Logic and combinatorics and topology

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。该项目将侧重于逻辑之间的连接，一方面，拓扑学和组合学，另一方面。它的目的是连接弗赖斯理论，从逻辑融合理论，与深层次的问题的同质性的通用连续统和可能的发展同源性理论。此外，它的目的是建立拉姆齐理论，一个组合学的分支，拓扑动力学，代数拓扑学和某些订单之间的联系发挥了重要作用的一部分集Theory.The项目将制定一个演示拉姆齐理论的代数拓扑概念-simplicialcomplex和simplicial maps。这种介绍应结合有限和无限拉姆齐理论，它应捕捉拉姆齐理论的陈述与顺从的子群的置换群的一套自然数。该项目还将揭示幺半群作用的动力学对拉姆齐理论的影响。该项目还将探索一种方法来解决拓扑动力学和拓扑学中的某些问题，使用纯粹的组合/模型理论方法。得到了一些重要的紧拓扑空间作为有限结构族的一般逆极限--射影Fraisse极限的标准等价式。拓扑同质性问题将使用这样的介绍进行调查。另一个目标将是发展正确的概念的单形和边界操作的同调理论的投影弗莱斯限制。这里的测试案例是通过射影弗莱斯极限发展普遍的门格尔极限。该项目的另一个目标是探索一个固定点属性的群体行动，测量现象的浓度和几何子措施之间的联系。

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.