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.

该研究计划是在纯数学的两个领域。第一个是组合集合论，重点是强迫和大基数。第二个是可测组合学。我们依次描述这些领域。集合论的现代研究始于哥德尔对可构造论域L的发展和科恩对强迫方法的发明。综合这些技术，证明了连续统假设与ZFC公理无关。连续统假设是断言自然数的所有子集的集合，它的幂集，具有最小的基数可能。哥德尔对L的构造是内模型理论现代研究的开端。科恩的强迫发展现在是最常用的技术，以产生独立的结果。集合论中的拟议项目可以根据它们所涉及的主题进行划分。(1)关于奇异基数幂集的基数性的问题。这是连续统假设研究的现代实例。(2)关于紧凑性原则的问题。紧性原理是这样的断言，即给定一个结构，如果所有较小的基数子结构都具有某种性质，那么整个结构都具有相同的性质。(3)关于V和遗传序集类HOD之间基数概念的不同的问题。这是内在模型理论的一个方面。可测图组合学最近从应用到几何悖论的老问题的兴趣激增。例如，塔斯基的圆平方问题：给定一个圆盘和一个正方形在平面上具有相同的面积，是否有可能将圆盘分割成100多块，这些块可以通过等距移动来分割正方形？ 1990年，Laczkovich使用选择公理解决了这个问题。这是问旅行车，如果同样是可能的博雷尔件。最近的一个定理Grabowski，Mathe和Pikhurko表明，这是可能的，无论是勒贝格可测或Baire可测件。在这个结果之后不久，我们与安德鲁·马克斯（Andrew Marks）共同证明了博雷尔版本。这个结果依赖于对R^2上某些由平移生成的局部有限Borel图的分析。特别地，Borel圆平方定理等价于其中一个图中存在Borel完美匹配。波莱尔图何时具有波莱尔完美匹配的问题是可测组合学中的一个问题。在可测量组合学中，问题的答案往往与经典的答案大不相同，需要新的技术。在这一领域的研究建议既包含一般性的问题，从可测组合学和问题，这是应用程序，如圆平方定理。