Combinatorial set theory and measurable combinatorics

组合集合论和可测组合学

• 批准号: RGPIN-2021-03549
• 负责人: Unger, Spencer
• 金额: $ 1.68万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758296
• 关键词: Combinatorial; set; theory; measurable; combinatorics

This research program is in two areas of pure mathematics. The first is combinatorial set theory with a focus on forcing and large cardinals. The second is measurable combinatorics. We describe each of these areas in turn. The modern study of set theory began with Godel's development of the constructible universe L and Cohen's invention of the method of forcing. Taken together these techniques give a proof that the continuum hypothesis is independent of the axioms of ZFC. The continuum hypothesis is the assertion that the collection of all subsets of the natural numbers, its powerset, has the smallest cardinality possible. Godel's construction of L is beginning of the modern study of inner model theory. Cohen's development of forcing is now the most used technique for producing independence results. The proposed projects in set theory can be divided by the themes that they address. (1) Questions about the cardinality of the powerset of singular cardinals. This is the modern instance of the study of the continuum hypothesis. (2) Questions about compactness principles. A compactness principle is the assertion that given a structure if all smaller cardinality substructures have some property, then the whole structure has the same property. (3) Questions about how the notions of cardinality differ between V and the class of hereditarily ordinal sets, HOD. This is an aspect of inner model theory. Measurable graph combinatorics has seen a recent surge in interest from applications to old questions about geometric paradoxes. For instance, Tarski's circle squaring problem: Given a disk and a square in the plane with the same area, is it possible to partition the disk in to finitely many pieces which can be moved by isometries to partition the square? This was solved positively by Laczkovich in 1990 using the axiom of choice. It was asked by Wagon if the same is possible with Borel pieces. A recent theorem of Grabowski, Mathe and Pikhurko showed that this is possible with either Lebesgue measurable or Baire measurable pieces. Soon after this result, we proved a Borel version in joint work with Andrew Marks. This result depends on an analysis of certain locally finite Borel graphs on R^2 generated by translations. In particular, the Borel circle squaring theorem is equivalent to the existence Borel perfect matching in one of these graphs. The question of when Borel graphs have Borel perfect matchings is an example of a question from measurable combinatorics. Answers to questions in measurable combinatorics are often quite different from their classical counterparts, requiring new techniques. The proposed research in this area contains both general questions from measurable combinatorics and questions which are applications like the circle squaring theorems.

该研究计划位于纯数学领域。第一个是组合集理论，重点是强迫和大型红衣主教。第二个是可测量的组合学。我们依次描述这些领域的每个领域。现代的集合理论研究始于Godel对可构造宇宙的发展和Cohen对强迫方法的发明。综上所述，这些技术可以证明连续假设与ZFC的公理无关。连续假设是断言自然数的所有子集的收集，即其powerset，具有最小的基数。 Godel的L结构是内在模型理论的现代研究的开始。科恩（Cohen）的强迫发展现在是产生独立成果的最常用技术。设定理论中提出的项目可以由它们所涉及的主题分开。 （1）关于奇异主要枢机主教的基数的问题。这是连续假设研究的现代实例。 （2）有关紧凑原则的问题。紧凑性原理是主张，即如果所有较小的基数子结构都具有一定的属性，则整个结构具有相同的属性。 （3）关于V与遗传条件集的类别中的基数概念如何不同的问题，HOD。这是内部模型理论的一个方面。可衡量的图形组合学已经看到了最近从应用到有关几何悖论的旧问题的兴趣激增。例如，塔斯基（Tarski）的圆形平方问题：给定平面上的磁盘和一个正方形，是否有可能将磁盘分开以有限的许多可以通过等法移动的零件来分区正方形？ Laczkovich在1990年使用选择的公理对此进行了积极解决。货车询问borel碎片是否可以使用。 Grabowski，Mathe和Pikhurko的最新定理表明，Lebesgue可测量或可衡量的零件是可能的。结果后不久，我们证明了与安德鲁·马克斯（Andrew Marks）共同合作的Borel版本。该结果取决于对翻译生成的r^2的某些局部有限鲍尔图的分析。特别是，Borel Circle平方定理等于存在其中一个图中的Borel Perfect匹配。 Borel图何时具有Borel完美匹配的问题是可测量组合学问题的一个示例。可衡量的组合学中问题的答案通常与他们的经典同行有很大不同，需要新技术。该领域的拟议研究既包含了可衡量的组合主义者的一般问题，又包含诸如圆形定理之类的应用。