Algebraic, Geometric, and Asymptotic Properties of Branch Groups

DMS-0308985 Grigorchuk，Rostislav摘要题目：分支群的代数、几何和渐近性质本项目将集中在分支群的性质上。分支群是抽象的数学对象，在解决数学不同部分交叉的主要问题中起着重要作用，如代数、数论、分析、几何和概率论。提议者将研究这些群体及其与现代数学中一些重要和长期存在的问题的关系，例如群论中的伯恩赛德和米尔诺问题，拓扑学中的阿蒂亚猜想，泛函分析中的凯迪森-卡普兰斯基猜想。该项目的研究成果在控制论、计算机科学、密码学和算法理论等领域有着重要的应用。组合群论是现代数学中发展最快的部分之一。从工程庞加莱和克莱因这一领域收到的贡献，一些最突出的数学家的二十世纪，如冯诺依曼，米尔诺和其他人。近几十年来，组合群论由于与统计物理学、量子混沌理论和计算机科学的新发现的联系而成为特别深入研究的对象。数学研究中出现了一些新的有前途的方向，其中之一就是分支群理论。这个理论与现代数学中的几个主流开放问题有很强的联系。近年来取得的巨大进展表明，对这些联系的探索导致了最富有成效的想法和成果。

DMS-0308985Grigorchuk, RostislavAbstractTitle: Algebraic, Geometric, and Asymptotic Properties of Branch GroupsThis project will focus on properties of branch groups.Branch groups are abstract mathematical objects that play an important role in solving major problems on the intersection of different parts of mathematics, such as Algebra, Number Theory, Analysis, Geometry and Probability Theory. The proposer will study these groups and their relation to a number of important and long-standing problems in modern mathematics, such as the Burnside and Milnor Problems in Group Theory, Atiyah Conjecture in Topology, Kadison-Kaplansky Conjecture in Functional Analysis. The results obtained in this project have useful applications in Cybernetics and Computer Science, as well as Cryptography and Theory of Algorithms.Combinatorial Group Theory is one of the most rapidly developing parts of modern mathematics. Starting with works by Poincare and Klein this area received contribution from some of the most prominent mathematicians of the twentieth century, such as von Neumann, Milnor and others. In recent decades Combinatorial Group Theory became an object of especially intensive research due to newly found connections with Statistical Physics, Quantum Chaos Theory and Computer Science. New promising directions of mathematical research have been developped, one of them being the theory of branch groups. This theory has strong connections to several mainstream open problems in modern mathematics. Tremendous progress acheived in recent years indicates that the exploration of these connections leads to most fruitful ideas and results.