Rigidity and Unitary Representations

AbstractYehuda Shalom该项目关注的新方向，其中酉表示可用于研究半单群及其离散子群的几何，动力学和遍历理论。 刚性现象的框架内的表示论将提出，这有直接的影响刚性的行动和同态格，无论是在秩一和更高的秩简单李群。 新的工具，在刚性理论的准等距顺从组也将被介绍和调查，使用这种方法。对称性的概念是根本的重要性，在数学分析，并根据许多自然的秘密。在研究和使用这个概念时，李群理论以及它们作为几何对象（如球）的对称性出现和运行的方式是一个中心主题。 独立地，希尔伯特空间的数学理论--高度对称的无限维类似于球（例如，由数列组成）--在不同的科学领域中找到了显着的应用，例如设计有效的网络，量子力学和基本粒子物理学。虽然上述两个主题之间的联系是已知的，并加以利用，但该项目的目的是开辟它们之间关系的全新方向。在此过程中，将使用这两种理论的技术，并引入新的工具，这将使人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.