Logic and combinatorics and topology

逻辑、组合学和拓扑

• 批准号: 1700426
• 负责人: Slawomir Solecki
• 金额: $ 30.4万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2017-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700426&HistoricalAwards=false
• 关键词: Logic; combinatorics; topology

The project will develop new mathematical methods and establish new connections between diverse areas of mathematics. The project will focus on connections between Logic, on the one hand, and Topology and Combinatorics, on the other. It will aim at connecting Fraisse theory, an amalgamation theory from Logic, with deep questions on homogeneity of the generic continuum and with possible development of a homology theory. Further, it will aim at establishing connections between Ramsey theory, a branch of Combinatorics, with Topological Dynamics, Algebraic Topology, and certain orders playing an important role in parts of Set Theory.The project will develop a presentation of Ramsey theory in terms of algebraic topological notions - simplicialcomplexes and simplicial maps. This presentation should incorporate both finite and infinite Ramsey theory, and it should capture Ramsey theoretic statements associated with amenability of subgroups of the permutation group of the set of natural numbers. The project will also uncover implications of the dynamics of monoid actions to Ramsey theory. The project will also explore an approach to certain problems in topological dynamics and topology that uses purely combinatorial/model theoretic methods. Some important compact topological spaces are obtained as canonical quotients of generic inverse limits of families of finite structures - projective Fraisse limits. Topological homogeneity questions will be investigated using such presentations. Another aim will be to develop the right notion of the simplex and the boundary operation for homology theory of projective Fraisse limits. The test case here is the development of universal Menger compacta through projective Fraisse limits. Another goal of the project is to explore connections between a fixed point property of group actions, concentration of measure phenomenon, and geometry of submeasures.

该项目将开发新的数学方法，并在数学的不同领域之间建立新的联系。该项目将集中于逻辑，一方面以及另一方面的拓扑和组合学之间的联系。它的目的是将弗拉伊斯理论（Fraisse Theory）连接起来，这是逻辑中的融合理论，以及关于通用连续体的同质性以及同源理论的发展的深刻问题。此外，它将旨在建立拉姆西理论（Ramsey Theory），组合学的分支，拓扑动力学，代数拓扑结构和某些订单在集合理论的一部分中起着重要作用。该项目将根据代数拓扑概念 - 简单性复杂图和简单图。本演讲应同时结合有限的和无限的拉姆西理论，并应捕获与自然数集合集合组的亚组的不合适性相关的拉姆西理论陈述。该项目还将揭示Monoid Actions动力学对Ramsey理论的影响。该项目还将探讨使用纯粹使用组合/模型理论方法的拓扑动态和拓扑问题的某些问题的方法。获得了一些重要的紧凑型拓扑空间，作为有限结构家族的通用逆极限的典型商 - 投影性的Fraisse限制。拓扑均匀性问题将使用此类演示进行研究。另一个目的是开发单纯形的正确概念和投影式Fraisse限制的同源性的边界操作。这里的测试案例是通过投射的Fraisse限制开发通用Menger Compacta。该项目的另一个目标是探索群体动作的固定点特性，测量现象的浓度和层面的几何形状之间的联系。