Mathematical Sciences: Ergodic Theory, Groups and Geometry

这个雄心勃勃的数学研究项目与李群及其某些离散子群在流形上的作用有关。这种情况的一个非常简单的例子是三维空间围绕通过原点的轴的一组旋转，作用于以原点为中心的球体。在更高的维度中，这些例子变得更加奇特，更重要的是，正如齐默教授所发现的那样，游戏规则规定了什么样的流形会以惊人而根本的方式发生变化，会发生什么样的行动。一个全新的学科，可以被称为非线性表示理论，正在成长为取代它的位置，旁边是更古老的，更好地理解向量空间上的群表示理论。这项研究直接涉及数学的四个核心领域：分析、几何、代数和拓扑学。更具体地说，齐默教授将通过描述流形的基本群的表示理论的特征来研究允许半单李群或离散子群的作用的流形的拓扑。这些团体的行动的算术性质将被调查。将继续研究上面提到的现象，这些现象使较高维度的情况与较低维度的情况在性质上不同。

This ambitious mathematical research project has to do with actions of Lie groups, and certain of their discrete subgroups, on manifolds. A very simple example of this situation is the group of rotations of three-dimensional space about axes passing through the origin, acting on a sphere centered at the origin. In higher dimensions, the examples become more exotic, and more importantly, as Professor Zimmer has discovered, the rules of the game that say what sorts of actions can occur on what sorts of manifolds change in striking and fundamental ways. A whole new subject, which might be called nonlinear representation theory, is growing up to take its place next to the older and much better understood theory of group representations on vector spaces. This research impinges directly on four core areas of mathematics: analysis, geometry, algebra, and topology. More specifically, Professor Zimmer will investigate the topology of manifolds admitting an action of a semisimple Lie group or discrete subgroup by describing features of the representation theory of the manifold's fundamental group. The arithmetic nature of actions of of these groups will be investigated. There will be continued study of the phenomena alluded to above that make the higher-dimensional case qualitatively different from the lower-dimensional.