Mathematical Sciences: Lie Groups

沃尔夫教授将继续他对李群、调和分析和表示论的研究，并将一些应用于复分析、黎曼几何和控制论。他将开发他在各种表示构造方法的等价性方面的最新结果的应用。他将研究从他与R.Herb推导的公式确定表示的性质的可能性。他将完成他在一般半单李群的Schwartz空间上的工作，并研究厄米特型群的离散级数表示。最后，Wolf教授将研究黎曼齐次空间上与不变微分算子相关的发展方程的能观性和能控性问题。群论基本上是对称论。举一个简单的例子，当所讨论的系统在空间原点的位置改变时是不变的，自然就产生了翻译群。虽然群是抽象的对象，但特殊情况需要对称群的具体实现或“表示”。

Professor Wolf will continue his research on Lie groups, harmonic analysis, and representation theory, with some applications to complex analysis, Riemannian geometry, and control theory. He will develop applications of his recent results on the equivalence of various methods of construction of representations. He will study the possibility of determining the character of a representation from a formula he has derived with R. Herb. He will complete his work on the Schwartz space of a general semisimple Lie group and study discrete series representations of groups of hermitian type. Finally, Professor Wolf will study the observability and controllability problems for evolution equations associated to invariant differential operators on Riemannian homogeneous spaces. Group theory is basically the theory of symmetry. To take a simple example, when the system in question is invariant under a change in the position of the origin of space, the group of translations naturally arises. While groups are abstract objects, particular situations demand concrete realizations or "representations" of the symmetry group.