Locally symmetric spaces and transfer operators

在过去的几十年里，人们使用越来越多的方法来研究黎曼流形上的数学量子混沌，这些方法侧重于流形的动力学而不是它们的(静态)几何。在这些动力学方法中，传递算子技术的潜力还远未被完全理解。PI极大地推动了黎曼双曲曲面传递算子技术的发展。对于广泛的非紧致曲面，它们现在提供了Maass尖点形式的纯动力学表征，从而显示了这些Laplace本征函数与周期测地线之间比任何其他方法所能建立的更深层次的关系。这个项目的主要主题是利用到目前为止所获得的见解来进一步扩展双曲曲面的研究(特别是共鸣、非酉挠等)，并发展高维双曲空间以及一些高阶黎曼局部对称空间的转移算子方法。

Over the last few decades, mathematical quantum chaos on Riemannian manifolds has been studied using an ever increasing number of methods which focus on the dynamics of the manifolds rather than on their (static) geometry. Among these dynamical methods are transfer operator techniques whose potential is still far from being fully understood.The PI has advanced significantly the development of transfer operator techniques for Riemannian hyperbolic surfaces. For a wide range of noncompact surfaces they now provide, e.g., a purely dynamical characterization of Maass cusp forms, and hence show a deeper relation between these Laplace eigenfunctions and periodic geodesics than could be established by any other method. The main theme of this project is to use the insights gained to far to further extend the investigations for hyperbolic surfaces (in particular, to resonances, non-unitary twists, etc.), and to develop transfer operator methods for higher-dimensional hyperbolic spaces and also for some higher rank Riemannian locally symmetric spaces.