Topological combinatorics and amenability

拓扑组合学和适应性

• 批准号: 283169120
• 负责人: Professor Dr. Friedrich Martin Schneider
• 金额: --
• 依托单位: Institut für Diskrete Mathematik und Algebra
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/283169120?language=en
• 关键词: Topological; combinatorics; amenability

Amenability ranges among the most powerful and well-recognized concepts in modern mathematics -- drawing exciting connections between group theory, functional analysis, probability theory, topological dynamics, ergodic theory, and combinatorics. Recent developments in Ramsey theory revealed the necessity of a better understanding of amenability for non-locally compact topological groups. The aim of the proposed project is to investigate amenability for topological groups and dynamical systems by means of asymptotic combinatorial properties of the underlying uniform spaces with regard to the respective group actions. In other words, I want to study topological groups and dynamical systems via approximating them by finite combinatorial objects, such as bipartite graphs and simplicial complexes. This approach shall be utilized to establish general topological versions of classical theorems for locally compact groups due to Følner, Tarski, Cohen, and Grigorchuk. From this, I expect both new insight into those classical results (for instance in terms of alternative proofs) and interesting applications to areas where local compactness is a rare property (for example operator theory or infinite combinatorics). Furthermore, I aim at finding a suitable generalization of the Ornstein-Weiss theorem for arbitrary amenable topological groups. This is supposed to serve as a basis for defining dynamical invariants (e.g. topological entropy) for continuous actions of amenable topological groups, which in turn would provide a first step towards a general ergodic theory for amenable topological groups. Since the Department of Mathematics at the University of Auckland encompasses outstanding expertise in general topology, functional analysis, group theory, and combinatorics, the intended research fellowship at Auckland would significantly advance the proposed project and lay the foundations for promising long-term collaborations concerning combinatorial methods in topological group theory.

顺从性是现代数学中最强大和最受认可的概念之一-在群论，泛函分析，概率论，拓扑动力学，遍历理论和组合学之间建立了令人兴奋的联系。Ramsey理论的最新发展揭示了更好地理解非局部紧拓扑群的顺从性的必要性。该项目的目的是研究拓扑群和动力系统的顺从性，通过相应的群作用的基本一致空间的渐近组合性质。换句话说，我想研究拓扑群和动力系统，通过用有限的组合对象来近似它们，比如二分图和单纯复形。这种方法将被用来建立一般拓扑版本的经典定理局部紧群由于福尔纳，塔斯基，科恩，和格里戈丘克。从这一点上，我希望新的洞察力，这些经典的结果（例如在替代证明方面）和有趣的应用领域，局部紧性是一个罕见的财产（例如运营商理论或无限组合）。此外，我的目标是找到一个合适的推广的Ornstein-Weiss定理任意顺从拓扑群。这被认为是为顺从拓扑群的连续作用定义动力学不变量（例如拓扑熵）的基础，这反过来又为顺从拓扑群的一般遍历理论迈出了第一步。由于数学系在奥克兰大学包括一般拓扑学，泛函分析，群论和组合学的杰出专业知识，在奥克兰的预期研究奖学金将显着推进拟议的项目，并奠定了基础，有前途的长期合作有关组合方法在拓扑群论。