Combinatorial set theory and measurable combinatorics

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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.