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.

该研究计划涉及纯数学的两个领域。第一个是组合集合论，重点关注强迫和大基数。第二个是可测量的组合学。我们依次描述每个领域。现代集合论的研究始于哥德尔对可构造宇宙L的发展和科恩对强迫方法的发明。综合起来，这些技术证明了连续统假设独立于 ZFC 公理。连续统假设是这样的断言：自然数的所有子集的集合（其幂集）具有可能的最小基数。哥德尔对L的构造是现代内模型理论研究的开端。科恩发展的强迫现在是产生独立结果最常用的技术。集合论中提出的项目可以根据它们所涉及的主题来划分。 (1) 关于奇异基数幂集的基数问题。这是连续统假说研究的现代实例。 (2)关于紧致性原则的问题。紧致性原则是这样的断言：给定一个结构，如果所有较小的基数子结构都具有某些属性，则整个结构具有相同的属性。 (3) 关于 V 和遗传序数集 HOD 类之间基数概念有何不同的问题。这是内模型理论的一个方面。最近，人们对可测图组合学的兴趣激增，从应用到有关几何悖论的老问题。例如，塔斯基的圆平方问题：给定一个圆盘和平面上面积相同的正方形，是否可以将圆盘划分为有限多个可以通过等距移动来划分正方形的块？ Laczkovich 在 1990 年使用选择公理积极解决了这个问题。 Wagon 询问 Borel 件是否也可以实现同样的效果。 Grabowski、Mathe 和 Pikhurko 最近的定理表明，无论是 Lebesgue 可测块还是 Baire 可测块，这都是可能的。在得到这个结果后不久，我们与 Andrew Marks 合作证明了 Borel 版本。该结果取决于对平移生成的 R^2 上某些局部有限 Borel 图的分析。特别是，Borel 圆平方定理相当于这些图中之一存在 Borel 完美匹配。 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。可测量组合学中问题的答案通常与经典问题的答案截然不同，需要新技术。该领域拟议的研究既包含可测量组合学的一般问题，也包含圆平方定理等应用问题。