Rigidity and Unitary Representations

这个项目涉及到利用么正表示研究半单群及其离散子群的几何、动力学和遍历理论的新方向。给出了表示论框架中的刚性现象，这些现象对一阶和高阶单李群中作用的刚性和格的同态都有直接的影响。利用这种方法，我们还将介绍和研究柔群的拟等距刚性理论中的新工具。对称性的概念在数学分析中具有基本的重要性，它是许多自然秘密的基础。在学习和使用这一概念时，李群的理论以及它们作为几何对象(如球)的对称性产生和运行的方式是一个中心主题。独立地说，希尔伯特空间的数学理论--高度对称的无限维类似于球(比方说，由数字序列组成)--在不同的科学领域得到了显著的应用，例如设计高效的网络、量子力学和基本粒子物理。虽然上述两个主题之间的联系是已知的，并得到了利用，但该项目的目的是开辟它们相关的全新方向。在这样做的过程中，将使用这两种理论的技术，并将引入新的工具，这将使我们能够更好地理解这两个重要的主题。

AbstractYehuda ShalomThe project is concerned with new directions in which unitary representations may be used to study the geometry, dynamics and ergodic theory of semisimple groups and their discrete subgroups. Rigidity phenomena in the framework of representation theory will be presented, which have direct implications to rigidity of actions and homomorphisms of lattices, both in rank one and in higher rank simple Lie groups. New tools in the rigidity theory of quasi-isometries of amenable groups will be introduced and investigated as well, using this approach.The concept of symmetry is of fundamental importance in mathematical analysis, and underlies many of nature's secrets. In studying and using this concept, the theory of Lie groups and the way they arise and operate as symmetries of geometric objects (such as a ball), is a central theme. Independently, the mathematical theory of Hilbert spaces -highly symmetric infinite dimensional analogous of balls (consisting of, say, sequences of numbers)- has found remarkable applications in diverse scientific areas, such as designing efficient networks, quantum mechanics and elementary particle physics. While connections between the above two themes are known, and exploited, the aim of this project is to open completely new directions in which they relate. In doing so, techniques from both theories will be used, and new tools will be introduced, which will enable a better understanding of these two important subjects.