Rigidity and Unitary Representations

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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.