Counting Problems and Semisimple Groups

一些计数问题允许额外的对称性，例如在半单李群作用下的不变性。这些问题有时可以通过涉及遍历理论的技术来解决。我们正在研究数论、几何学和动力学的例子。对称现象是某些计数问题的核心。通常，除非利用所有隐藏的对称性，否则这些问题无法解决。所提出的方法包括研究遍历理论的对称模式和混沌现象之间的相互作用。

Some counting problems admit extra symmetries, such as invariance under actions of a semisimple Lie group. These problems can sometimes be solved by techniques involving ergodic theory. We are working on examples from number theory, geometry and dynamics. The phenomenon of symmetry is central to certain counting problems. Typically these problems cannot be solved unless all of the hidden symmetry is exploited. The proposed methods consist of studying the interplay between the symmetric patterns and the chaotic phemomena of ergodic theory.