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不同的数学领域之间建立新的联系。该项目将一方面关注逻辑与拓扑和组合学之间的联系。它将旨在将弗赖斯理论（一种来自逻辑学的融合理论）与通用连续体同质性的深刻问题以及同源理论的可能发展联系起来。此外，它将旨在建立拉姆齐理论（组合学的一个分支）与拓扑动力学、代数拓扑以及在集合论的某些部分中发挥重要作用的某些阶之间的联系。该项目将用代数拓扑概念（单纯复形和单纯映射）来表达拉姆齐理论。该演示文稿应包含有限和无限拉姆齐理论，并且应捕获与自然数集的置换群的子群的顺应性相关的拉姆齐理论陈述。该项目还将揭示幺半群行为的动力学对拉姆齐理论的影响。该项目还将探索一种使用纯组合/模型理论方法来解决拓扑动力学和拓扑中某些问题的方法。一些重要的紧拓扑空间是作为有限结构族的一般逆极限（射影弗赖斯极限）的正则商而获得的。将使用此类演示来研究拓扑同质性问题。另一个目标是发展射影弗赖斯极限同调论的单纯形和边界运算的正确概念。这里的测试用例是通过投影 Fraisse 极限开发通用门格尔紧致体。该项目的另一个目标是探索群体行为的定点属性、测量现象的集中度和子测量的几何形状之间的联系。