Mathematical Sciences: Representation Theory of Infinite Dimensional Classical Groups

博耶教授将研究表征理论 与算子代数相关的经典群。重点 将是对紧凑的群体，这是下一个归纳极限 有限维群后的复杂性阶分析 将使用近似有限的框架- 维C*-代数（特别是K-函子及其阶 结构），动力系统及其遍历测度，以及 近似群的组合学一个具体目标 是把归纳极限酉的表示 群，它推广了著名的拟自由表示 正则（反）对易关系。 该项目是目前正在进行的许多努力的一部分， 四分之一，以扩展概念和结果长期熟悉的 李群研究（以挪威著名的 数学家Sophus Lie）到更奇异的物体。 通常意义上的李群的一个例子是 球体的旋转，其中组运算包括 一个动作接着另一个动作的详细知识 旋转群在分析任何 球对称的数学或物理情形 礼物我们需要的是这个群体的一个重要特征 只有有限数量的参数（在这种情况下是三个）， 指定其任何成员。同样，我们可以考虑 在更高的维度中旋转组。没有真正的新现象发生 只要维度是有限的，但如果我们让它 无限制地增加，可以说取极限，我们得到 一种无限维李群许多相同的 对于这个归纳极限旋转组，问题是有意义的， 对于普通的三维模型。博耶教授将在 在这个问题上需要新的方法来回答他们， 困难的设置。

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.