Logic and combinatorics and topology

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。该项目将侧重于逻辑与拓扑学和组合学之间的联系。它的目的是将来自逻辑的融合理论Frisse理论与关于类属连续体的同质性的深层次问题以及同调理论的可能发展联系起来。此外，它将致力于在组合学的分支Ramsey理论与拓扑动力学、代数拓扑学和在集合论的某些部分中扮演重要角色的某些阶之间建立联系。该项目将以代数拓扑概念-单纯复形和单纯映射的形式来表示Ramsey理论。这个演示文稿应该同时包含有限和无限的Ramsey理论，并且它应该捕捉到与自然数集合的置换群的子群的可修饰性相关的Ramsey理论陈述。该项目还将揭示么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使用纯粹的组合/模型理论方法来解决拓扑动力学和拓扑学中某些问题的方法。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Frisse极限的典范商.将使用这样的演示来研究拓扑同质性问题。另一个目的是发展射影Frisse极限的同调理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射影Frisse极限发展万能Menger紧致。该项目的另一个目标是探索群体作用的不动点性质、测量现象的集中度和子测量的几何之间的联系。

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.