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，特别强调重要的算术群类。大部分研究的目的是通过一个具体的一般猜想，将算术群理论中的不同结果锐化、重塑和统一到一个新的背景中，特别是作为特例，特别强调已有30年历史的Marguis-Zimmer猜想(关于公度子群的结构)。这项拟议工作中的一个关键概念是和谐性，它是与这一猜想相关的核心成分，也是与群上的Lipschitz调和函数有关的不同建议的刚性方向(其动机来自于它们最近在PI与Terry Tao的联合工作中的使用)。所谓的算术群理论是现代数学中研究最深入、最美丽、最有成果和长期研究的理论之一。它涉及纯数学的大多数领域，从表示、数字和K理论的代数方面，到各种形式的几何，再到遍历理论和动力学的分析方面。拟议的工作旨在解决一些基本问题，并建议重塑该理论的部分内容，从而为该领域带来新的曙光和促进新的发展。特别强调的是格雷戈里·马古利斯和罗伯特·齐默提出的一个30年前的猜想。预计它的解决将涉及新的动力学概念，这些概念与和谐性(粗略地说，在平均下是稳定的状态)密切相关。和谐性的新研究和应用有望在拟议工作的其他方向带来进展。

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.