HARMONICITY AND RIGIDITY OF DISCRETE AND ARITHMETIC GROUPS

拟议活动的主要目的是加强离散群刚性理论的研究，特别强调算术群这一重要类别。该研究的实质部分旨在通过具体的一般猜想，将算术群理论中的各种结果锐化、重铸和统一到一个新的环境中，特别强调30年前的马古利斯-齐默猜想（关于代数子群的结构）作为特例。一个关键的概念，在拟议的工作是调和性，这是出现的核心成分有关thisconjecture，以及在不同的建议方向的刚性有关的Lipschitz调和函数的群体（动机是他们在PI与Terry Tao最近的联合工作中的使用）。所谓的算术群理论是最深刻的，最美丽的，富有成果的，和长期研究的现代数学。它涉及纯数学的大多数领域，从代数方面的代表性，数量和K理论，通过几何的各种形式，分析方面的遍历理论和动力学。拟议的工作旨在解决一些基本问题，并建议对理论的部分内容进行重铸，从而为该领域带来新的启发并促进新的发展。特别强调的是一个30岁的猜想，由于格雷戈里马古利斯和罗伯特齐默。它的解决方案预计将涉及新的动力学思想，这是密切相关的调和性（粗略地说，一个国家是稳定的平均）。新的调查和应用的谐波预计将带来进展的其他方向的拟议工作。

The principal aim of the proposed activity is to enhance research in the rigidity theory of discrete groups, with particular emphasis on the important class of arithmetic groups. Substantial parts of the research aim to sharpen, recast and unify diverse results in the theory of arithmetic groups into one novel setting, through a concrete general conjecture, with particular emphasis on the 30-year old Margulis-Zimmer conjecture (concerning the structure of commensurated subgroups), as a special case. A key notion in the proposed work is that of harmonicity, which emerges as central ingredient in relation to thisconjecture, as well as in a different suggested direction of rigidity pertaining to Lipschitz harmonic functions on groups (motivated by their use in recent joint work of the PI with Terry Tao).The theory of so-called arithmetic groups is one of the deepest, most beautiful, fruitful, and long studied ones in modern mathematics. It involves most fields of pure mathematics, from the algebraic side of representation, number, and K-theory, through geometry in its various forms, to the analytic side of ergodic theory and dynamics. The proposed work aims to address some fundamental issues and suggest a recast of parts of the theory, thereby shedding new light and enhancing new developments in the field. Of particular emphasis is a 30-year old conjecture due to Gregory Margulis and Robert Zimmer. Its resolution is expected to involve new dynamical ideas, which are strongly related to harmonicity (roughly speaking, a state which is stable under averaging). Novel investigations and applications of harmonicity are expected to bring progress in other directions of the proposed work.