Combinatorial set theory and measurable combinatorics

这个研究项目涉及纯数学的两个领域。第一个是组合集理论，重点是强迫和大基数。第二个是可测量组合学。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宇宙L的发展和科恩对强迫方法的发明。综上所述，这些方法证明了连续统假设是独立于ZFC公理的。连续统假设是断言自然数的所有子集的集合，它的幂集，具有尽可能小的基数。哥德尔关于L的构造是现代内模理论研究的开端。Cohen的强迫发展是现在产生独立结果最常用的技术。集合理论中提出的项目可以根据它们所处理的主题进行划分。(1)关于奇异基数幂集的基数性问题。这是研究连续体假说的现代实例。(2)关于紧密性原则的问题。紧致性原则是这样一种断言：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，则整个结构具有相同的性质。(3)关于V和遗传序数集类（HOD）之间的基数概念有何不同的问题。这是内模理论的一个方面。最近，人们对可测量图组合学的兴趣激增，从应用到关于几何悖论的老问题。例如，塔斯基的圆平方问题：给定平面上的一个圆盘和一个面积相同的正方形，是否有可能将圆盘划分为有限多块，这些块可以通过等边线移动来划分正方形？1990年，Laczkovich用选择公理（axiom of choice）积极地解决了这个问题。Wagon问Borel的作品是否也有同样的可能。Grabowski， Mathe和Pikhurko最近的一个定理表明，这对于Lebesgue可测量块或Baire可测量块都是可能的。在这个结果之后不久，我们与Andrew Marks共同证明了Borel版本。这一结果依赖于对由平移生成的R^2上的某些局部有限Borel图的分析。特别地，Borel圆平方定理等价于这些图中的存在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例子。在可测量组合学中，问题的答案往往与它们的经典对应物大不相同，需要新的技术。在这一领域提出的研究既包括可测组合学的一般问题，也包括圆平方定理等应用问题。

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.