so to speak take the limit, we obtain a kind of infinite-dimensional Lie group. Many of the same questions make sense for this inductive limit rotation group as for the ordinary three-dimensional one. Professor Boyer will work on the new methods that are needed to answer them in this more difficult setting.