## Departmental PhD Thesis Exam – Xiao Ming

Thursday, June 25, 2020
2:00 p.m.

PhD Candidate:  Xiao Ming
Supervisor:   Stevo Todorcevic
Thesis title:  Borel Chain Conditions

***

The subject matter of this Thesis is an instance of the Chain Condition Method of coarse classification of Boolean algebras and partially ordered sets. This method has played an important role in the measure theory, the theory of forcing, and the theory of Martin type axioms.

We focus on the posets that are Borel definable in Polish spaces and investigate the connections between the chain condition method and the chromatic numbers, a classification scheme for graphs. We then introduce Borel version of some classical chain condition and show that the Borel poset $T(\pi\mathbb{Q})$, the Borel example Todorcevic used to distinguish $\sigma$-finite chain condition and $\sigma$-bounded chain condition, cannot be decomposed into countably many Borel pieces witnessing the $\sigma$-finite chain condition, despite the fact that the non-Borel such partition exists. Starting from there, we use the variations on the $G_0$-dichotomy analyzed to construct a number of examples of Borel posets of the form $\mathbb{D}(G)$ that the new hierarchy of Borel chain conditions is proper.

A copy of the thesis can be found here:  Thesis_revised_202006210421