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。该项目将侧重于一方面是逻辑学，另一方面是拓扑学和组合学之间的联系。本文的目的是将逻辑学中的融合理论弗拉斯理论与一般连续统的同质性问题以及同质论的可能发展联系起来。此外，它将旨在建立拉姆齐理论之间的联系，拉姆齐理论是组合学的一个分支，具有拓扑动力学，代数拓扑，以及在集合论中扮演重要角色的某些顺序。该项目将从代数拓扑概念——简单复合体和简单映射——的角度阐述拉姆齐理论。本文应结合有限和无限的拉姆齐理论，并抓住与自然数集合的置换群的子群的易受性有关的拉姆齐理论命题。该项目还将揭示单线动作对拉姆齐理论的影响。该项目还将探索使用纯组合/模型理论方法解决拓扑动力学和拓扑中的某些问题的方法。作为有限结构族的一般逆极限-射影frisse极限的正则商，得到了一些重要的紧拓扑空间。拓扑同质性问题将使用这样的报告进行调查。另一个目的是为射影frisse极限的同调理论发展单纯形的正确概念和边界运算。这里的测试案例是通过射影弗拉斯极限的通用门格尔压缩的发展。该项目的另一个目标是探索群体行为的不动点性质、测度现象的集中和子测度几何之间的联系。

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.